Introduction介绍

On the cosmic scale, gravitation dominates the universe. Nuclear and electromagnetic forces account for the detailed processes that allow stars to shine and astronomers to see them. But it is gravitation that shapes the universe, determining the geometry of space and time and thus the large-scale distribution of galaxies. Providing insight into gravitation – its effects, its nature and its causes – is therefore rightly seen as one of the most important goals of physics and astronomy.在宇宙尺度上，引力主宰着宇宙。核力和电磁力解释了恒星发光和天文学家看到它们的详细过程。但正是引力塑造了宇宙，决定了空间和时间的几何形状，从而决定了星系的大尺度分布。因此，深入了解引力——它的影响、性质和原因——被正确地视为物理学和天文学最重要的目标之一。

Through more than a thousand years of human history the common explanation of gravitation was based on the Aristotelian belief that objects had a natural place in an Earth-centred universe that they would seek out if free to do so. For about two and a half centuries the Newtonian idea of gravity as a force held sway. Then, in the twentieth century, came Einstein’s conception of gravity as a manifestation of spacetime curvature. It is this latter view that is the main concern of this book.在一千多年的人类历史中，对万有引力的常见解释是基于亚里士多德的信念，即物体在以地球为中心的宇宙中拥有一个自然的位置，如果可以自由地这样做，它们就会寻找这个位置。大约两个半世纪以来，牛顿引力作为一种力的观念一直占据主导地位。然后，在二十世纪，爱因斯坦提出了引力作为时空曲率表现的概念。本书主要关注的正是后一种观点。

The story of Einsteinian gravitation begins with a failure. Einstein’s theory of special relativity, published in 1905 while he was working as a clerk in the Swiss Patent Office in Bern, marked an enormous step forward in theoretical physics and soon brought him academic recognition and personal fame. However, it also showed that the Newtonian idea of a gravitational force was inconsistent with the relativistic approach and that a new theory of gravitation was required. Ten years later, Einstein’s general theory of relativity met that need, highlighting the important role of geometry in accounting for gravitational phenomena and leading on to concepts such as black holes and gravitational waves. Within a year and a half of its completion, the new theory was providing the basis for a novel approach to cosmology – the science of the universe – that would soon have to take account of the astronomy of galaxies and the physics of cosmic expansion. The change in thinking demanded by relativity was radical and profound. Its mastery is one of the great challenges and greatest delights of any serious study of physical science.爱因斯坦引力的故事始于一次失败。 1905 年，爱因斯坦在伯尔尼的瑞士专利局担任职员时发表了狭义相对论，标志着理论物理学向前迈出了一大步，并很快为他带来了学术认可和个人声誉。然而，它也表明牛顿的引力思想与相对论方法不一致，需要一种新的引力理论。十年后，爱因斯坦的广义相对论满足了这一需求，强调了几何学在解释引力现象中的重要作用，并引出了黑洞和引力波等概念。在完成后的一年半内，新理论为宇宙学(宇宙科学)的新方法提供了基础，该方法很快就必须考虑星系天文学和宇宙膨胀物理学。相对论所要求的思维变革是彻底而深刻的。掌握它是任何严肃的物理科学研究的巨大挑战和最大的乐趣之一。

This book begins with two chapters devoted to special relativity. These are followed by a mainly mathematical chapter that provides the background in geometry that is needed to appreciate Einstein’s subsequent development of the theory. Chapter 4 examines the basic principles and assumptions of general relativity – Einstein’s theory of gravity – while Chapters 5 and 6 apply the theory to an isolated spherical body and then extend that analysis to non-rotating and rotating black holes. Chapter 7 concerns the testing of general relativity, including the use of astronomical observations and gravitational waves. Finally, Chapter 8 examines modern relativistic cosmology, setting the scene for further and ongoing studies of observational cosmology.本书开头有两章专门讨论狭义相对论。接下来是主要是数学的章节，提供理解爱因斯坦随后理论发展所需的几何背景。第 4 章探讨了广义相对论(爱因斯坦的引力理论)的基本原理和假设，而第 5 章和第 6 章将该理论应用于孤立的球体，然后将该分析扩展到非旋转和旋转黑洞。第七章涉及广义相对论的检验，包括天文观测和引力波的使用。最后，第八章探讨了现代相对论宇宙学，为进一步和正在进行的观测宇宙学研究奠定了基础。

The text before you is the result of a collaborative effort involving a team of authors and editors working as part of the broader effort to produce the Open University course S383 The Relativistic Universe. Details of the team’s membership and responsibilities are listed elsewhere but it is appropriate to acknowledge here the particular contributions of Jim Hague regarding Chapters 1 and 2, Derek Capper concerning Chapters 3, 4 and 7, and Aiden Droogan in relation to Chapters 5, 6 and 8. Robert Lambourne was responsible for planning and producing the final unified text which benefited greatly from the input of the S383 Course Team Chair, Andrew Norton, and the attention of production editor您面前的文本是作者和编辑团队共同努力的结果，作为制作开放大学课程 S383 相对论宇宙的更广泛努力的一部分。团队成员和职责的详细信息在其他地方列出，但在此应感谢 Jim Hague 对第 1 章和第 2 章的特殊贡献，Derek Capper 对第 3、4 和 7 章的特殊贡献，以及 Aiden Droogan 对第 5、6 和 8 章的特殊贡献。Robert Lambourne 负责规划和制作最终的统一文本，这得益于 S383 课程团队主席 Andrew Norton 的投入以及制作编辑的关注

Peter Twomey. The whole team drew heavily on the work and wisdom of an earlier Open University Course Team that was responsible for the production of the course S357 Space, Time and Cosmology.彼得·图梅.整个团队很大程度上借鉴了早期开放大学课程团队的工作和智慧，该团队负责制作 S357 空间、时间和宇宙学课程。

A major aim for this book is to allow upper-level undergraduate students to develop the skills and confidence needed to pursue the independent study of the many more comprehensive texts that are now available to students of relativity, gravitation and cosmology. To facilitate this the current text has largely adopted the notation used in the outstanding book by Hobson et al.本书的主要目的是让高年级本科生培养独立研究相对论、引力和宇宙学学生现在可以使用的许多更全面的文本所需的技能和信心。为了方便这一点，当前的文本很大程度上采用了 Hobson 等人的杰出著作中使用的符号。

General Relativity: An Introduction for Physicists, M. P. Hobson, G. Efstathiou and A. N. Lasenby, Cambridge University Press, 2006.《广义相对论：物理学家简介》，M. P. Hobson、G. Efstathiou 和 A. N. Lasenby，剑桥大学出版社，2006 年。

Other books that provide valuable further reading are (roughly in order of increasing mathematical demand):其他提供有价值的进一步阅读的书籍是(大致按照数学需求增加的顺序排列)：

An Introduction to Modern Cosmology, A. Liddle, Wiley,现代宇宙学导论，A. Liddle，Wiley，

1999.1999 年。

Relativity, Gravitation and Cosmology: A Basic Introduction, T-P. Cheng, Oxford University Press: 2005. Introducing Einstein’s Relativity, R. d’Inverno, Oxford University Press, 1992. Relativity: Special, General and Cosmological, W. Rindler, Oxford University Press, 2001. Cosmology, S. Weinberg, Cambridge University Press,相对论、引力和宇宙学：基本介绍，T-P。 Cheng，牛津大学出版社：2005。介绍爱因斯坦的相对论，R. d’Inverno，牛津大学出版社，1992。相对论：狭义、广义和宇宙学，W. Rindler，牛津大学出版社，2001。宇宙学，S. Weinberg，剑桥大学出版社，

2008.2008年。

Two useful sources of reprints of original papers of historical significance are:具有历史意义的原始论文重印的两个有用来源是：

The Principle of Relativity, A. Einstein et al., Dover, New York, 1952. Cosmological Constants, edited by J. Bernstein and G. Feinberg, Columbia University Press, 1986.《相对论原理》，A. Einstein 等人，多佛，纽约，1952 年。《宇宙学常数》，J. Bernstein 和 G. Feinberg 编辑，哥伦比亚大学出版社，1986 年。

Those wishing to undertake background reading in astronomy, physics and mathematics to support their study of this book or of any of the others listed above might find the following particularly helpful:那些希望进行天文学、物理和数学背景阅读以支持他们对本书或上述任何其他书籍的研究的人可能会发现以下内容特别有帮助：

An Introduction to Galaxies and Cosmology, edited by M. H. Jones and R. J. A. Lambourne, Cambridge University Press, 2003. The seven volumes in the series The Physical World, edited by R. J. A. Lambourne, A. J. Norton et al., Institute of Physics Publishing, 2000. (Go to www.physicalworld.org for further details.) The paired volumes Basic Mathematics for the Physical Sciences, edited by R. J. A. Lambourne and M. H. Tinker, Wiley, 2000. Further Mathematics for the Physical Sciences, edited by M. H. Tinker and R. J. A. Lambourne, Wiley, 2000.《星系和宇宙学导论》，由 M. H. Jones 和 R. J. A. Lambourne 编辑，剑桥大学出版社，2003 年。《物理世界》系列的七卷，由 R. J. A. Lambourne、A. J. Norton 等人编辑，Institute of Physical Publishing，2000 年。(更多详细信息，请访问 www.physicalworld.org。) R. J. A. Lambourne 和 M. H. Tinker，Wiley，2000。《物理科学的进一步数学》，M. H. Tinker 和 R. J. A. Lambourne 编辑，Wiley，2000 年。

Chapter 1 Special relativity and spacetime第一章狭义相对论与时空

Introduction介绍

In two seminal papers in 1861 and 1864, and in his treatise of 1873, James Clerk Maxwell (Figure 1.1), Scottish physicist and genius, wrote down his revolutionary unified theory of electricity and magnetism, a theory that is now summarized in the equations that bear his name. One of the deep results of the theory introduced by Maxwell was the prediction that wave-like excitations of combined electric and magnetic fields would travel through a vacuum with the same speed as light. It was soon widely accepted that light itself was an electromagnetic disturbance propagating through space, thus unifying electricity and magnetism with optics.苏格兰物理学家和天才詹姆斯·克拉克·麦克斯韦(James Clerk Maxwell)(图 1.1)在 1861 年和 1864 年的两篇开创性论文以及 1873 年的论文中写下了他革命性的电学和磁学统一理论，该理论现在被总结为以他的名字命名的方程。麦克斯韦提出的理论的深刻成果之一是预测电场和磁场组合的波状激发将以与光速相同的速度穿过真空。很快人们就广泛接受光本身是一种通过空间传播的电磁扰动，从而将电、磁与光学统一起来。

The fundamental work of Maxwell opened the way for an understanding of the universe at a much deeper level. Maxwell himself, in common with many scientists of the nineteenth century, believed in an all-pervading medium called the ether, through which electromagnetic disturbances travelled, just as ocean waves travelled through water. Maxwell’s theory predicted that light travels with the same speed in all directions, so it was generally assumed that the theory predicted the results of measurements made using equipment that was at rest with respect to the ether. Since the Earth was expected to move through the ether as it orbited the Sun, measurements made in terrestrial laboratories were expected to show that light actually travelled with different speeds in different directions, allowing the speed of the Earth’s movement through the ether to be determined.麦克斯韦的基础工作为更深层次地理解宇宙开辟了道路。麦克斯韦本人与 19 世纪的许多科学家一样，相信有一种称为以太的无所不在的介质，电磁扰动通过它传播，就像海浪在水中传播一样。麦克斯韦理论预测光以相同的速度向各个方向传播，因此人们普遍认为该理论预测了使用相对于以太静止的设备进行的测量结果。由于预计地球绕太阳运行时会穿过以太，因此地面实验室进行的测量预计会表明光实际上以不同的速度向不同方向传播，从而可以确定地球穿过以太的运动速度。

However, the failure to detect any variations in the measured speed of light, most notably by A. A. Michelson and E. W. Morley in 1887, prompted some to suspect that measurements of the speed of light in a vacuum would always yield the same result irrespective of the motion of the measuring equipment. Explaining how this could be the case was a major challenge that prompted ingenious proposals from mathematicians and physicists such as Henri Poincaré, George Fitzgerald and Hendrik Lorentz. However, it was the young Albert Einstein who first put forward a coherent and comprehensive solution in his 1905 paper ‘On the electrodynamics of moving bodies’, which introduced the special theory of relativity. With the benefit of hindsight, we now realize that Maxwell had formulated the first major theory that was consistent with special relativity, a revolutionary new way of thinking about space and time.然而，未能检测到所测量的光速的任何变化，尤其是 1887 年 A. A. 迈克尔逊和 E. W. 莫利的测量，促使一些人怀疑真空中光速的测量总是会产生相同的结果，而与测量设备的运动无关。解释为什么会出现这种情况是一项重大挑战，促使亨利·庞加莱、乔治·菲茨杰拉德和亨德里克·洛伦兹等数学家和物理学家提出了巧妙的建议。然而，年轻的阿尔伯特·爱因斯坦在他1905年的论文《论运动物体的电动力学》中首先提出了一个连贯且全面的解决方案，该论文引入了狭义相对论。事后看来，我们现在意识到麦克斯韦提出了第一个与狭义相对论相一致的主要理论，这是一种革命性的思考空间和时间的新方式。

This chapter reviews the implications of special relativity theory for the understanding of space and time. The narrative covers the fundamentals of the theory, concentrating on some of the major differences between our intuition about space and time and the predictions of special relativity. By the end of this chapter, you should have a broad conceptual understanding of special relativity, and be able to derive its basic equations, the Lorentz transformations, from the postulates of special relativity. You will understand how to use events and intervals to describe properties of space and time far from gravitating bodies. You will also have been introduced to Minkowski spacetime, a four-dimensional fusion of space and time that provides the natural setting for discussions of special relativity.本章回顾狭义相对论对于理解空间和时间的意义。叙述涵盖理论基础，重点讨论我们关于空间和时间的直觉与狭义相对论预言之间的一些主要差异。读完本章后，你应当对狭义相对论有宽泛的概念性理解，并能够从狭义相对论的公设推导出它的基本方程，即洛伦兹变换。你将理解如何用事件和间隔来描述远离引力体处的空间和时间性质。你还会接触到闵可夫斯基时空，即空间与时间的四维融合，它为讨论狭义相对论提供了自然背景。

1.1 Basic concepts of special relativity1.1 狭义相对论的基本概念

1.1.1 Events, frames of reference and observers1.1.1 事件、参照系和观察者

When dealing with special relativity it is important to use language very precisely in order to avoid confusion and error. Fundamental to the precise description of physical phenomena is the concept of an event, the spacetime analogue of a point in space or an instant in time.在处理狭义相对论时，非常精确地使用语言以避免混淆和错误非常重要。精确描述物理现象的基础是事件的概念，即空间中的一点或时间中的瞬间的时空模拟。

Events事件

An event is an instantaneous occurrence at a specific point in space.事件是在空间中特定点瞬时发生的事件。

An exploding firecracker or a small light that flashes once are good approximations to events, since each happens at a definite time and at a definite position.爆炸的鞭炮或闪烁一次的小光都是对事件的很好的近似，因为每一个事件都发生在确定的时间和确定的位置。

To know when and where an event happened, we need to assign some coordinates to it: a time coordinate t and an ordered set of spatial coordinates such as the Cartesian coordinates (x, y, z), though we might equally well use spherical coordinates (r, \(\theta\), \(\phi\)) or any other suitable set. The important point is that we should be able to assign a unique set of clearly defined coordinates to any event. This leads us to our second important concept, a frame of reference.要知道事件发生的时间和地点，我们需要为其分配一些坐标：时间坐标 t 和一组有序的空间坐标，例如笛卡尔坐标 (x, y, z)，尽管我们同样可以使用球坐标 (r, \(\theta\), \(\phi\)) 或任何其他合适的集合。重要的一点是，我们应该能够为任何事件分配一组唯一的、明确定义的坐标。这引出了我们的第二个重要概念，即参考系。

Frames of reference参考系

A frame of reference is a system for assigning coordinates to events. It consists of a system of synchronized clocks that allows a unique value of the time to be assigned to any event, and a system of spatial coordinates that allows a unique position to be assigned to any event.参考系是为事件分配坐标的系统。它由一个同步时钟系统和一个空间坐标系统组成，同步时钟系统允许为任何事件分配唯一的时间值，而空间坐标系统允许为任何事件分配唯一的位置。

In much of what follows we shall make use of a Cartesian coordinate system with axes labelled x, y and z. The precise specification of such a system involves selecting an origin and specifying the orientation of the three orthogonal axes that meet at the origin. As far as the system of clocks is concerned, you can imagine that space is filled with identical synchronized clocks all ticking together (we shall need to say more about how this might be achieved later). When using a particular frame of reference, the time assigned to an event is the time shown on the clock at the site of the event when the event happens. It is particularly important to note that the time of an event is not the time at which the event is seen at some far off point — it is the time at the event itself that matters.在接下来的大部分内容中，我们将使用笛卡尔坐标系，其轴标记为 x、y 和 z。这种系统的精确规范涉及选择原点并指定在原点相交的三个正交轴的方向。就时钟系统而言，您可以想象空间中充满了相同的同步时钟，所有这些时钟都一起滴答作响(稍后我们将需要更多地讨论如何实现这一点)。当使用特定参考系时，分配给事件的时间是事件发生时事件现场的时钟上显示的时间。特别重要的是要注意，事件发生的时间并不是在某个遥远的地方看到该事件的时间——重要的是事件本身的时间。

Reference frames are often represented by the letter S. Figure 1.2 provides what we hope is a memorable illustration of the basic idea, in this case with just two spatial dimensions. This might be called the frame S gnome.参考系通常用字母 S 表示。图 1.2 提供了我们希望的基本思想的令人难忘的说明，在本例中只有两个空间维度。这可能被称为框架 S 侏儒。

Among all the frames of reference that might be imagined, there is a class of frames that is particularly important in special relativity. This is the class of inertial frames. An inertial frame of reference is one in which a body that is not subject to any net force maintains a constant velocity. Equivalently, we can say the following.在所有可能想象到的参考系中，有一类参考系在狭义相对论中特别重要。这是一类惯性系。惯性参考系是指不受任何净力作用的物体保持恒定速度的参考系。同样，我们可以说如下。

Inertial frames of reference惯性参考系

An inertial frame of reference is a frame of reference in which Newton’s first law of motion holds true.惯性参考系是牛顿第一运动定律成立的参考系。

Any frame that moves with constant velocity relative to an inertial frame will also be an inertial frame. So, if you can identify or establish one inertial frame, then you can find an infinite number of such frames each having a constant velocity relative to any of the others. Any frame that accelerates relative to an inertial frame cannot be an inertial frame. Since rotation involves changing velocity, any frame that rotates relative to an inertial frame is also disqualified from being inertial.任何相对于惯性系以匀速运动的坐标系也将是惯性系。因此，如果您可以识别或建立一个惯性系，那么您就可以找到无数个这样的框架，每个框架相对于任何其他框架都具有恒定的速度。任何相对于惯性系加速的坐标系都不能是惯性系。由于旋转涉及速度的变化，因此任何相对于惯性系旋转的坐标系也被取消惯性系的资格。

One other concept is needed to complete the basic vocabulary of special relativity. This is the idea of an observer.需要另一个概念来完成狭义相对论的基本词汇。这是观察者的想法。

Observers观察者

An observer is an individual dedicated to using a particular frame of reference for recording events.观察者是致力于使用特定参考系来记录事件的个人。

We might speak of an observer O using frame S, or a different observer \(O'\) (read as ‘O-prime’) using frame \(S'\) (read as ‘S-prime’).我们可能会说一个使用框架 S 的观察者 O，或者使用框架 \(S'\)(读作“S-素数”)的不同观察者 \(O'\)(读作“O-素数”)。

Though you may think of an observer as a person, just like you or me, at rest in their chosen frame of reference, it is important to realize that an observer’s location is of no importance for reporting the coordinates of events in special relativity. The position that an observer assigns to an event is the place where it happened. The time that an observer assigns is the time that would be shown on a clock at the site of the event when the event actually happened, and where the clock concerned is part of the network of synchronized clocks always used in that observer’s frame of reference. An observer might see the explosion of a distant star tonight, but would report the time of the explosion as the time long ago when the explosion actually occurred, not the time at which the light from the explosion reached the observer’s location. To this extent, ‘seeing’ and ‘observing’ are very different processes. It is best to avoid phrases such as ‘an observer sees... ’ unless that is what you really mean. An observer measures and observes.尽管您可能将观察者视为一个人，就像您或我一样，在他们选择的参考系中休息，但重要的是要认识到观察者的位置对于报告狭义相对论中事件的坐标并不重要。观察者为事件指定的位置就是事件发生的地点。观察者分配的时间是事件实际发生时事件现场的时钟上显示的时间，并且相关时钟是始终在该观察者的参考系中使用的同步时钟网络的一部分。观察者今晚可能会看到一颗遥远恒星的爆炸，但会将爆炸时间报告为很久以前爆炸实际发生的时间，而不是爆炸发出的光到达观察者位置的时间。从这个意义上说，“看”和“观察”是非常不同的过程。最好避免使用诸如“观察者看到……”之类的短语，除非这就是您真正的意思。观察者进行测量和观察。

Any observer who uses an inertial frame of reference is said to be an inertial observer. Einstein’s special theory of relativity is mainly concerned with observations made by inertial observers. That’s why it’s called special relativity — the term ‘special’ is used in the sense of ‘restricted’ or ‘limited’. We shall not really get away from this limitation until we turn to general relativity in Chapter 4.任何使用惯性参考系的观察者都被称为惯性观察者。爱因斯坦的狭义相对论主要涉及惯性观察者的观察。这就是为什么它被称为狭义相对论——“特殊”一词的意思是“受限”或“有限”。直到我们在第四章转向广义相对论之前，我们才能真正摆脱这个限制。

Exercise 1.1 For many purposes, a frame of reference练习 1.1 出于多种目的，参考系

fixed in a laboratory on the Earth provides a good approximation to an inertial frame. However, such a frame is not really an inertial frame. How might its true, non-inertial, nature be revealed experimentally, at least in principle?固定在地球上的实验室中，可以很好地近似惯性系。然而，这样的框架并不是真正的惯性系。其真实的、非惯性的性质如何通过实验(至少在原则上)得以揭示？

1.1.2 The postulates of special relativity1.1.2 狭义相对论的假设

Physicists generally treat the laws of physics as though they hold true everywhere and at all times. There is some evidence to support such an assumption, though it is recognized as a hypothesis that might fail under extreme conditions. To the extent that the assumption is true, it does not matter where or when observations are made to test the laws of physics since the time and place of a test of fundamental laws should not have any influence on its outcome.物理学家通常认为物理定律在任何地方、任何时候都适用。有一些证据支持这种假设，尽管它被认为是在极端条件下可能失败的假设。如果假设是正确的，那么在何时何地进行观察来检验物理定律并不重要，因为基本定律检验的时间和地点不应对其结果产生任何影响。

Where and when laws are tested might not influence the outcome, but what about motion? We know that inertial and non-inertial observers will not agree about Newton’s first law. But what about different inertial observers in uniform relative motion where one observer moves at constant velocity with respect to the other? A pair of inertial observers would agree about Newton’s first law; might they also agree about other laws of physics?法律的检验地点和时间可能不会影响结果，但是运动呢？我们知道惯性和非惯性观察者不会同意牛顿第一定律。但是，当一个观察者相对于另一个观察者以恒定速度移动时，处于匀速相对运动的不同惯性观察者又会怎样呢？一对惯性观察者会同意牛顿第一定律；他们是否也同意其他物理定律？

It has long been thought that they would at least agree about the laws of mechanics. Even before Newton’s laws were formulated, the great Italian physicist Galileo Galilei (1564–1642) pointed out that a traveller on a smoothly moving boat had exactly the same experiences as someone standing on the shore. A ball game could be played on a uniformly moving ship just as well as it could be played on shore. To the early investigators, uniform motion alone appeared to have no detectable consequences as far as the laws of mechanics were concerned. An observer shut up in a sealed box that prevented any observation of the outside world would be unable to perform any mechanics experiment that would reveal the uniform velocity of the box, even though any acceleration could be easily detected. (We are all familiar with the feeling of being pressed back in our seats when a train or car accelerates forward.) These notions provided the basis for the first theory of relativity, which is now known as Galilean relativity in honour of Galileo’s original insight. This theory of relativity assumes that all inertial observers will agree about the laws of Newtonian mechanics.长期以来人们一直认为他们至少会同意力学定律。甚至在牛顿定律制定之前，伟大的意大利物理学家伽利略·伽利雷(Galileo Galilei，1564-1642)就指出，在平稳移动的船上的旅行者与站在岸上的人有着完全相同的经历。球类运动可以在均匀移动的船上进行，就像在岸上进行一样。对于早期的研究人员来说，就力学定律而言，仅匀速运动似乎没有可察觉的后果。一个观察者被关在一个无法观察外界的密封盒子里，将无法进行任何揭示盒子匀速的力学实验，即使任何加速度都可以很容易地检测到。(我们都熟悉当火车或汽车加速前进时被压在座位上的感觉。)这些概念为第一个相对论提供了基础，该理论现在被称为伽利略相对论，以纪念伽利略的原始见解。这种相对论假设所有惯性观察者都同意牛顿力学定律。

Einstein believed that inertial observers would agree about the laws of physics quite generally, not just in mechanics. But he was not convinced that Galilean relativity was correct, which brought Newtonian mechanics into question. The only statement that he wanted to presume as a law of physics was that all inertial observers agreed about the speed of light in a vacuum. Starting from this minimal assumption, Einstein was led to a new theory of relativity that was markedly different from Galilean relativity. The new theory, the special theory of relativity, supported Maxwell’s laws of electromagnetism but caused the laws of mechanics to be substantially rewritten. It also provided extraordinary new insights into space and time that will occupy us for the rest of this chapter.爱因斯坦相信，惯性观察者会普遍同意物理定律，而不仅仅是力学方面的。但他不相信伽利略相对论是正确的，这使牛顿力学受到质疑。他想假定为物理定律的唯一陈述是所有惯性观察者都同意真空中的光速。从这个最小的假设出发，爱因斯坦提出了一种与伽利略相对论明显不同的新相对论。新理论，即狭义相对论，支持麦克斯韦电磁定律，但导致力学定律被大幅改写。它还提供了对空间和时间的非同寻常的新见解，这将占据我们本章余下的部分。

Einstein based the special theory of relativity on two postulates, that is, two statements that he believed to be true on the basis of the physics that he knew. The first postulate is often referred to as the principle of relativity.爱因斯坦将狭义相对论建立在两个假设的基础上，也就是说，他根据他所知道的物理学相信这两个陈述是正确的。第一个假设通常被称为相对性原理。

The first postulate of special relativity狭义相对论第一公设

The laws of physics can be written in the same form in all inertial frames.物理定律可以在所有惯性系中以相同的形式书写。

This is a bold extension of the earlier belief that observers would agree about the laws of mechanics, but it is not at first sight exceptionally outrageous. It will, however, have profound consequences.这是对早期观点的大胆延伸，即观察者会同意力学定律，但乍一看这并不是特别离谱。然而，这将产生深远的影响。

The second postulate is the one that gives primacy to the behaviour of light, a subject that was already known as a source of difficulty. This postulate is sometimes referred to as the principle of the constancy of the speed of light.第二个假设将光的行为置于首要地位，这是一个众所周知的困难来源的主题。该假设有时被称为光速恒定原理。

The second postulate of special relativity狭义相对论第二公设

The speed of light in a vacuum has the same constant value, c = \(3\times10^{8}\) \(\mathrm{m\,s^{-1}}\), in all inertial frames.在所有惯性系中，真空中的光速具有相同的常数值，c = \(3\times10^{8}\) \(\mathrm{m\,s^{-1}}\)。

This postulate certainly accounts for Michelson and Morley’s failure to detect any variations in the speed of light, but at first sight it still seems crazy. Our experience with everyday objects moving at speeds that are small compared with the speed of light tells us that if someone in a car that is travelling forward at speed v throws something forward at speed w relative to the car, then, according to an observer standing on the roadside, the thrown object will move forward with speed v + w. But the second postulate tells us that if the traveller in the car turns on a torch, effectively throwing forward some light moving at speed c relative to the car, then the roadside observer will also find that the light travels at speed c, not the v + c that might have been expected. Einstein realized that for this to be true, space and time must behave in previously unexpected ways.这个假设当然可以解释迈克尔逊和莫利未能检测到光速的任何变化，但乍一看它仍然看起来很疯狂。我们对日常物体以小于光速的速度运动的经验告诉我们，如果一辆以速度 v 向前行驶的汽车中的某个人以相对于汽车的速度 w 向前扔东西，那么，根据站在路边的观察者的说法，扔出的物体将以速度 v + w 向前移动。但第二个假设告诉我们，如果车里的旅行者打开手电筒，有效地向前投射一些相对于汽车以 c 速度移动的光，那么路边观察者也会发现光以 c 速度传播，而不是预期的 v + c 速度。爱因斯坦意识到，要实现这一点，空间和时间必须以以前意想不到的方式运行。

The second postulate has another important consequence. Since all observers agree about the speed of light, it is possible to use light signals (or any other electromagnetic signal that travels at the speed of light) to ensure that the network of clocks we imagine each observer to be using is properly synchronized. We shall not go into the details of how this is done, but it is worth pointing out that if an observer sent a radar signal (which travels at the speed of light) so that it arrived at an event just as the event was happening and was immediately reflected back, then the time of the event would be midway between the times of transmission and reception of the radar signal. Similarly, the distance to the event would be given by half the round trip travel time of the signal, multiplied by the speed of light.第二个假设还有另一个重要的结果。由于所有观察者都同意光速，因此可以使用光信号(或以光速传播的任何其他电磁信号)来确保我们想象的每个观察者使用的时钟网络正确同步。我们不会详细说明这是如何完成的，但值得指出的是，如果观察者发送雷达信号(以光速传播)，使其在事件发生时到达事件并立即反射回来，那么事件发生的时间将介于雷达信号的传输时间和接收时间之间。类似地，到事件的距离将由信号往返传播时间的一半乘以光速得出。

1.2 Coordinate transformations1.2 坐标变换

A theory of relativity concerns the relationship between observations made by observers in relative motion. In the case of special relativity, the observers will be inertial observers in uniform relative motion, and their most fundamental observations will be the time and space coordinates of events.相对论涉及观察者在相对运动中所做的观察之间的关系。在狭义相对论中，观察者将是匀速相对运动的惯性观察者，他们最基本的观察将是事件的时间和空间坐标。

For the sake of definiteness and simplicity, we shall consider two inertial observers O and \(O'\) whose respective frames of reference, S and \(S'\), are arranged in the following standard configuration (see Figure 1.3):为了明确和简单起见，我们考虑两个惯性观测器 O 和 \(O'\)，其各自的参考系 S 和 \(S'\) 按以下标准配置排列(见图 1.3)：

1. The origin of frame \(S'\) moves along the x -axis of1. 坐标系\(S'\)的原点沿x轴移动

frame S, in the direction of increasing values of x, with constant velocity V as measured in S.坐标系 S，沿 x 值增加的方向，具有在 S 中测量的恒定速度 V。

2. The x -, y - and z -axes of frame S are always parallel2. 框架S的x、y、z轴始终平行

to the corresponding \(x'\)-, \(y'\)- and \(z'\)-axes of frame \(S'\).到框架 \(S'\) 相应的 \(x'\)-、\(y'\)- 和 \(z'\)- 轴。

3. The event at which the origins of S and \(S'\) coincide3. S与\(S'\)的起源重合的事件

occurs at time t = 0 in frame S and at time \(t'\) = 0 in frame \(S'\).发生在帧 S 中的时间 t = 0 处以及帧 \(S'\) 中的时间 \(t'\) = 0 处。

We shall make extensive use of ‘standard configuration’ in what follows. The arrangement does not entail any real loss of generality since any pair of inertial frames in uniform relative motion can be placed in standard configuration by choosing to reorientate the coordinate axes in an appropriate way and by resetting the clocks appropriately.下面我们将大量使用“标准配置”。该布置不会造成任何真正的通用性损失，因为通过选择以适当的方式重新定向坐标轴并通过适当地重置时钟，可以将任何一对匀速相对运动的惯性系置于标准配置中。

In general, the observers using the frames S and \(S'\) will not agree about the coordinates of an event, but since each observer is using a well-defined frame of coordinates (t, x, y, z) reference, there must exist a set of equations relating the assigned to a particular event by observer O, to the coordinates assigned to the same event by observer \(O'\). The set of equations that performs the task of relating the two sets of coordinates is called a coordinate transformation. This section considers first the Galilean transformations that provide the basis of Galilean relativity, and then the Lorentz transformations on which Einstein’s special relativity is based.一般来说，使用坐标系 S 和 \(S'\) 的观察者不会就事件的坐标达成一致，但由于每个观察者都使用明确定义的坐标系 (t, x, y, z) 参考，因此必须存在一组方程，将观察者 O 分配给特定事件的坐标与观察者 \(O'\) 分配给同一事件的坐标相关联。执行关联两组坐标任务的方程组称为坐标变换。本节首先考虑提供伽利略相对论基础的伽利略变换，然后考虑爱因斯坦狭义相对论所基于的洛伦兹变换。

1.2.1 The Galilean transformations1.2.1 伽利略变换

Before the introduction of special relativity, most physicists would have said that the coordinate transformation between S and \(S'\) was ‘obvious’, and they would have written down the following Galilean transformations:在引入狭义相对论之前，大多数物理学家都会说S和\(S'\)之间的坐标变换是“显而易见的”，他们会写下以下伽利略变换：

where V = | V | is the relative speed of \(S'\) with respect to S.其中 V = | V |是 \(S'\) 相对于 S 的相对速度。

1.2.2 The Lorentz transformations1.2.2 洛伦兹变换

Rather than rely on intuition and run the risk of making unjustified assumptions, Einstein chose to set out his two postulates and use them to deduce the appropriate coordinate transformation between S and \(S'\). A derivation will be given later, but before that let’s examine the result that Einstein found. The equations that he derived had already been obtained by the Dutch physicist Hendrik Lorentz (Figure 1.4) in the course of his own investigations into light and electromagnetism. For that reason, they are known as the Lorentz transformations even though Lorentz did not interpret or utilize them in the same way that Einstein did. Here are the equations:爱因斯坦没有依靠直觉并冒着做出不合理假设的风险，而是选择提出他的两个假设并用它们来推导出 S 和 \(S'\) 之间适当的坐标变换。稍后将给出推导，但在此之前我们先来看看爱因斯坦发现的结果。他推导出来的方程已经由荷兰物理学家亨德里克·洛伦兹(图 1.4)在他自己对光和电磁学的研究过程中获得。因此，它们被称为洛伦兹变换，尽管洛伦兹并没有像爱因斯坦那样解释或利用它们。以下是方程式：

It is clear that the Lorentz transformations are very different from the Galilean transformations. They indicate a thorough mixing together of space and time, since the \(t'\)-coordinate of an event now depends on both t and x, just as the \(x'\)-coordinate does. According to the Lorentz transformations, the two observers do not generally agree about the time of events, even though they still agree about the time at which the origins of their respective frames coincided. So, time is no longer an absolute quantity that all observers agree about. To be meaningful, statements about the time of an event must now be associated with a particular observer. Also, the extent to which the observers - disagree about the position of an event has been modified by a factor of 1/1 − \(V^{2}\)/\(c^2\). In fact, this multiplicative factor is so common in special relativity that it is usually referred to as the the symbol \(\gamma(V)\), Lorentz factor or gamma factor and is represented by emphasizing that its value depends on the relative speed V of the two frames. Using this factor, the Lorentz transformations can be written in the following compact form.显然，洛伦兹变换与伽利略变换有很大不同。它们表明空间和时间的彻底混合，因为事件的 \(t'\) 坐标现在取决于 t 和 x，就像 \(x'\) 坐标一样。根据洛伦兹变换，两个观察者对于事件发生的时间总体上并不达成一致，尽管他们仍然同意各自框架的起源重合的时间。因此，时间不再是所有观察者都同意的绝对量。为了有意义，有关事件时间的陈述现在必须与特定的观察者相关联。此外，观察者对事件位置的分歧程度已按 1/1 系数修改 - \(V^{2}\)/\(c^2\)。事实上，这个乘法因子在狭义相对论中非常常见，通常被称为符号 \(\gamma(V)\)、洛伦兹因子或伽马因子，并通过强调其值取决于两个框架的相对速度 V 来表示。使用这个因子，洛伦兹变换可以写成以下紧凑的形式。

The Lorentz transformations洛伦兹变换

where在哪里

Figure 1.5 shows how the Lorentz factor grows as the relative speed V of the two frames increases. For speeds that are small compared with the speed of light, \(\gamma(V)\) ≈ 1, and the Lorentz transformations approximate the Galilean transformations provided that x is not too large. As the relative speed of the two frames approaches the speed of light, however, the Lorentz factor grows rapidly and so do the discrepancies between the Galilean and Lorentz transformations.图 1.5 显示了洛伦兹因子如何随着两坐标系相对速度 V 的增加而增长。对于与光速相比较小的速度，\(\gamma(V)\) ≈ 1，并且洛伦兹变换近似于伽利略变换，前提是 x 不太大。然而，当两个框架的相对速度接近光速时，洛伦兹因子迅速增长，伽利略变换和洛伦兹变换之间的差异也随之增大。

Exercise 1.2 Compute the Lorentz factor \(\gamma(V)\) when the relative speed V is练习1.2 计算相对速度V为时的洛伦兹因子γ(V)

(a) 10% of the speed of light, and (b) 90% of the speed of light.(a) 10% 光速，(b) 90% 光速。

The Lorentz transformations are so important in special relativity that you will see them written in many different ways. They are often presented in matrix form, as洛伦兹变换在狭义相对论中非常重要，您会看到它们以多种不同的方式书写。它们通常以矩阵形式呈现，如

You should convince yourself that this matrix multiplication gives equations equivalent to the Lorentz transformations. (The equation for transforming the time coordinate is multiplied by c.) We can also represent this relationship by the equation您应该说服自己，该矩阵乘法给出的方程相当于洛伦兹变换。 (时间坐标变换方程乘以c。)我们也可以用方程来表示这种关系

where we use the symbol \([x^{\mu}]\) to represent the column vector with components (\(x^0\), \(x^1\), \(x_{2}\), \(x^{3}\)) = (ct, x, y, z), and the symbol [\(\Lambda\) \(\mu\)] to represent the Lorentz transformation matrix其中我们使用符号 \([x^{\mu}]\) 表示分量为 (\(x^0\), \(x^1\), \(x_{2}\), \(x^{3}\)) = (ct, x, y, z) 的列向量，并使用符号 [\(\Lambda\) \(\mu\)] 表示洛伦兹变换矩阵

At this stage, when dealing with an individual matrix element \(\Lambda\) \(\mu\) \(\nu\), you can simply regard the first index as indicating the row to which it belongs and the second index as indicating the column. It then makes sense that each of the elements x \(\mu\) in the column vector \([x^{\mu}]\) should have a raised index. However, as you will see later, in the context of relativity the positioning of these indices actually has a much greater significance.此时，在处理单个矩阵元素 \(\Lambda\) \(\mu\) \(\nu\) 时，可以简单地将第一个索引视为表示其所属的行，第二个索引表示其所属的列。那么，列向量 \([x^{\mu}]\) 中的每个元素 x \(\mu\) 应该具有升高的索引，这是有意义的。然而，正如您稍后将看到的，在相对论的背景下，这些指数的定位实际上具有更大的意义。

The quantity \([x^{\mu}]\) is sometimes called the four-position since its four components (ct, x, y, z) describe the position of the event in time and space. Note that by using ct to convey the time information, rather than just t, all four components of the four-position are measured in units of distance. Also note that the Greek indices \(\mu\) and \(\nu\) take the values 0 to 3. It is conventional in special and general relativity to start the indexing of the vectors and matrices from zero, where \(x^0\) = ct. This is because the time coordinate has special properties.量 \([x^{\mu}]\) 有时称为四位置，因为它的四个分量 (ct、x、y、z) 描述了事件在时间和空间中的位置。请注意，通过使用 ct 而不仅仅是 t 来传达时间信息，四位置的所有四个分量均以距离为单位进行测量。另请注意，希腊指数 \(\mu\) 和 \(\nu\) 的值为 0 到 3。在狭义相对论和广义相对论中，通常从零开始对向量和矩阵进行索引，其中 \(x^0\) = ct。这是因为时间坐标具有特殊的属性。

Using the individual components of the four-position, another way of writing the Lorentz transformation is in terms of summations:使用四位的各个分量，洛伦兹变换的另一种编写方式是求和：

This one line really represents four different equations, one for each value of \(\mu\). When an index is used in this way, it is said to be a free index, since we are free to give it any value between 0 and 3, and whatever choice we make indicates a different equation. The index \(\nu\) that appears in the summation is not free, since whatever value we choose for \(\mu\), we are required to sum over all possible values of \(\nu\) to obtain the final equation. This means that we could replace all appearances of \(\nu\) by some other index, \(\alpha\) say, without actually changing anything. An index that is summed over in this way is said to be a dummy index.这一行实际上代表了四个不同的方程，每个方程对应 \(\mu\) 的每个值。当以这种方式使用索引时，它被称为自由索引，因为我们可以自由地给它提供 0 到 3 之间的任何值，并且无论我们做出什么选择都表示不同的方程。求和中出现的索引 \(\nu\) 不是自由的，因为无论我们为 \(\mu\) 选择什么值，我们都需要对 \(\nu\) 的所有可能值求和以获得最终方程。这意味着我们可以用其他索引(例如 \(\alpha\))替换 \(\nu\) 的所有外观，而无需实际更改任何内容。以这种方式求和的索引被称为虚拟索引。

Familiarity with the summation form of the Lorentz transformations is particularly useful when beginning the discussion of general relativity; you will meet many such sums. Before moving on, you should convince yourself that you can easily switch between the use of separate equations, matrices (including the use of four-positions) and summations when representing Lorentz transformations.在开始讨论广义相对论时，熟悉洛伦兹变换的求和形式特别有用。你会遇到很多这样的金额。在继续之前，您应该说服自己，在表示洛伦兹变换时，您可以轻松地在使用单独的方程、矩阵(包括使用四位)和求和之间切换。

Given the coordinates of an event in frame S, the Lorentz transformations tell us the coordinates of that same event as observed in frame \(S'\). It is equally important that there is some way to transform coordinates in frame \(S'\) back into the coordinates in frame S. The transformations that perform this task are known as the inverse Lorentz transformations.给定帧 S 中事件的坐标，洛伦兹变换告诉我们在帧 \(S'\) 中观察到的同一事件的坐标。同样重要的是，有某种方法可以将框架 \(S'\) 中的坐标变换回框架 S 中的坐标。执行此任务的变换称为洛伦兹逆变换。

The inverse Lorentz transformations洛伦兹逆变换

Note that the only difference between the Lorentz transformations and their inverses is that all the primed and unprimed quantities have been interchanged, and the relative speed of the two frames, V, has been replaced by the quantity − V. (This changes the transformations but not the value of the write that as \(\gamma(V)\).) Lorentz factor, which depends only on \(V^{2}\), so we can still This relationship between the transformations is expected, since frame \(S'\) is moving with speed V in the positive x -direction as measured in frame S, while frame S is moving with speed V in the negative \(x'\)-direction as measured in frame \(S'\). You should confirm that performing a Lorentz transformation and its inverse transformation in succession really does lead back to the original coordinates, i.e. (ct, x, y, z) → (\(ct'\), \(x'\), \(y'\), \(z'\)) → (ct, x, y, z).请注意，洛伦兹变换及其逆变换之间的唯一区别在于，所有启动和未启动的量都已互换，并且两个框架的相对速度 V 已被量 - V 取代。(这会改变变换，但不会改变写入的值，如 \(\gamma(V)\)。)洛伦兹因子，仅取决于 \(V^{2}\)，因此我们仍然可以预期变换之间的这种关系，因为帧 \(S'\) 在帧 S 中测量的正 x 方向上以速度 V 移动，而帧 S 帧 S 中测量的负 \(x'\) 方向上以速度 V 移动AAAQQ4ZZZ。您应该确认连续执行洛伦兹变换及其逆变换确实会返回原始坐标，即 (ct, x, y, z) → (\(ct'\), \(x'\), \(y'\), \(z'\)) → (ct, x, y, z)。

• ● An event occurs at coordinates (ct = 3 m, x = 4 m● 事件发生在坐标 (ct = 3 m, x = 4 m

, y = 0, z = 0) in frame S according to an observer O. What are the coordinates of the same with speed V = 3 c/4 event in frame \(S'\) according to an observer \(O'\), moving in the positive x -direction, as measured in S?，y = 0，z = 0)，根据观察者 O，在坐标系 S 中。根据观察者 \(O'\)，在坐标系 \(S'\) 中，速度 V = 3 c/4 的相同事件的坐标是多少，在 S 中测量，沿正 x 方向移动？

❍ First, the Lorentz factor \(\gamma(V)\) should be computed:❍ 首先，应计算洛伦兹因子 \(\gamma(V)\)：

The new coordinates are then given by the Lorentz transformations: \(ct'\) = cγ (3 c/4)(t − 3 x/4 c) = (4/7)(3 m − 3 c × 4 m/4 c) = 0 m,然后通过洛伦兹变换给出新坐标： \(ct'\) = cγ (3 c/4)(t − 3 x/4 c) = (4/7)(3 m − 3 c × 4 m/4 c) = 0 m，

\(x'\) = \(\gamma(3 c/4)\)(x − 3 tc/4) = (4/7)(4 m − 3 × 3 m/4) = 7 m,\(x'\) = \(\gamma(3 c/4)\)(x − 3 tc/4) = (4/7)(4 m − 3 × 3 m/4) = 7 m，

\(y'\) = y = 0 m,\(y'\) = y = 0 米，

\(z'\) = z = 0 m.\(z'\) = z = 0 米。

Exercise 1.3 The matrix equation练习1.3 矩阵方程

can be inverted to determine the coordinates (ct, x) in terms of (\(ct'\), \(x'\)). Show that inverting the 2 × 2 matrix leads to the inverse Lorentz transformations in Equations 1.14 and 1.15.可以反转以确定以 (\(ct'\), \(x'\)) 表示的坐标 (ct, x)。证明 2 × 2 矩阵的反转会导致方程式 1.14 和 1.15 中的洛伦兹逆变换。

1.2.3 A derivation of the Lorentz transformations1.2.3 洛伦兹变换的推导

This subsection presents a derivation of the Lorentz transformations that relates the coordinates of an event in two inertial frames, S and \(S'\), that are in standard configuration. It mainly ignores the y - and z -coordinates and just considers the transformation of the t - and x -coordinates of an event. A general transformation relating the coordinates (\(t'\), \(x'\)) of an event in frame \(S'\) to the coordinates (t, x) of the same event in frame S may be written as本小节介绍了洛伦兹变换的推导，该变换将标准配置中的两个惯性系 S 和 \(S'\) 中的事件坐标联系起来。它主要忽略 y 和 z 坐标，只考虑事件的 t 和 x 坐标的变换。将帧 \(S'\) 中事件的坐标 (\(t'\), \(x'\)) 与帧 S 中同一事件的坐标 (t, x) 相关的一般变换可以写为

where the dots represent additional terms involving higher powers of x or t.其中点代表涉及 x 或 t 的更高幂的附加项。

Now, we know from the definition of standard configuration that the event marking the coincidence of the origins of frames S and \(S'\) has the coordinates (t, x) = (0, 0) in S and (\(t'\), \(x'\)) = (0, 0) in \(S'\). It follows from Equations 1.18 and 1.19 that the constants a 0 and b 0 are zero.现在，我们从标准配置的定义中知道，标记帧S和\(S'\)的原点重合的事件在S中的坐标为(t, x) = (0, 0)，在\(S'\)中的坐标为(\(t'\), \(x'\)) = (0, 0)。从方程1.18和1.19可知，常数a 0 和b 0 为零。

The transformations in Equations 1.18 and 1.19 can be further simplified by the requirement that the observers are using inertial frames of reference. Since Newton’s first law must hold in all inertial frames of reference, it is necessary that an object not accelerating in one set of coordinates is also not accelerating in the other set of coordinates. If the higher-order terms in and were not zero, then an object observed to have no acceleration in S (such as a spaceship with its thrusters off moving on the line, shown in the upper part of Figure 1.6) would be observed to accelerate in terms of \(x'\) and \(t'\) (i.e. \(x'_{3}\) = \(v'\) \(t'\), as indicated in the lower part of Figure 1.6). Observer O would report no force on the spaceship, while observer O would report some unknown force acting on it. In this way, the two observers would register different laws of physics, violating the first postulate of special relativity. The higher-order terms are therefore inconsistent with the required physics and must be removed, leaving only a linear transformation.方程 1.18 和 1.19 中的变换可以通过观察者使用惯性参考系的要求进一步简化。由于牛顿第一定律必须适用于所有惯性参考系，因此在一组坐标中不加速的物体在另一组坐标中也必须不加速。如果和中的高阶项不为零，则观察到 S 中没有加速度的物体(例如推进器关闭的宇宙飞船沿直线移动，如图 1.6 的上半部分所示)将观察到以 \(x'\) 和 \(t'\) 形式加速(即 \(x'_{3}\) = \(v'\) \(t'\)，如图下半部分所示) 1.6)。观察者 O 会报告宇宙飞船上没有受到任何力，而观察者 O 会报告有一些未知的力作用在其上。这样，两个观察者就会记录到不同的物理定律，违反狭义相对论第一假设。因此，高阶项与所需的物理特性不一致，必须删除，只留下线性变换。

So we expect the special relativistic coordinate transformation between two frames in standard configuration to be represented by linear equations of the form因此，我们期望标准配置中两个框架之间的狭义相对论坐标变换可以由以下形式的线性方程表示

The remaining task is to determine the coefficients a 1, a 2, b 1 and b 2.剩下的任务是确定系数 a 1、a 2、b 1 和 b 2。

To do this, use is made of known relations between coordinates in both frames of reference. The first step is to use the fact that at any time t, the origin of \(S'\) (which is always at \(x'\) = 0 in \(S'\)) will be at x = V t in S. It follows from Equation 1.21 that为此，需要利用两个参考系中的坐标之间的已知关系。第一步是利用这样一个事实：在任何时间 t，\(S'\) 的原点(始终位于 \(S'\) 中的 \(x'\) = 0)将位于 S 中的 x = V t。从公式 1.21 可以得出：

from which we see that从中我们看到

Dividing Equation 1.21 by Equation 1.20, and using Equation将方程 1.21 除以方程 1.20，并使用方程

Now, as a second step we can use the fact that at any time \(t'\), the origin of frame S (which is always at x = 0 in S) will be at \(x'\) = − V \(t'\) in \(S'\). Substituting these values for x and \(x'\) into Equation 1.23 gives现在，作为第二步，我们可以使用以下事实：在任何时间 \(t'\)，框架 S 的原点(始终位于 S 中的 x = 0)将位于 \(S'\) 中的 \(x'\) = − V \(t'\)。将 x 和 \(x'\) 的这些值代入公式 1.23 得出

from which it follows that由此可见

If we now substitute a 1 = b 1 into Equation 1.23 and divide the numerator and denominator on the right-hand side by t, then如果我们现在将 a 1 = b 1 代入方程 1.23，并将右侧的分子和分母除以 t，则

As a third step, the coefficient a 2 can be found using the principle of the constancy of the speed of light. A pulse of light emitted in the positive x -direction from (ct = 0, x = 0) has speed c = \(x'\)/\(t'\) and also c = x/t. Substituting these values into Equation 1.25 gives第三步，可以利用光速恒定原理求出系数a 2。从 (ct = 0, x = 0) 沿正 x 方向发射的光脉冲的速度为 c = \(x'\)/\(t'\) 并且 c = x/t。将这些值代入公式 1.25 得出

which can be rearranged to give可以重新排列以给出

Now that a 2, b 1 and b 2 are known in terms of a 1, the coordinate transformations between the two frames can be written as既然a 2、b 1 和b 2 都以a 1 的形式已知，那么两个坐标系之间的坐标变换可以写为

All that remains for the fourth step is to find an expression for a 1. To do this, we first write down the inverse transformations to Equations第四步剩下的就是找到 1 的表达式。为此，我们首先写下方程的逆变换

1.27 and 1.28, which1.27 和 1.28，其中

are found by exchanging primes and replacing V by − V. (We are implicitly assuming that a 1 depends only on some even power of通过交换素数并用 − V 替换 V 可以找到。(我们隐含地假设 1 仅取决于

Substituting Equations 1.29 and 1.30 into Equation 1.28 gives将方程 1.29 和 1.30 代入方程 1.28 得出

The second and third terms involving a 1 V \(t'\) cancel in this expression, leaving an expression in which the \(x'\) cancels on both sides:涉及 1 V \(t'\) 的第二项和第三项在此表达式中取消，留下 \(x'\) 在两侧取消的表达式：

By rearranging this equation and taking the positive square root, the coefficient a is determined to be通过重新排列该方程并取正平方根，系数 a 确定为

Thus a 1 is seen to be the Lorentz factor \(\gamma(V)\), which completes the derivation.因此，1 被视为洛伦兹因子 \(\gamma(V)\)，从而完成了推导。

Some further arguments allow the Lorentz transformations to be extended to one time and three space dimensions. There can be no y and z contributions to the transformations for \(t'\) and \(x'\) since the y - and z -axes could be oriented in any of the perpendicular directions without affecting the events on the x -axis. Similarly, there can be no contributions to the transformations for \(y'\) and \(z'\) from any other coordinates, as space would become distorted in a non-symmetric manner.一些进一步的论点允许洛伦兹变换扩展到一维和三维空间。 \(t'\) 和 \(x'\) 的变换不会有 y 和 z 贡献，因为 y 轴和 z 轴可以沿任何垂直方向定向，而不影响 x 轴上的事件。类似地，任何其他坐标对 \(y'\) 和 \(z'\) 的变换都没有贡献，因为空间会以非对称方式扭曲。

1.2.4 Intervals and their transformation rules1.2.4 区间及其变换规则

Knowing how the coordinates of an event transform from one frame to another, it is relatively simple to determine how the coordinate intervals that separate pairs of events transform. As you will see in the next section, the rules for transforming intervals are often very useful.了解事件的坐标如何从一帧变换到另一帧，确定分隔事件对的坐标间隔如何变换就相对简单。正如您将在下一节中看到的，转换间隔的规则通常非常有用。

Intervals间隔

An interval between two events, measured along a specified axis in a given frame of reference, is the difference in the corresponding coordinates of the two events.在给定参考系中沿指定轴测量的两个事件之间的间隔是两个事件的相应坐标的差值。

To develop transformation rules for intervals, consider the Lorentz transformations for the coordinates of two events labelled 1 and 2:要制定间隔的变换规则，请考虑标记为 1 和 2 的两个事件的坐标的洛伦兹变换：

Subtracting the transformation equation for \(t'_{1}\) from that for \(t'_{2}\), and subtracting the transformation equation for \(x'_{1}\) from that for \(x'_{2}\), and so on, gives the following transformation rules for intervals:从 \(t'_{2}\) 的变换方程中减去 \(t'_{1}\) 的变换方程，从 \(x'_{2}\) 的变换方程中减去 \(x'_{1}\) 的变换方程，依此类推，给出以下区间变换规则：

where \(\Delta t = t\)− t, \(\Delta x = x\)− x, \(\Delta y = y\)− y and \(\Delta z = z\)− z denote the various time and space intervals between the events. The inverse transformations V: for intervals have the same form, with V replaced by −其中 \(\Delta t = t\)− t、\(\Delta x = x\)− x、\(\Delta y = y\)− y 和 \(\Delta z = z\)− z 表示事件之间的各种时间和空间间隔。区间的逆变换 V: 具有相同的形式，其中 V 替换为 -

The transformation rules for intervals are useful because they depend only on coordinate differences and not on the specific locations of events in time or space.间隔的变换规则很有用，因为它们仅取决于坐标差异，而不取决于事件在时间或空间中的具体位置。

1.3 Consequences of the Lorentz transformations1.3 洛伦兹变换的结果

In this section, some of the extraordinary consequences of the Lorentz transformations will be examined. In particular, we shall consider the findings of different observers regarding the rate at which a clock ticks, the length of a rod and the simultaneity of a pair of events. In each case, the trick for determining how the relevant property transforms between frames of reference is to carefully specify how intuitive concepts such as length or duration should be defined consistently in different frames of reference. This is most easily done by identifying each concept with an appropriate interval between two events: 1 and在本节中，将研究洛伦兹变换的一些非凡后果。特别是，我们将考虑不同观察者关于时钟走动的速度、杆的长度和一对事件的同时性的发现。在每种情况下，确定相关属性如何在参考系之间转换的技巧是仔细指定如何在不同参考系中一致地定义直观概念(例如长度或持续时间)。最容易做到这一点的方法是通过两个事件之间的适当间隔来识别每个概念：1 和

2. Once this has been achieved, we can determine which2. 一旦实现了这一点，我们就可以确定

intervals are known and then use the interval transformation rules (Equations 1.32–1.35 and 1.36–1.39) to find relationships between them. The rest of this section will give examples of this process.区间已知，然后使用区间变换规则(方程 1.32-1.35 和 1.36-1.39)来查找它们之间的关系。本节的其余部分将给出此过程的示例。

1.3.1 Time dilation1.3.1 时间膨胀

One of the most celebrated consequences of special relativity is the finding that ‘moving clocks run slow’. More precisely, any inertial observer must observe that the clocks used by another inertial observer, in uniform relative motion, will run slow. Since clocks are merely indicators of the passage of time, this is really the assertion that any inertial observer will find that time passes more slowly for any other inertial observer who is in relative motion. Thus, according to special relativity, if you and I are inertial observers, and we are in uniform relative motion, then I can perform measurements that will show that time is passing more slowly for you and, simultaneously, you can perform measurements that will show that time is passing more slowly for me. Both of us will be right because time is a relative quantity, not an absolute one. To show how this effect follows from the Lorentz transformations, it is essential to introduce clear, unambiguous definitions of the time intervals that are to be related.狭义相对论最著名的结论之一是发现“移动的时钟运行缓慢”。更准确地说，任何惯性观察者都必须观察到另一个惯性观察者使用的时钟在匀速相对运动中会走慢。由于时钟仅仅是时间流逝的指示器，这实际上是这样的断言：任何惯性观察者都会发现，对于任何其他相对运动的惯性观察者来说，时间过得更慢。因此，根据狭义相对论，如果你和我是惯性观察者，并且我们处于匀速相对运动，那么我可以进行测量，表明时间对你来说流逝得更慢，同时，你也可以进行测量来表明时间对我来说流逝得更慢。我们俩都是对的，因为时间是一个相对量，而不是绝对量。为了展示洛伦兹变换如何产生这种效应，有必要引入相关时间间隔的清晰、明确的定义。

Rather than deal with ticking clocks, our discussion here will refer to short-lived sub-nuclear particles of the sort routinely studied at CERN and other particle physics laboratories. For the purpose of the discussion, a short-lived particle is considered to be a point-like object that is created at some event, labelled 1, and subsequently decays at some other event, labelled 2. The time interval between these two events, as measured in any particular inertial frame, is the lifetime of the particle in that frame. This interval is analogous to the time between successive ticks of a clock.我们在这里讨论的不是滴答作响的时钟，而是欧洲核子研究中心和其他粒子物理实验室常规研究的那种短命亚核粒子。出于讨论的目的，短寿命粒子被认为是在某个事件中创建的点状物体，标记为 1，随后在某个其他事件中衰变，标记为 2。在任何特定惯性系中测量的这两个事件之间的时间间隔就是该系中粒子的寿命。该间隔类似于时钟连续滴答之间的时间。

We shall consider the lifetime of a particular particle as observed by two different inertial observers O and \(O'\). Observer O uses a frame S that is fixed in the laboratory, in which the particle travels with constant speed V in the positive x -direction. We shall call this the laboratory frame. Observer \(O'\) uses a frame S that moves with the particle. Such a frame is called the rest frame of the particle since the particle is always at rest in that frame. (You can think of the observer O as riding on the particle if you wish.)我们将考虑由两个不同的惯性观察者 O 和 \(O'\) 观察到的特定粒子的寿命。观察者 O 使用固定在实验室中的框架 S，其中粒子以恒定速度 V 沿正 x 方向行进。我们将其称为实验室框架。观察者 \(O'\) 使用随粒子移动的坐标系 S。这样的坐标系称为粒子的静止坐标系，因为粒子在该坐标系中始终处于静止状态。 (如果你愿意的话，你可以将观察者 O 想象成骑在粒子上。)

According to observer \(O'\), the birth and decay of the (stationary) particle happen at the same place, so if event 1 occurs at (\(t'\), \(x'\)), then event 2 occurs at (\(t'\), \(x'\)), and the lifetime of the particle will be \(\Delta t'\) = \(t'_{2}\) − \(t'_{1}\). In special relativity, the time between two events measured in a frame in which the events happen at the same position is called the proper time between the events and is usually denoted by the symbol \(\Delta \tau\). So, in this case, we can say that in frame \(S'\) the intervals of time and space that separate the two events are \(\Delta t'\) = \(\Delta \tau =\)\(t'\) − \(t'\) and \(\Delta x'\) = 0.根据观察者 \(O'\)，(静止)粒子的诞生和衰变发生在同一个地方，因此如果事件 1 发生在(\(t'\)，\(x'\))，那么事件 2 发生在(\(t'\)，\(x'\))，粒子的寿命将为 \(\Delta t'\) = \(t'_{2}\) − \(t'_{1}\)。在狭义相对论中，在事件发生在同一位置的框架中测量的两个事件之间的时间称为事件之间的固有时间，通常用符号 \(\Delta \tau\) 表示。因此，在这种情况下，我们可以说，在帧 \(S'\) 中，分隔两个事件的时间和空间间隔为 \(\Delta t'\) = \(\Delta \tau =\)\(t'\) − \(t'\) 且 \(\Delta x'\) = 0。

According to observer O in the laboratory frame S, event 1 occurs at (t 1, \(x^1\)) and event 2 at (\(t_{2}\), \(x_{2}\)), and the lifetime of the particle is \(\Delta t = t_{2}\)− t 1, which we shall call \(\Delta T\). Thus in frame S the intervals of time and space that separate the two events are \(\Delta t =\)\(\Delta T = t\)− t and \(\Delta x = x\)− x.根据实验室坐标系 S 中的观察者 O，事件 1 发生在 (t 1, \(x^1\))，事件 2 发生在 (\(t_{2}\), \(x_{2}\))，粒子的寿命为 \(\Delta t = t_{2}\)− t 1，我们将其称为 \(\Delta T\)。因此，在坐标系 S 中，分隔两个事件的时间和空间间隔为 \(\Delta t =\)\(\Delta T = t\)− t 和 \(\Delta x = x\)− x。

These events and intervals are represented in Figure 1.7, and everything we know about them is listed in Table 1.1. Such a table is helpful in establishing which of the interval transformations will be useful.这些事件和间隔如图 1.7 所示，表 1.1 列出了我们对它们的了解。这样的表有助于确定哪些区间变换有用。

Table 1.1 A tabular approach to time dilation. The coordinates of the events are listed and the intervals between them worked out, taking account of any known values. The last row is used to show which of the intervals relates to a named quantity (such as the lifetimes \(\Delta T\) and \(\Delta \tau\)) or has a known value (such as \(\Delta x'\) = 0). Any interval that is neither known nor related to a named quantity is shown as a question mark.表 1.1 时间膨胀的表格方法。列出事件的坐标并计算出事件之间的间隔，同时考虑到任何已知值。最后一行用于显示哪个间隔与命名量相关(例如寿命 \(\Delta T\) 和 \(\Delta \tau\))或具有已知值(例如 \(\Delta x'\) = 0)。任何既不已知也不与命名量相关的区间都显示为问号。

are listed and the intervals between them worked out, taking account of any known values. The last row is used to show which of the intervals relates to a named quantity (such as the lifetimes \(\Delta T\) and \(\Delta \tau\)) or has a known value (such as \(\Delta x'\) = 0). Any interval that is neither known nor related to a named quantity is shown as a question mark.列出并计算出它们之间的间隔，同时考虑到任何已知值。最后一行用于显示哪个间隔与命名量相关(例如寿命 \(\Delta T\) 和 \(\Delta \tau\))或具有已知值(例如 \(\Delta x'\) = 0)。任何既不已知也不与命名量相关的区间都显示为问号。

Event S (laboratory) \(S'\) (rest frame)事件S(实验室)\(S'\)(休息架)

Intervals间隔

Relation to known intervals (\(\Delta T\),?)与已知间隔的关系(\(\Delta T\)，？)

Each of the interval transformation rules that were introduced in the previous section involves three intervals. Only Equation 1.36 involves the three known intervals. Substituting the known intervals into that equation gives上一节中介绍的每个区间变换规则都涉及三个区间。只有方程 1.36 涉及三个已知区间。将已知区间代入该方程可得出

\(\Delta T =\)\(\gamma(V)\)(\(\Delta \tau + 0\)). Therefore the particle lifetimes measured in S and \(S'\) are related by\(\Delta T =\)\(\gamma(V)\)(\(\Delta\tau + 0\))。因此，以 S 和 \(S'\) 测量的颗粒寿命与下式相关：

Since \(\gamma(V)\) > 1, this result tells us that the particle is observed to live longer in the laboratory frame than it does in its own rest frame. This is an example of the effect known as time dilation. A process that occupies a (proper) time \(\Delta \tau\) in its own rest frame has a longer duration \(\Delta T\) when observed from some other frame that moves relative to the rest frame. If the process is the ticking of a clock, then a consequence is that moving clocks will be observed to run slow.由于 \(\gamma(V)\) > 1，该结果告诉我们，观察到粒子在实验室框架中的寿命比在其自身静止框架中的寿命更长。这是时间膨胀效应的一个例子。当从相对于静止坐标系移动的某个其他坐标系观察时，在其自身静止坐标系中占据(适当)时间 \(\Delta \tau\) 的过程具有较长的持续时间 \(\Delta T\)。如果这个过程是一个时钟的滴答声，那么结果就是移动的时钟会被观察到运行缓慢。

The time dilation effect has been demonstrated experimentally many times. It provides one of the most common pieces of evidence supporting Einstein’s theory of special relativity. If it did not exist, many experiments involving short-lived particles, such as muons, would be impossible, whereas they are actually quite routine.时间膨胀效应已被多次实验证明。它提供了支持爱因斯坦狭义相对论的最常见的证据之一。如果它不存在，许多涉及短寿命粒子(例如\(\mu\)子)的实验将是不可能的，而它们实际上是相当常规的。

It is interesting to note that the French mathematician Henri Poincaré (Figure 1.8) proposed an effect similar to time dilation shortly before Einstein formulated special relativity.有趣的是，法国数学家亨利·庞加莱(Henri Poincaré)(图 1.8)在爱因斯坦提出狭义相对论之前不久就提出了类似于时间膨胀的效应。

Exercise 1.4 A particular muon lives for \(\Delta \tau = 2\).2练习 1.4 一个特定的 \(\mu\) 子存在于 \(\Delta \tau = 2\).2 中

\(\mu\) s in its own rest frame. If that muon is travelling with speed V = 3 c/5 relative to an observer on Earth, what is its lifetime as measured by that observer?\(\mu\) 位于其自己的休息框架中。如果该 \(\mu\) 子相对于地球上的观察者以 V = 3 c/5 的速度行进，那么该观察者测得的它的寿命是多少？

1.3.2 Length contraction1.3.2 长度收缩

There is another curious relativistic effect that relates to the length of an object observed from different frames of reference. For the sake of simplicity, the object that we shall consider is a rod, and we shall start our discussion with a definition of the rod’s length that applies whether or not the rod is moving.还有另一种奇怪的相对论效应，它与从不同参考系观察到的物体的长度有关。为了简单起见，我们要考虑的对象是一根杆，我们将从杆长度的定义开始讨论，无论杆是否移动，该定义都适用。

In any inertial frame of reference, the length of a rod is the distance between its \(x^1\) " 1 end-points at a single time as measured in that frame.在任何惯性参考系中，杆的长度是在该参考系中测量的单个时间点之间的 \(x^1\) " 1 端点之间的距离。

Thus, in an inertial frame S in which the rod is oriented along the x -axis and moves along that axis with constant speed V, the length L of the rod can be related to two events, 1 and 2, that happen at the ends of the rod at the same time t. If event 1 is at (t, \(x^1\)) and event 2 is at (t, \(x_{2}\)), then the length of the rod, as measured in S at time t, is given by L = \(\Delta x = x_{2}\)− x因此，在惯性系 S 中，杆沿 x 轴定向并以恒定速度 V 沿该轴移动，杆的长度 L 可以与在同一时间 t 发生在杆两端的两个事件 1 和 2 相关。如果事件 1 发生在 (t, \(x^1\))，事件 2 发生在 (t, \(x_{2}\))，则在时间 t 时在 S 中测量的杆长度由下式给出： L = \(\Delta x = x_{2}\)− x

1.1.

Now consider these same two events as observed in an inertial frame S in which the rod is oriented along the \(x'\)-axis but is always at rest. In this case we still know that event 1 and event 2 occur at the end-points of the rod, but we have no reason to suppose that they will occur at the same time, so we shall describe them by the coordinates (\(t'\), \(x'\)) and (\(t'\), \(x'\)). Although these events may not be simultaneous, we know that in frame \(S'\) the rod is not moving, so its end-points are always at \(x'_{1}\) and \(x'_{2}\). Consequently, we can say that the length of the rod in its own rest frame — a quantity sometimes referred to as the proper length of the rod and denoted L — is given by L = \(\Delta x'\) = \(x'\) − \(x'\).现在考虑在惯性系 S 中观察到的这两个相同的事件，其中杆沿着 \(x'\) 轴定向，但始终处于静止状态。在这种情况下，我们仍然知道事件 1 和事件 2 发生在杆的端点，但我们没有理由假设它们会同时发生，因此我们将通过坐标 (\(t'\), \(x'\)) 和 (\(t'\), \(x'\)) 来描述它们。尽管这些事件可能不是同时发生的，但我们知道在 \(S'\) 坐标系中，杆没有移动，因此其端点始终位于 \(x'_{1}\) 和 \(x'_{2}\)。因此，我们可以说，杆在其自身静止框架中的长度(有时称为杆的适当长度并表示为 L)由下式给出：L = \(\Delta x'\) = \(x'\) – \(x'\)。

These events and intervals are represented in Figure 1.9, and everything we know about them is listed in Table 1.2.这些事件和间隔如图 1.9 所示，表 1.2 列出了我们对它们的了解。

Table 1.2 Events and intervals for length contraction.表 1.2 长度收缩的事件和间隔。

Event S (laboratory) \(S'\) (rest frame)事件S(实验室)\(S'\)(休息架)

Intervals间隔

Relation to known intervals (0, L)与已知区间 (0, L) 的关系

On this occasion, the one unknown interval is \(\Delta t'\), so the interval transformation rule that relates the three known intervals is Equation 1.33. Substituting the known intervals into that equation gives L P = \(\gamma(V)\)(L − 0). So the lengths measured in S and \(S'\) are related by此时，一个未知区间为\(\Delta t'\)，因此将三个已知区间关联起来的区间变换规则为公式1.33。将已知区间代入该方程可得出 L P = \(\gamma(V)\)(L − 0)。因此，以 S 和 \(S'\) 测量的长度之间的关系为

Since \(\gamma(V)\) > 1, this result tells us that the rod is observed to be shorter in the laboratory frame than in its own rest frame. In short, moving rods contract. This is an example of the effect known as length contraction. The effect is not limited to rods. Any moving body will be observed to contract along its direction of motion, though it is particularly important in this case to remember that this does not mean that it will necessarily be seen to contract. There is a substantial body of literature relating to the visual appearance of rapidly moving bodies, which generally involves factors apart from the observed length of the body.由于 \(\gamma(V)\) > 1，该结果告诉我们，观察到杆在实验室框架中比在其自身的静止框架中更短。简而言之，移动的杆收缩。这是长度收缩效应的一个例子。效果不限于棒。任何移动的物体都会被观察到沿着其运动方向收缩，尽管在这种情况下特别重要的是要记住，这并不意味着它一定会被看到收缩。有大量关于快速移动物体的视觉外观的文献，这些文献通常涉及除观察到的物体长度之外的因素。

Length contraction is sometimes known as Lorentz–Fitzgerald contraction after the physicists (Figure 1.4 and Figure 1.11) who first suggested such a phenomenon, though their interpretation was rather different from that of Einstein.长度收缩有时被称为洛伦兹-菲茨杰拉德收缩，这是由首先提出这种现象的物理学家(图 1.4 和图 1.11)命名的，尽管他们的解释与爱因斯坦的解释截然不同。

Exercise 1.5 There is an alternative way of defining length in frame S based练习 1.5 有一种基于 S 帧定义长度的替代方法

on two events, 1 and 2, that happen at different times in that frame. Suppose that event 1 occurs at x = 0 as the front end of the rod passes that point, and event 2 also occurs at x = 0 but at the later time when the rear end passes. Thus event 1 is at (t 1, 0) and event 2 is at (\(t_{2}\), 0). Since the rod moves with uniform speed V in frame S, we can define the length of the rod, as measured in S, by the relation L = V (\(t_{2}\) − t 1). Use this alternative definition of length in frame S to establish that the length of a moving rod is less than its proper length. (The events are represented in Figure 1.10.)两个事件(1 和 2)在该帧中的不同时间发生。假设事件 1 发生在杆前端经过该点时的 x = 0 处，事件 2 也发生在 x = 0 处，但发生在杆后端经过该点的较晚时间。因此，事件 1 位于 (t 1, 0)，事件 2 位于 (\(t_{2}\), 0)。由于杆在坐标系 S 中以匀速 V 移动，因此我们可以通过关系式 L = V (\(t_{2}\) − t 1) 来定义在 S 中测量的杆的长度。使用 S 系中长度的替代定义来确定移动杆的长度小于其正确长度。 (这些事件如图 1.10 所示。)

1.3.3 The relativity of simultaneity1.3.3 同时性的相对性

It was noted in the discussion of length contraction that two events that occur at the same time in one frame do not necessarily occur at the same time in another frame. Indeed, looking again at Figure 1.9 and Table 1.2 but now calling on the interval transformation rule of Equation 1.32, it is clear that if the events 1 and 2 are observed to occur at the same time in frame S (so \(\Delta t = 0\)) but are separated by a distance L along the x -axis, then in frame \(S'\) they will be separated by the time在长度收缩的讨论中注意到，在一帧中同时发生的两个事件不一定在另一帧中同时发生。事实上，再次查看图 1.9 和表 1.2，但现在调用方程 1.32 的区间变换规则，很明显，如果观察到事件 1 和 2 在帧 S 中同时发生(因此 \(\Delta t = 0\))，但沿 x 轴相隔距离 L，则在帧 \(S'\) 中它们将被时间间隔

Two events that occur at the same time in some frame are said to be simultaneous in that frame. The above result shows that the condition of being simultaneous is a relative one not an absolute one; two events that are simultaneous in one frame are not necessarily simultaneous in every other frame. This consequence of the Lorentz transformations is referred to as the relativity of simultaneity.在某个帧中同时发生的两个事件被称为在该帧中同时发生。上述结果表明，同时的条件是相对的，而不是绝对的；一帧中同时发生的两个事件不一定在其他帧中同时发生。洛伦兹变换的这一结果被称为同时性相对性。

1.3.4 The Doppler effect1.3.4 多普勒效应

A physical phenomenon that was well known long before the advent of special relativity is the Doppler effect. This accounts for the difference between the emitted and received frequencies (or wavelengths) of radiation arising from the relative motion of the emitter and the receiver. You will have heard an example of the Doppler effect if you have listened to the siren of a passing ambulance: the frequency of the siren is higher when the ambulance is approaching (i.e. travelling towards the receiver) than when it is receding (i.e. travelling away from the receiver).在狭义相对论出现之前很久就众所周知的物理现象是多普勒效应。这解释了由于发射器和接收器的相对运动而产生的辐射的发射和接收频率(或波长)之间的差异。如果您听过经过的救护车的警报声，您就会听到多普勒效应的一个例子：当救护车接近(即朝接收器行驶)时警报器的频率比救护车后退(即远离接收器行驶)时的频率更高。

Astronomers routinely use the Doppler effect to determine the speed of approach or recession of distant stars. They do this by measuring the received wavelengths of narrow lines in the star’s spectrum, and comparing their results with the proper wavelengths of those lines that are well known from laboratory measurements and represent the wavelengths that would have been emitted in the star’s rest frame.天文学家通常使用多普勒效应来确定遥远恒星接近或后退的速度。他们通过测量恒星光谱中窄线的接收波长，并将其结果与实验室测量中众所周知的那些线的正确波长进行比较，这些线代表了恒星静止框架中发射的波长。

Despite the long history of the Doppler effect, one of the consequences of special relativity was the recognition that the formula that had traditionally been used to describe it was wrong. We shall now obtain the correct formula.尽管多普勒效应有着悠久的历史，但狭义相对论的后果之一是人们认识到传统上用来描述它的公式是错误的。现在我们将获得正确的公式。

Consider a lamp at rest at the origin of an inertial frame S emitting electromagnetic waves of proper frequency f em as measured in S. Now suppose that the lamp is observed from another inertial frame \(S'\) that is in standard configuration with S, moving away at constant speed V (see Figure 1.12). A detector fixed at the origin of \(S'\) will show that the radiation from the receding lamp is received with frequency f rec as measured in \(S'\). Our aim is to find the relationship between f rec and f em.考虑一盏静止在惯性系 S 原点的灯，发射在 S 中测量的适当频率 f em 的电磁波。现在假设从另一个与 S 处于标准配置的惯性系 \(S'\) 观察该灯，以恒定速度 V 移动(见图 1.12)。固定在 \(S'\) 原点的检测器将显示来自后退灯的辐射以 \(S'\) 测量的频率 f rec 被接收。我们的目标是找到 f rec 和 fem 之间的关系。

The emitted waves have regularly positioned nodes that are separated by a proper wavelength \(\lambda\) em = \(f_{\rm em}\)/c as measured in S. In that frame the time interval between the emission of one node and the next, \(\Delta t\), represents the proper period of the wave, T em, so we can write \(\Delta t = T em\)= 1/\(f_{\rm em}\).发射的波具有规则定位的节点，这些节点以适当的波长 \(\lambda\) em = \(f_{\rm em}\)/c 分隔开，如在 S 中测量的。在该帧中，一个节点与下一个节点发射之间的时间间隔 \(\Delta t\) 代表波的适当周期 T em，因此我们可以写成 \(\Delta t = T em\)= 1/\(f_{\rm em}\)。

Due to the phenomenon of time dilation, an observer in frame \(S'\) will find that the time separating the emission of successive nodes is \(\Delta t' = \gamma(V)\Delta t\). However, this is not the time that separates the arrival of those nodes at the detector because the detector is moving away from the emitter at a constant rate. In fact, during the interval \(\Delta t'\) the detector will increase its distance from the emitter by \(V\Delta t'\) as measured in \(S'\), and this will cause the reception of the two nodes to be separated by a total time \(\Delta t' + V\Delta t'/c\) as measured in \(S'\). This represents the received period of the wave and is therefore the reciprocal of the received frequency, so we can write由于时间膨胀现象，在 \(S'\) 坐标系中的观察者会发现连续节点发射的时间间隔为 \(\Delta t'\) = \(\gamma(V)\) \(\Delta t\)。然而，这并不是这些节点到达检测器的时间间隔，因为检测器以恒定的速率远离发射器。事实上，在时间间隔 \(\Delta t'\) 内，检测器与发射器的距离将增加 V \(\Delta t'\)(以 \(S'\) 为单位测量)，这将导致两个节点的接收间隔总时间 \(\Delta t'\) + V \(\Delta t'\)/c(以 \(S'\) 为单位测量)。这代表了波的接收周期，因此是接收频率的倒数，所以我们可以写

We can now identify - \(\Delta t\) with the reciprocal of the emitted frequency and use the identity \(\gamma(V)\) = 1/(1 − V/c)(1 + V/c) to write现在我们可以用发射频率的倒数来表示 - \(\Delta t\)，并使用恒等式 \(\gamma(V)\) = 1/(1 − V/c)(1 + V/c) 来写出

which can be rearranged to give可以重新排列以给出

This shows that the radiation received from a receding source will have a frequency that is less than the proper frequency with which the radiation was emitted. It follows that the received wavelength \(\lambda\) rec = c/\(f_{\rm rec}\) will be greater than the proper wavelength \(\lambda\) em. Consequently, the spectral lines seen in the light of receding stars will be shifted towards the red end of the spectrum; a phenomenon known as redshift (see Figure 1.13). In a similar way, the spectra of approaching stars will be subject to a blueshift described by an equation similar to Equation 1.42 but with V replaced by − V throughout. The correct interpretation of these Doppler shifts is of great importance.这表明从后退源接收到的辐射的频率将小于发射辐射的适当频率。由此可见，接收波长 \(\lambda\) rec = c/\(f_{\rm rec}\) 将大于适当波长 \(\lambda\) em。因此，在后退恒星的光线下看到的光谱线将向光谱的红端移动；这种现象称为红移(见图 1.13)。以类似的方式，接近恒星的光谱将受到蓝移的影响，该蓝移由类似于方程 1.42 的方程描述，但 V 始终替换为 - V。对这些多普勒频移的正确解释非常重要。

Exercise 1.6 Some astronomers are studying an unusual phenomenon, close练习 1.6 一些天文学家正在研究一种不寻常的现象，接近

to the centre of our galaxy, involving a jet of material containing sodium. The jet is moving almost directly along the line between the Earth and the galactic centre. In a laboratory, a stationary sample of sodium vapour absorbs light of wavelength \(\lambda\) = \(5850\times10^{-10}\) m. Spectroscopic studies show that the wavelength of the sodium absorption line in the jet’s spectrum is \(\lambda'\) = \(4483\times10^{-10}\) m. Is the jet approaching or receding? What is the speed of the jet relative to Earth? (Note that the main challenge in this question is the mathematical one of using Equation 1.42 to obtain an expression for V in terms of \(\lambda\)/\(\lambda'\).)到我们银河系的中心，涉及到一股含有钠的物质射流。这架喷射机几乎直接沿着地球和银河中心之间的线移动。在实验室中，钠蒸气的固定样品吸收波长为 \(\lambda\) = \(5850\times10^{-10}\) m 的光。光谱研究表明，射流光谱中钠吸收线的波长为 \(\lambda'\) = \(4483\times10^{-10}\) m。喷气式飞机正在接近还是正在后退？喷气机相对于地球的速度是多少？ (请注意，此问题的主要挑战是使用公式 1.42 获得以 \(\lambda\)/\(\lambda'\) 表示的 V 表达式。)

1.3.5 The velocity transformation1.3.5 速度变换

Suppose that an object is observed to be moving with velocity v = (\(v_{x}\), \(v_{y}\), \(v_{z}\)) in an inertial frame S. What will its velocity be in a frame \(S'\) that is in standard configuration with S, travelling with uniform speed V in the positive x -direction? The Galilean transformation would lead us to expect \(v'\) = (\(v_{x}\) − V, \(v_{y}\), \(v_{z}\)), but we know that is not consistent with the observed behaviour of light. Once again we shall use the interval transformation rules that follow directly from the Lorentz transformations to find the velocity transformation rule according to special relativity.假设观察到一个物体在惯性系 S 中以速度 v = (\(v_{x}\), \(v_{y}\), \(v_{z}\)) 移动。在与 S 处于标准配置的系 \(S'\) 中，该物体以匀速 V 沿正 x 方向行进，其速度是多少？伽利略变换会让我们期望 \(v'\) = (\(v_{x}\) − V, \(v_{y}\), \(v_{z}\))，但我们知道这与观察到的光行为不一致。我们将再次使用直接源自洛伦兹变换的区间变换规则来根据狭义相对论找到速度变换规则。

We know from Equations 1.32 and 1.33 that the time and space intervals between two events 1 and 2 that occur on the x -axis in frame S, transform according to从方程 1.32 和 1.33 可知，S 帧中 x 轴上发生的两个事件 1 和 2 之间的时间和空间间隔，根据以下变换：

Dividing the second of these equations by the first gives将这些方程中的第二个方程除以第一个方程得到

Dividing the upper and lower expressions on the right-hand side of this equation by \(\Delta t\), and cancelling the Lorentz factors, gives将该方程右侧的上式和下式表达式除以 \(\Delta t\)，并取消洛伦兹因子，得到

Now, if we suppose that the two events that we are considering are very close together — indeed, if we consider the limit as \(\Delta t\) and \(\Delta x\) go to zero — then the quantities \(\Delta x\)/\(\Delta t\) and \(\Delta x'\)/\(\Delta t'\) will become the instantaneous velocity components \(v_{x}\) and v \(x'\) of a moving object that passes through the events 1 and 2. Extending these arguments to three dimensions by considering events that are not confined to the x -axis leads to the following velocity transformation rules:现在，如果我们假设我们正在考虑的两个事件非常接近 - 事实上，如果我们将极限视为 \(\Delta t\) 和 \(\Delta x\) 趋近于零 - 那么量 \(\Delta x\)/\(\Delta t\) 和 \(\Delta x'\)/\(\Delta t'\) 将成为穿过事件 1 和 2 的移动物体的瞬时速度分量 \(v_{x}\) 和 v \(x'\)。通过考虑不限于 x 轴的事件将这些参数扩展到三个维度，从而得出以下速度变换规则：

These equations may look rather odd at first sight but they make good sense in the context of special relativity. When \(v_{x}\) and V are small compared to the speed of light c, the term \(v_{x}\) V/\(c^2\) is very small and the denominator is approximately 1. In \(x'\) = \(v_{x}\) − V, is recovered such cases, the Galilean velocity transformation rule, v as a low-speed approximation to the special relativistic result. At high speeds the situation is even more interesting, as the following question will show.这些方程乍一看可能相当奇怪，但在狭义相对论的背景下它们很有意义。当 \(v_{x}\) 和 V 与光速 c 相比较小时，\(v_{x}\) V/\(c^2\) 项非常小，分母约为 1。在 \(x'\) = \(v_{x}\) − V 中，恢复了这种情况下的伽利略速度变换规则，v 作为狭义相对论结果的低速近似。在高速情况下，情况甚至更加有趣，正如下面的问题所示。

• ● An observer has established that two objects are receding● 观察者已确定两个物体正在后退

in opposite directions. Object 1 has speed c, and object 2 has speed V. Using the velocity transformation, compute the velocity with which object 1 recedes as measured by an observer travelling on object 2.在相反的方向。对象 1 的速度为 c，对象 2 的速度为 V。使用速度变换，计算在对象 2 上行进的观察者测量到的对象 1 后退的速度。

❍ Let the line along which the objects are travelling be the x -axis of the original observer’s frame, S. We can then suppose that a frame of reference \(S'\) that has its origin on object 2 is in standard configuration with frame S, and apply the velocity transformation to the velocity components of object 1 with v = (− c, 0, 0) (see Figure 1.14). The velocity transformation predicts that as 0, 0), where observed in \(S'\), the velocity of object 2 is \(v'\) = (\(v'\),❍ 设物体沿其移动的线为原始观察者坐标系 S 的 x 轴。然后，我们可以假设原点位于物体 2 上的参考系 \(S'\) 与坐标系 S 处于标准配置，并将速度变换应用于物体 1 的速度分量，其中 v = (− c, 0, 0)(见图 1.14)。速度变换预测为 0, 0)，在 \(S'\) 中观察到，物体 2 的速度为 \(v'\) = (\(v'\),

So, as observed from object 2, object 1 is travelling in the − \(x'\)-direction at the speed of light, c. This was inevitable, since the second postulate of special relativity (which was used in the derivation of the Lorentz transformations) tells us that all observers agree about the speed of light. It is nonetheless pleasing to see how the velocity transformation delivers the required result in this case. It is worth noting that this result does not depend on the value of V.因此，从物体 2 观察到，物体 1 正在以光速 c 沿 - \(x'\) 方向行进。这是不可避免的，因为狭义相对论的第二假设(用于洛伦兹变换的推导)告诉我们所有观察者都同意光速。尽管如此，在这种情况下，看到速度变换如何提供所需的结果还是令人高兴的。值得注意的是，这个结果并不依赖于V的值。

Exercise 1.7 According to an observer on a spacestation,练习 1.7 根据空间站观察员的说法，

two spacecraft are moving away, travelling in the same direction at different speeds. The nearer spacecraft is moving at speed c/2, the further at speed 3 c/4. What is the speed of one of the spacecraft as observed from the other?两艘航天器正在远离，以不同的速度朝同一方向行驶。较近的航天器以 c/2 的速度移动，较远的航天器以 3 c/4 的速度移动。从其中一艘航天器观察到另一艘航天器的速度是多少？

1.4 Minkowski spacetime1.4 闵可夫斯基时空

In 1908 Einstein’s former mathematics teacher, Hermann Minkowski (Figure 1.15), gave a lecture in which he introduced the idea of spacetime. He said in the lecture: ‘Henceforth space by itself, and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality’. This section concerns that four-dimensional union of space and time, the set of all possible events, which is now called Minkowski spacetime.1908 年，爱因斯坦的前数学老师赫尔曼·闵可夫斯基(Hermann 闵可夫斯基，图 1.15)在一次演讲中介绍了时空的概念。他在演讲中说：“从此以后，空间本身和时间本身都注定会消失在纯粹的阴影中，只有两者的某种结合才能保留一个独立的现实”。本节涉及空间和时间的四维联合，即所有可能事件的集合，现在称为闵可夫斯基时空。

1.4.1 Spacetime diagrams, lightcones and causality1.4.1 时空图、光锥和因果关系

We have already seen how the Lorentz transformations lead to some very counter-intuitive consequences. This subsection introduces a graphical tool known as a spacetime diagram or a Minkowski diagram that will help you to visualize events in Minkowski spacetime and thereby develop a better intuitive understanding of relativistic effects. The spacetime diagram for a frame of reference S is usually presented as a plot of ct against x, and each point on the diagram represents a possible event as observed in frame S. The y - and z -coordinates are usually ignored.我们已经看到洛伦兹变换如何导致一些非常违反直觉的后果。本小节介绍一种称为时空图或闵可夫斯基图的图形工具，它将帮助您可视化闵可夫斯基时空中的事件，从而更好地直观地理解相对论效应。参考系 S 的时空图通常表示为 ct 相对于 x 的图，图中的每个点代表在 S 系中观察到的一个可能事件。y 和 z 坐标通常被忽略。

Given two inertial frames, S and \(S'\), in standard configuration, it is instructive to plot the \(ct'\)- and \(x'\)-axes of frame \(S'\) on the spacetime diagram for frame S. The \(x'\)-axis of frame \(S'\) is defined by the set of events for which \(ct'\) = 0, and the \(ct'\)-axis is defined by the set of events for which \(x'\) = 0. The coordinates of these events in S are related to their coordinates in \(S'\) by the following Lorentz transformations. (Note that the time transformation of Equation 1.5 has been multiplied by c so that each coordinate can be measured in units of length.)给定两个惯性系 S 和 \(S'\)，在标准配置中，在系 S 的时空图上绘制系统 \(S'\) 的 \(ct'\) 轴和 \(x'\) 轴是有益的。系统 \(x'\) 的 \(x'\) 轴由 \(ct'\) = 0 的事件集定义，并且\(ct'\) 轴由 \(x'\) = 0 的事件集定义。这些事件在 S 中的坐标通过以下洛伦兹变换与它们在 \(S'\) 中的坐标相关。 (请注意，公式 1.5 的时间变换已乘以 c，以便可以以长度单位测量每个坐标。)

Setting \(ct'\) = 0 in the first of these equations gives 0 = \(\gamma(V)\)(ct − V x/c). This shows that in the spacetime diagram for frame S, the \(ct'\)-axis of frame \(S'\) is represented by the line ct = (V/c) x, a straight line through the origin with gradient V/c. Similarly, setting \(x'\) = 0 in the second transformation equation gives 0 = \(\gamma(V)\)(x − V t), showing that the \(x'\)-axis of frame \(S'\) is represented by the line ct = (c/V) x, a straight line through the origin with gradient c/V in the spacetime diagram of S. These lines are shown in Figure在第一个方程中设置 \(ct'\) = 0 得出 0 = \(\gamma(V)\)(ct − V x/c)。这表明，在坐标系 S 的时空图中，坐标系 \(S'\) 的 \(ct'\) 轴由线 ct = (V/c) x 表示，这是一条通过原点且梯度为 V/c 的直线。类似地，在第二个变换方程中设置 \(x'\) = 0 得到 0 = \(\gamma(V)\)(x − V t)，表明框架 \(S'\) 的 \(x'\) 轴由线 ct = (c/V) x 表示，这条直线穿过原点，在 S 的时空图中具有梯度 c/V。这些线如图所示

1.16.1.16。

There is another feature of interest in the diagram. The straight line through the origin of gradient 1 links all the events where x = ct and thus shows the path of a light ray that passes through x = 0 at time t =该图中还有另一个有趣的特征。通过梯度 1 原点的直线联络了 x = ct 处的所有事件，因此显示了在时间 t = 时穿过 x = 0 的光线的路径

0. Using the inverse0. 使用逆

Lorentz transformations shows that this line also passes through all the events where \(\gamma(V)\)(\(x'\) + V \(t'\)) = \(\gamma(V)\)(\(ct'\) + V \(x'\)/c), that is (after some cancelling and rearranging), where \(x'\) = \(ct'\). So the line of gradient 1 passing through the origin also represents the path of a light ray that passes through the origin of frame \(S'\) at \(t'\) = 0. In fact, any line with gradient 1 on a spacetime diagram must always represent the possible path of a light ray, and thanks to the second postulate of special relativity, we can be sure that all observers will agree about that.洛伦兹变换表明，这条线还经过 \(\gamma(V)\)(\(x'\) + V \(t'\)) = \(\gamma(V)\)(\(ct'\) + V \(x'\)/c) 的所有事件，即(经过一些取消和重新排列后)，其中 \(x'\) = \(ct'\)。因此，穿过原点的梯度 1 的线也代表光线在 \(t'\) = 0 处穿过坐标系 \(S'\) 的原点的路径。事实上，时空图上任何梯度为 1 的线都必须始终代表光线的可能路径，并且由于狭义相对论的第二假设，我们可以确信所有观察者都会同意这一点。

As the relative speed V of the frames S and \(S'\) increases, the lines representing the \(x'\)- and \(ct'\)-axes of \(S'\) close in on the line of gradient 1 from either side, rather like the closing of a clapper board. This behaviour reflects the fact that Lorentz transformations will generally alter the coordinates of events but will not change the behaviour of light on which all observers must agree.随着框架 S 和 \(S'\) 的相对速度 V 的增加，表示 \(S'\) 的 \(x'\) 轴和 \(ct'\) 轴的线从两侧向梯度 1 线靠拢，就像关闭拍板一样。这种行为反映了这样一个事实：洛伦兹变换通常会改变事件的坐标，但不会改变所有观察者必须同意的光的行为。

In the somewhat unusual case when we include a second spatial axis (the y -axis, say) in the spacetime diagram, the original line of gradient 1 is seen to be part of a cone, as indicated in Figure 1.17. This cone, which connects the event at the origin to all those events, past and future, that might be linked to it by a signal travelling at the speed of light, is an example of a lightcone. A horizontal slice (at ct = constant) through the (pseudo) three-dimensional diagram at any particular time shows a circle, but in a fully four-dimensional diagram with all three spatial axes included, such a fixed-time slice would be a sphere, and would represent a spherical shell of light surrounding the origin. At times earlier than t = 0, the shell would represent incoming light signals closing in on the origin. At times later than t = 0, the shell would represent outgoing light signals travelling away from the origin. Although observers O and \(O'\), using frames S and \(S'\), would not generally agree about the coordinates of events, they would agree about which events were on the lightcone, which were inside the lightcone and which were outside. This agreement between observers makes lightcones very useful in discussions about which events might cause, or be caused by, other events.在有点不寻常的情况下，当我们在时空图中包含第二个空间轴(例如 y 轴)时，原始梯度线 1 被视为圆锥体的一部分，如图 1.17 所示。这个锥体将起源处的事件与所有过去和未来的事件联络起来，这些事件可能通过以光速传播的信号与其相连，这是光锥体的一个例子。在任何特定时间通过(伪)三维图的水平切片(ct = 常数)显示一个圆，但在包含所有三个空间轴的完整四维图中，这样的固定时间切片将是一个球体，并且表示围绕原点的光的球壳。在 t = 0 之前的时间，壳层将代表接近原点的传入光信号。在 t = 0 以后的时间，壳将代表离开原点的传出光信号。尽管使用框架 S 和 \(S'\) 的观察者 \(O'\) 和 \(O'\) 通常不会就事件的坐标达成一致，但他们会同意哪些事件在光锥上、哪些事件在光锥内部、哪些在光锥外部。观察者之间的这种一致使得光锥在讨论哪些事件可能导致其他事件或由其他事件引起时非常有用。

Going back to an ordinary two-dimensional spacetime diagram of the kind shown in Figure 1.18, it is straightforward to read off the coordinates of an event in frame S or in frame \(S'\). The event 1 in the diagram clearly has coordinates (ct 1, \(x^1\)) in frame S. In frame \(S'\), it has a different set of coordinates. These can be determined by drawing construction lines parallel to the lines representing the primed axes. Where a construction line parallel to one primed axis intersects the other primed axis, the coordinate can be found. By doing this on both axes, both coordinates are found. In the case of Figure 1.18, the dashed construction lines show that, as observed in frame \(S'\), event 1 occurs at the same time as event 2, and at the same position as event 3.回到图 1.18 所示的普通二维时空图，可以直接读取 S 系或 \(S'\) 系中事件的坐标。图中的事件 1 显然在 S 帧中具有坐标 (ct 1, \(x^1\))。在 \(S'\) 帧中，它具有一组不同的坐标。这些可以通过绘制与代表带底漆的轴的线平行的构造线来确定。当平行于一个主轴的构造线与另一个主轴相交时，可以找到坐标。通过在两个轴上执行此操作，可以找到两个坐标。在图 1.18 的情况下，虚线表示，如在帧 \(S'\) 中观察到的，事件 1 与事件 2 同时发生，并且与事件 3 发生在同一位置。

Another lesson that can be drawn from Figure 1.18 concerns the order of events. Starting from the bottom of the ct -axis and working upwards, it is clear that in frame S, the four events occur in the order 0, 2, 3 and 1. But it is equally clear from the dashed construction lines that in frame \(S'\), event 3 happens at the same time as event 0, and both happen at an earlier time than event 2 and event 1, which are also simultaneous in \(S'\). This illustrates the relativity of simultaneity, but more importantly it also shows that the order of events 2 and 3 will be different for observers O and \(O'\).从图 1.18 中可以得出的另一个教训涉及事件的顺序。从 ct 轴底部开始向上，可以清楚地看出，在帧 S 中，四个事件按 0、2、3 和 1 的顺序发生。但从虚线构造线同样可以清楚地看出，在帧 \(S'\) 中，事件 3 与事件 0 同时发生，并且两者发生的时间早于事件 2 和事件 1，而事件 2 和事件 1 在 \(S'\) 中也是同时发生的。这说明了同时性的相对性，但更重要的是，它还表明对于观察者 O 和 \(O'\) 来说，事件 2 和 3 的顺序会不同。

At first sight it is quite shocking to learn that the relative motion of two observers can reverse the order in which they observe events to happen. This has the potential to overthrow our normal notion of causality, the principle that all observers must agree that any effect is preceded by its cause. It is easy to imagine observing the pressing of a plunger and then observing the explosion that it causes. It would be very shocking, however, if some other observer, simply by moving sufficiently fast in the right direction, was able to observe the explosion first and then the pressing of the plunger that caused it. (It is important to remember that we are discussing observing, not seeing.)乍一看，两个观察者的相对运动可以颠倒他们观察事件发生的顺序，这真是令人震惊。这有可能推翻我们正常的因果关系概念，即所有观察者都必须同意任何结果都先有其原因的原则。很容易想象观察柱塞的按压，然后观察它引起的爆炸。然而，如果其他观察者仅仅通过在正确的方向上足够快地移动，能够首先观察到爆炸，然后观察到导致爆炸的柱塞的按压，那将是非常令人震惊的。 (重要的是要记住，我们正在讨论观察，而不是看到。)

Fortunately, such an overthrow of causality is not permitted by special relativity, provided that we do not allow signals to travel at speeds greater than c. Although observers will disagree about the order of some events, they will not disagree about the order of any two events that might be linked by a light signal or any signal that travels at less than the speed of light. Such events are said to be causally related.幸运的是，狭义相对论不允许这种因果关系的推翻，前提是我们不允许信号以大于 c 的速度传播。尽管观察者会对某些事件的顺序有不同意见，但他们不会对可能通过光信号或任何以低于光速传播的信号联系起来的任何两个事件的顺序有不同意见。据说这些事件是有因果关系的。

To see how the order of causally related events is preserved, look again at Figure 1.18, noting that all the events that are causally related to event 0 are contained within its lightcone, and that includes event要了解如何保留因果相关事件的顺序，请再次查看图 1.18，注意与事件 0 因果相关的所有事件都包含在其光锥内，其中包括事件

2. Events that are not2. 不存在的事件

causally related to event 0, such as event 1 and event 3, are outside the lightcone of event 0 and could only be linked to that event by signals that travel faster than light. Now, remember that as the relative speed V of the observers O and \(O'\) increases, the line representing the \(ct'\)-axis closes in on the lightcone. As a result, there will not be any value of V that allows the causally related events 0 and 2 to change their order. Event 2 will always be at a higher value of \(ct'\) than event 0. However, when you examine the corresponding behaviour of events 0 and 3, which are not causally related, the conclusion is quite different. Figure 1.18 shows the condition in which event 0 and event 3 occur at the same time \(t'\) = 0, according to \(O'\). When O and \(O'\) have a lower relative speed, event 3 occurs after event 0, but as V increases and the line representing the \(x'\)-axis (where all events occur at \(ct'\) = 0) closes in on the lightcone, we see that there will be a value of V above which the order of event 0 and event 3 is reversed.与事件 0 因果相关的事件(例如事件 1 和事件 3)位于事件 0 的光锥之外，并且只能通过传播速度快于光速的信号与该事件相关联。现在，请记住，随着观察者 O 和 \(O'\) 的相对速度 V 的增加，表示 \(ct'\) 轴的线会靠近光锥。因此，不会有任何 V 值允许因果相关的事件 0 和 2 更改其顺序。事件 2 的 \(ct'\) 值始终高于事件 0。但是，当您检查没有因果关系的事件 0 和 3 的相应行为时，结论会大不相同。图1.18显示了根据\(O'\)，事件0和事件3同时发生\(t'\) = 0的情况。当 O 和 \(O'\) 的相对速度较低时，事件 3 在事件 0 之后发生，但随着 V 的增加，代表 \(x'\) 轴的线(其中所有事件发生在 \(ct'\) = 0 处)在光锥上闭合，我们看到将存在一个 V 值，高于该值时事件 0 和事件 3 的顺序相反。

So, if event 0 represents the pressing of a plunger and event 2 and event 3 represent explosions, all observers will agree that event 0 might have caused event 2, which happened later. However, those same observers will not agree about the order of event 0 and event 3, though they will agree that event 0 could not have caused event 3 unless bodies or signals can travel faster than light. It is the desire to preserve causal relationships that is the basis for the requirement that no material body or signal of any kind should be able to travel faster than light.因此，如果事件 0 代表柱塞的按压，而事件 2 和事件 3 代表爆炸，则所有观察者都会同意事件 0 可能导致了稍后发生的事件 2。然而，这些观察者不会就事件 0 和事件 3 的顺序达成一致，尽管他们会同意事件 0 不可能导致事件 3，除非物体或信号的传播速度比光快。维护因果关系的愿望是要求任何物质体或任何类型的信号都不能比光传播得更快的要求的基础。

• ● Is event 1 in Figure 1.18 causally related to event 0● 图1.18中的事件1与事件0有因果关系吗

? Is event 1 causally related to event 3? Justify your answers.？事件 1 与事件 3 有因果关系吗？证明你的答案合理。

❍ Event 1 is outside the lightcone of event 0, so the two cannot be causally related. The diagram does not show the lightcone of event 3, but if you imagine a line of gradient 1, parallel to the shown lightcone, passing through event 3, it is clear that event 1 is inside the lightcone of event 3, so those two events are causally related. The earlier event may have caused the later one, and all observers will agree about that.❍ 事件 1 在事件 0 的光锥之外，因此两者不可能存在因果关系。该图未显示事件 3 的光锥，但如果您想象一条与所示光锥平行的梯度 1 线穿过事件 3，则很明显事件 1 位于事件 3 的光锥内部，因此这两个事件存在因果关系。较早的事件可能导致了较晚的事件，所有观察者都会同意这一点。

An important lesson to learn from this question is the significance of drawing lightcones for events other than those at the origin. Every event has a lightcone, and that lightcone is of great value in determining causal relationships.从这个问题中学到的一个重要教训是，为除原点以外的事件绘制光锥的重要性。每个事件都有一个光锥，而该光锥对于确定因果关系具有很大的价值。

1.4.2 Spacetime separation and the Minkowski metric1.4.2 时空分离和闵可夫斯基度规

In three-dimensional space, the separation between two points (\(x^1\), y 1, z 1) and (\(x_{2}\), \(y^{2}\), \(z^{2}\)) can be conveniently described by the square of the distance \(\Delta l\) between them:在三维空间中，两点 (\(x^1\), y 1, z 1) 和 (\(x_{2}\), \(y^{2}\), \(z^{2}\)) 之间的间隔可以方便地用它们之间的距离 \(\Delta l\) 的平方来描述：

where \(\Delta x = x\)− x, \(\Delta y = y\)− y and \(\Delta z = z\)− z. This quantity has the useful property of being unchanged by rotations of the coordinate system. So, if we choose to describe the points using a new coordinate system with axes \(x'\), \(y'\) and \(z'\), obtained by rotating the old system about one or more of its axes, then the spatial separation of the two points would still be described by an expression of the form其中 \(\Delta x = x\)− x、\(\Delta y = y\)− y 和 \(\Delta z = z\)− z。该量具有不随坐标系旋转而改变的有用特性。因此，如果我们选择使用具有 \(x'\)、\(y'\) 和 \(z'\) 轴的新坐标系(通过绕一个或多个轴旋转旧系统而获得)来描述点，那么两个点的空间分离仍将由以下形式的表达式来描述

and we would find in addition that此外我们还会发现

We describe this situation by saying that the spatial separation of two points is invariant under rotations of the coordinate system used to describe the positions of the two points.我们通过这样的说法来描述这种情况：在用于描述两点位置的坐标系旋转的情况下，两点的空间间隔是不变的。

These ideas can be extended to four-dimensional Minkowski spacetime, where the most useful expression for the spacetime separation of two events is the following.这些想法可以扩展到四维闵可夫斯基时空，其中两个事件的时空分离最有用的表达式如下。

Spacetime separation时空分离

The reason why this particular form is chosen is that it turns out to be invariant under Lorenz transformations. So, if O and \(O'\) are inertial observers using frames S and \(S'\), they will generally not agree about the coordinates that describe two events 1 and 2, or about the distance or the time that separates them, but they will agree that the two events have an invariant spacetime separation选择这种特殊形式的原因是它在洛伦兹变换下是不变的。因此，如果 O 和 \(O'\) 是使用坐标系 S 和 \(S'\) 的惯性观察者，他们通常不会就描述两个事件 1 和 2 的坐标，或者将它们分开的距离或时间达成一致，但他们会同意这两个事件具有不变的时空分离。

Exercise 1.8 Two events occur at (ct, x, y, z) = (3, 7, 0, 0) m and练习 1.8 两个事件发生在 (ct, x, y, z) = (3, 7, 0, 0) m 处，并且

(ct, x, y, z) = (5, 5, 0, 0) m. What is their spacetime separation?(ct, x, y, z) = (5, 5, 0, 0) m。他们的时空间隔是多少？

Exercise 1.9 In the case that \(\Delta y = 0 and\)\(\Delta z = 0\), use the interval练习 1.9 在 \(\Delta y = 0\)且 \(\Delta z = 0\)的情况下，使用区间

transformation rules to show that the spacetime separation given by Equation 1.49 really is invariant under Lorentz transformations.变换规则表明方程 1.49 给出的时空分离在洛伦兹变换下确实是不变的。

A convenient way of writing the spacetime separation is as a summation:写出时空分离的一种方便方法是求和：

where the four quantities \(\Delta x\) 0, \(\Delta x\) 1, \(\Delta x\) 2 and \(\Delta x\) 3 are the components of [\(\Delta x\) \(\mu\)] = (c \(\Delta t\), \(\Delta x\), \(\Delta y\), \(\Delta z\)), and the new quantities \(\eta\) that have been introduced are the sixteen components of an entity called the Minkowski metric, which can be represented as其中，四个量 \(\Delta x\) 0、\(\Delta x\) 1、\(\Delta x\) 2 和 \(\Delta x\) 3 是 [\(\Delta x\) \(\mu\)] = (c \(\Delta t\), \(\Delta x\), \(\Delta y\), \(\Delta z\)) 的分量，引入的新量 \(\eta\) 是称为闵可夫斯基度规的实体的 16 个分量，可以表示为

It’s worth noting that the Minkowski metric has been shown as a matrix only for convenience; Equation 1.51 is not a matrix equation, though it is a well-defined sum. The important point is that the quantity [\(\eta\) \(\mu\)\(\nu\)] has sixteen components, and from Equation 1.52 you can uniquely identify each of them. The metric provides a valuable reminder of how the spacetime separation is related to the coordinate intervals. Metrics will have a crucial role to play in the rest of this book. The Minkowski metric is just the first of many that you will meet.值得注意的是，为了方便起见，将闵可夫斯基度规显示为矩阵。方程 1.51 不是一个矩阵方程，尽管它是一个明确定义的和。重要的一点是，数量 [\(\eta\) \(\mu\)\(\nu\)] 有 16 个分量，从公式 1.52 中您可以唯一地识别它们中的每一个。该度规提供了关于时空分离与坐标间隔如何相关的有价值的提醒。指标将在本书的其余部分中发挥至关重要的作用。闵可夫斯基度规只是您将遇到的众多度规中的第一个。

The spacetime separation of two events is an important quantity for several reasons. Its sign alone tells us about the possible causal relationship between the events. In fact, we can identify three classes of relationship, corresponding to the cases (\(\Delta s\)) 2 > 0, (\(\Delta s\)) 2 = 0 and (\(\Delta s\)) 2 < 0.由于多种原因，两个事件的时空间隔是一个重要的量。它的符号本身就告诉我们事件之间可能存在的因果关系。事实上，我们可以识别三类关系，分别对应于 (\(\Delta s\)) 2 > 0、(\(\Delta s\)) 2 = 0 和 (\(\Delta s\)) 2 < 0 的情况。

Time-like, light-like and space-like separations似时间、似光、似空间的分离

Events with a positive spacetime separation, (\(\Delta s\)) 2 > 0, are said to be time-like separated. Such events are causally related, and there will exist a frame in which the two events happen at the same place but at different times.具有正时空分离 (\(\Delta s\)) 2 > 0 的事件被称为类时间分离。此类事件是有因果关系的，并且会存在一个框架，其中两个事件在同一地点但不同时间发生。

s) 2 = 0, are said to be Events with a zero (or null) spacetime separation, (\(\Delta l\)ight-like separated. Such events are causally related, and all observers will agree that they could be linked by a light signal.s) 2 = 0，被认为是零(或零)时空分离的事件，(\(\Delta l\)ight-like 分离)。此类事件是因果相关的，所有观察者都会同意它们可以通过光信号联系起来。

Events with a negative spacetime separation, (\(\Delta s\)) 2 < 0, are said to be space-like separated. Such events are not causally related, and there will exist a frame in which the two events happen at the same time but at different places.具有负时空分离 (\(\Delta s\)) 2 < 0 的事件被称为类空间分离。此类事件没有因果关系，并且会存在一个框架，其中两个事件同时发生但在不同地点。

These different kinds of spacetime separation correspond to different regions of spacetime defined by the lightcone of an event. Figure这些不同类型的时空分离对应于由事件的光锥定义的不同时空区域。数字

1.19 shows the lightcone of1.19 显示了光锥

event 0. All the events that have a time-like separation from event 0 are within the future or past lightcone of event 0; all the events that are light-like separated from event 0 are on its lightcone; and all the events that are space-like separated from event 0 are outside its lightcone. This emphasizes the role that lightcones play in revealing the causal structure of Minkowski spacetime.事件 0。与事件 0 具有类似时间间隔的所有事件都在事件 0 的未来或过去光锥内；所有与事件 0 分离的类似光的事件都在其光锥上；所有与事件 0 类似空间分离的事件都在其光锥之外。这强调了光锥在揭示闵可夫斯基时空因果结构中所发挥的作用。

Another reason why spacetime separation is important relates to proper time. You will recall that in the earlier discussion of time dilation, it was said that the proper time between two events was the time separating those events as measured in a frame where the events happen at the same position. In such a frame, the spacetime separation of the events is (\(\Delta s\)) 2 = \(c^2\) (\(\Delta t\)) 2 = \(c^2\) (\(\Delta \tau\)) 2. However, since the spacetime separation of events is an invariant quantity, we can use it to determine the proper time between two time-like separated events, irrespective of the frame in which the events are described. For two time-like separated events with positive spacetime separation (\(\Delta s\)) 2, the proper time \(\Delta \tau\) between those two events is given by the following.时空分离很重要的另一个原因与原时有关。您会记得，在前面关于时间膨胀的讨论中，据说两个事件之间的固有时间是在事件发生在同一位置的帧中测量的分隔这些事件的时间。在这样的坐标系中，事件的时空间隔为 (\(\Delta s\)) 2 = \(c^2\) (\(\Delta t\)) 2 = \(c^2\) (\(\Delta \tau\)) 2。然而，由于事件的时空分离是一个不变的量，我们可以用它来确定两个类似时间的分离事件之间的固有时间，而不管事件描述的框架如何。对于两个具有正时空间隔 (\(\Delta s\)) 2 的类时分离事件，这两个事件之间的本征时间 \(\Delta \tau\) 由下式给出。

Proper time related to spacetime separation与时空分离相关的本征时间

The relation between proper time and the invariant spacetime separation is extremely useful in special relativity. The reason for this relates to the length of a particle’s pathway through four-dimensional Minkowski spacetime. Such a pathway, with all its twists and turns, records the whole history of the particle and is sometimes called its world-line. (One well-known relativist called his autobiography My worldline.) By adding together the spacetime separations between successive events along a particle’s world-line, and dividing the sum by \(c^2\), we can determine the total time that has passed according to a clock carried by the particle. This simple principle will be used to help to explain a troublesome relativistic effect in the next subsection.原时与不变时空分离之间的关系在狭义相对论中极其有用。其原因与粒子穿过四维闵可夫斯基时空的路径长度有关。这样一条充满曲折的路径记录了粒子的整个历史，有时被称为它的世界线。(一位著名的相对论者将他的自传称为“我的世界线”。)通过将粒子世界线上连续事件之间的时空间隔相加，并将总和除以 \(c^2\)，我们可以根据粒子携带的时钟确定已经过去的总时间。这个简单的原理将用于帮助解释下一小节中麻烦的相对论效应。

In this book, a positive sign will always be associated with the square of the time interval in the spacetime separation, and a negative sign with the spatial intervals. This choice of sign is just a convention, and the opposite set of signs could have been used. The convention used here ensures that the spacetime separation of events on the world-line of an object moving slower than light is positive. Nonetheless, you will find that many authors adopt the opposite convention, so when consulting other works, always pay attention to the sign convention that they are using.在本书中，正号总是与时空间隔中的时间间隔的平方相关联，而负号则与空间间隔相关联。这种符号选择只是一种惯例，也可以使用相反的符号集。这里使用的约定确保运动速度慢于光的物体的世界线上事件的时空分离是正的。尽管如此，你会发现许多作者采用相反的约定，因此在查阅其他作品时，请始终注意他们使用的符号约定。

Exercise 1.10 Given two time-like separated events, show that the proper time练习 1.10 给定两个类似时间的分离事件，证明适当的时间

between those events is the least amount of time that any observer will measure between them.这些事件之间的间隔是任何观察者在它们之间测量的最短时间。

1.4.3 The twin effect1.4.3 孪生效应

We end this chapter with a discussion of a well-known relativistic effect, the twin effect. This caused a great deal of controversy early in the theory’s history. It is usually presented as a thought experiment concerning the phenomenon of time dilation. The thought experiment involves two twins, Astra and Terra. The twins are identical in every way, except that Astra likes to travel around very fast in her spaceship, while Terra prefers to stay at home on Earth.我们以对著名的相对论效应——孪生效应的讨论来结束本章。这在该理论的历史早期引起了很大的争议。它通常表现为关于时间膨胀现象的思想实验。这个思想实验涉及两个双胞胎，阿斯特拉和泰拉。这对双胞胎在各方面都完全相同，只是阿斯特拉喜欢在她的宇宙飞船中快速旅行，而泰拉更喜欢呆在地球上的家里。

As was demonstrated earlier in this chapter, fast-moving objects are subject to observable time dilation effects. This indicates that if Astra jets off in some fixed direction at close to the speed of light, then, as measured by Terra, she will age more slowly because ‘moving clocks run slow’. This is fine — it is just what relativity theory predicts, and agrees with the observed behaviour of high-speed particles. But now suppose that Astra somehow manages to turn around and return to Earth at equally high speed. It seems clear that Terra will again observe that Astra’s clock will run slow and will therefore not be surprised to find that on her return, Astra has aged less than her stay-at-home twin Terra.正如本章前面所演示的，快速移动的物体会受到可观察到的时间膨胀效应的影响。这表明，如果阿斯特拉以接近光速的速度向某个固定方向喷射，那么根据泰拉的测量，她的衰老速度会更慢，因为“移动的时钟走得很慢”。这很好——这正是相对论的预测，并且与观察到的高速粒子的行为一致。但现在假设阿斯特拉以某种方式设法转身并以同样高的速度返回地球。很明显，泰拉将再次观察到阿斯特拉的时钟会变慢，因此不会惊讶地发现，在她回来时，阿斯特拉的年龄比她待在家里的双胞胎泰拉要小。

The supposed problem arises when this process is examined from Astra’s point of view. Would it not be the case, some argued, that Astra would observe the same events apart from a reversal of velocities, so that Terra would be the travelling twin and it would be Terra’s clock that would be running slow during both parts of the journey? Consequently, shouldn’t Astra expect Terra to be the younger when they were reunited? Clearly, it’s not possible for each twin to be younger than the other when they meet at the same place, so if the arguments are equally sound, it was said, there must be something wrong with special relativity.当从 Astra 的角度审视这个过程时，就会出现所谓的问题。一些人认为，除了速度逆转之外，阿斯特拉是否会观察到相同的事件，因此泰拉将是旅行的双胞胎，而泰拉的时钟在旅程的两个部分都会变慢？因此，阿斯特拉不应该期望泰拉在他们重聚时变得更年轻吗？显然，当双胞胎在同一个地方相遇时，他们不可能比另一个年轻，所以如果论证同样合理，有人说，狭义相对论一定有问题。

In fact, the arguments are not equally sound. The basic problem is that the presumed symmetry between Terra’s view and Astra’s view is illusory. It is Astra who would be the younger at the reunion, as will now be explained with the aid of a spacetime diagram and a proper use of spacetime separations in Minkowski space.事实上，这些论点并不同样合理。基本问题是，泰拉的视图和阿斯特拉的视图之间假定的对称性是虚幻的。阿斯特拉在重聚时将是更年轻的，现在将借助时空图和正确使用闵可夫斯基空间中的时空分离进行解释。

The first point to make clear is that although velocity is a purely relative quantity, acceleration is not. According to the first postulate of special relativity, the laws of physics do not distinguish one inertial frame from another, so a traveller in a closed box cannot determine his or her speed by performing a physics experiment. However, such a traveller would certainly be able to feel the effect of any acceleration, as we all know from everyday experience. In order to leave the Solar System, jet around the galaxy and return, Astra must have undergone a change in velocity, and that would involve a detectable acceleration. To a first approximation, Terra does not accelerate (her velocity changes due to the rotation and revolution of the Earth are very small compared with Astra’s accelerations). A single inertial frame of reference is sufficient to represent Terra’s view of events, but no single inertial frame can adequately represent Astra’s view. There is no symmetry between these two observers; only Terra is an (approximately) inertial observer.首先要明确的是，虽然速度是一个纯粹的相对量，但加速度却不是。根据狭义相对论第一假设，物理定律无法区分一个惯性系和另一个惯性系，因此封闭盒子中的旅行者无法通过进行物理实验来确定他或她的速度。然而，正如我们从日常经验中知道的那样，这样的旅行者肯定能够感受到任何加速度的影响。为了离开太阳系，绕银河系喷射并返回，阿斯特拉必须经历速度变化，这将涉及可检测到的加速度。初步估计，Terra 并没有加速(与 Astra 的加速度相比，她因地球自转和公转而产生的速度变化非常小)。单一惯性参考系足以代表泰拉对事件的看法，但没有单一惯性系能够充分代表阿斯特拉的观点。这两个观察者之间不存在对称性；只有 Terra 是(近似)惯性观察者。

In order to be clear about what’s going on and to avoid the use of non-inertial frames, it is convenient to use three inertial frames when discussing the twin effect. The first is Terra’s frame, which we can treat as fixed on a non-rotating, non-revolving Earth. The second, which we shall call Astra’s frame, moves at a high but constant speed V relative to Terra’s frame. You can think of this as the frame of Astra’s spaceship, and you can think of Astra as simply jumping aboard her passing ship at the departure, event 0, when she leaves Terra to begin the outward leg of her journey. The third inertial frame, called Stella’s frame, belongs to another space traveller who happens to be approaching Earth at speed V along the same line that Astra leaves along. At some point, Stella’s ship will pass Astra’s, and at that point we can imagine that Astra jumps from her ship to Stella’s ship to make the return leg of her journey. Of course, this is unrealistic since the ‘jump’ would kill Astra, so you may prefer to imagine that Astra is actually a conscious robot or even that she can somehow ‘teleport’ from one ship to another. In any case, the important point is that the transfer is abrupt and has no effect on Astra’s age.为了清楚地了解发生了什么并避免使用非惯性系，在讨论孪生效应时可以方便地使用三个惯性系。第一个是 Terra 的框架，我们可以将其视为固定在不自转、不公转的地球上。第二个，我们称之为阿斯特拉框架，相对于泰拉框架以高但恒定的速度 V 移动。您可以将其视为阿斯特拉宇宙飞船的框架，并且您可以将阿斯特拉视为简单地在事件 0 出发时跳上她经过的飞船，当她离开泰拉开始她的旅程的向外旅程时。第三个惯性系，称为斯特拉系，属于另一位太空旅行者，他恰好沿着阿斯特拉离开的同一条线以速度 V 接近地球。在某个时刻，斯特拉的船将经过阿斯特拉的船，那时我们可以想象阿斯特拉从她的船上跳到斯特拉的船上，开始她的旅程的回程。当然，这是不现实的，因为“跳跃”会杀死阿斯特拉，所以你可能更愿意想象阿斯特拉实际上是一个有意识的机器人，甚至她可以以某种方式从一艘船“传送”到另一艘船。无论如何，重要的一点是，转会是突然的，对阿斯特拉的年龄没有影响。

The event at which Astra makes the transfer to Stella’s ship we shall call event 1, and the event at which Astra and Terra are eventually reunited we shall call event 2. Astra’s quick transfer from one ship to the other allows us to discuss the essential features of the twin effect without getting bogged down in details about the nature of the acceleration that Astra experiences. It is vital that Astra is accelerated, but exactly how that happens is unimportant. Note that we may treat each of these frames as being in standard configuration with either of the others. We can set up the frames in such a way that the origins of Terra’s frame and Astra’s frame coincide at event 0, the origins of Astra’s frame and Stella’s frame coincide at event 1, and the origins of Stella’s frame and Terra’s frame coincide at event 2.阿斯特拉转移到斯特拉飞船的事件我们称为事件 1，而阿斯特拉和泰拉最终团聚的事件我们称为事件 2。阿斯特拉从一艘船快速转移到另一艘船使我们能够讨论双生效应的基本特征，而不必陷入有关阿斯特拉所经历的加速性质的细节。Astra 的加速至关重要，但具体如何发生并不重要。请注意，我们可以将这些框架中的每一个视为与其他框架中的任何一个的标准配置。我们可以这样设置框架，使得 Terra 框架和 Astra 框架的原点在事件 0 处重合，Astra 框架和 Stella 框架的原点在事件 1 处重合，Stella 框架和 Terra 框架的原点在事件 2 处重合。

Figure 1.20 is a spacetime diagram for Terra’s frame, showing all these events and making clear the coordinates that Terra assigns to them.图1.20是Terra框架的时空图，显示了所有这些事件并明确了Terra分配给它们的坐标。

It is clear from the figure that the proper time between departure and reunion (both of which happen at Terra’s location) is T. A little calculation using the relation (\(\Delta \tau\)) 2 = (\(\Delta s\)) 2/\(c^2\) makes it equally clear that the proper time between event 0 and event 1 is given by从图中可以清楚地看出，出发和团聚(两者都发生在 Terra 所在位置)之间的正确时间为 T。使用关系式 (\(\Delta \tau\)) 2 = (\(\Delta s\)) 2/\(c^2\) 进行一点计算，同样可以清楚地看出，事件 0 和事件 1 之间的正确时间由下式给出：

So所以

Although we have arrived at this result using the coordinates assigned by Terra, it is important to note that proper time is an invariant, so all inertial observers will agree on the proper time between two events no matter how it is calculated.虽然我们是使用 Terra 指定的坐标得出这个结果，但值得注意的是，原时是一个不变量，因此所有惯性观察者都会就两个事件之间的原时达成一致，无论它是如何计算的。

A similar calculation for the proper time separating event对固有时间分隔事件的类似计算

So the total proper time that elapses along the world-line followed by Astra is \(\Delta \tau\) 0, 1 + \(\Delta \tau\) 1, 2 = T/γ. As expected, this shows that Astra will be the younger twin at the time of the reunion.因此，阿斯特拉沿世界线经过的总固有时间为 \(\Delta \tau\) 0, 1 + \(\Delta \tau\) 1, 2 = T/γ。不出所料，这表明阿斯特拉将成为重聚时的双胞胎中的弟弟。

How is it possible for Terra and Astra to disagree about the proper time between events 0 and 2? The answer to this question is that when the whole trip is considered, Astra is not an inertial observer; she undergoes an acceleration that Terra does not.Terra 和 Astra 怎么可能在事件 0 和 2 之间的正确时间上存在分歧？这个问题的答案是，当考虑整个行程时，Astra并不是一个惯性观察者；而是一个惯性观察者。她经历了泰拉没有经历的加速。

The analysis that we have just completed is really sufficient to settle any questions about the twin effect. However, it is still instructive to examine the same events from Astra’s frame (which she leaves at event 1). The spacetime diagram for Astra’s frame is shown in Figure 1.21. The coordinates of the events have been worked out from those given in Terra’s frame using the Lorentz transformations.我们刚刚完成的分析确实足以解决有关孪生效应的任何问题。然而，从阿斯特拉的框架中检查相同的事件(她在事件 1 中留下)仍然具有启发性。 Astra 框架的时空图如图 1.21 所示。事件的坐标是使用洛伦兹变换根据 Terra 框架中给出的坐标计算出来的。

• ● Confirm the coordinate assignments shown in Figure● 确认如图所示的坐标分配

1.21.1.21。

❍ In Terra’s frame, event 0 is at (ct, x) = (0, 0), event 1 at (cT/2, V T/2), and event 2 at (cT, 0). Treating Terra’s frame as frame S and Astra’s frame as \(S'\), and using the Lorentz transformations \(t'\) = \(\gamma(t - V x/\(c^2\))\) and \(x'\) = \(\gamma(x - V t)\), it follows immediately that in Astra’s frame, event 0 is at (\(ct'\), \(x'\)) = (0, 0), event 1 is at (\(ct'\), \(x'\)) = (cT/2 γ, 0) (remember that \(\gamma(V)\) = 1/1 − \(V^{2}\)/\(c^2\)), and event 2 is at (\(ct'\), \(x'\)) = (cγT, − γV T).❍ 在 Terra 的框架中，事件 0 位于 (ct, x) = (0, 0)，事件 1 位于 (cT/2, V T/2)，事件 2 位于 (cT, 0)。将 Terra 的坐标系视为坐标系 S，将 Astra 的坐标系视为 \(S'\)，并使用洛伦兹变换 \(t'\) = \(\gamma(t - V x/\(c^2\))\) 和 \(x'\) = \(\gamma(x - V t)\)，紧接着可以得出在 Astra 的坐标系中，事件 0 位于 (\(ct'\), \(x'\)) = (0, 0)，事件 1 位于 (\(ct'\), \(x'\)) = (cT/2 γ, 0)(记住 \(\gamma(V)\) = 1/1 − \(V^{2}\)/\(c^2\))，事件 2 位于 (\(ct'\), \(x'\)) = (cγT, − γV T)。

Note that again there is a kink in Astra’s world-line due to the acceleration that she undergoes. There is no such kink in Terra’s world-line since she is an inertial observer. Once again we can work out the proper time that Astra experiences while passing between the three events: this represents the time that would have elapsed according to a clock that Astra carries between each of the events. The, 1 = T/2 γ, since those proper time between event 0 and event 1 is simply \(\Delta \tau\) 0 events happen at the same place in Astra’s frame. The proper time between event 1 and event 2 is given by请注意，由于阿斯特拉经历的加速，她的世界线再次出现了扭结。泰拉的世界线中不存在这样的扭结，因为她是惯性观察者。我们可以再次计算出阿斯特拉在这三个事件之间经过时所经历的正确时间：这代表根据阿斯特拉在每个事件之间携带的时钟所经过的时间。1 = T/2 γ，因为事件 0 和事件 1 之间的正确时间就是 \(\Delta \tau\) 0 个事件发生在 Astra 框架中的同一位置。事件 1 和事件 2 之间的正确时间由下式给出

Since γ 2 (1 − \(V^{2}\)/\(c^2\)) = 1, the above expression simplifies to give由于 γ 2 (1 − \(V^{2}\)/\(c^2\)) = 1，上述表达式简化为

So once again the theory predicts that the time for the round trip recorded by Astra is \(\Delta \tau +\)\(\Delta \tau = T\)/γ.因此理论再次预测Astra记录的往返时间为Δτ+\(\Delta\tau=T\)/γ。

There is one other point to notice using Astra’s frame. Time dilation tells us that, as measured in Astra’s frame, Terra’s clock will be running slow. From Astra’s frame, a 1 -second tick of Terra’s clock will be observed to last γ seconds. But in Astra’s frame, it is also the case that the time of the reunion is γT, which is greater than the time of the reunion as observed in Terra’s frame. According to an observer who uses Astra’s frame, this longer journey time compensates for the slower ticking of Terra’s clock, with the result that such an observer will fully expect Terra to have aged by T while Astra herself has aged by only T/γ. Using the coordinates of event 0 and event 2 in Astra’s frame, it is easy to confirm that the proper time between them is T, which is another way of stating the same result.使用Astra 的框架还有一点需要注意。时间膨胀告诉我们，按照阿斯特拉的坐标系测量，泰拉的时钟将会走慢。从 Astra 的坐标系中，可以观察到 Terra 时钟的 1 秒滴答持续 γ 秒。但在阿斯特拉的框架中，团聚的时间也是γT，大于在泰拉的框架中观察到的团聚的时间。根据一位使用阿斯特拉框架的观察者的说法，更长的旅程时间补偿了泰拉时钟较慢的滴答声，结果是这样的观察者完全预计泰拉已经老化了 T，而阿斯特拉本身只老化了 T/γ。使用Astra框架中事件0和事件2的坐标，很容易确认它们之间的正确时间是T，这是陈述相同结果的另一种方式。

Exercise 1.11 Using the velocity transformation, show练习1.11 使用速度变换，显示

that Astra observes the \(V^{2}\)/\(c^2\)). speed of approach of Stella’s spaceship to be 2 V/(1 +Astra 观察到 \(V^{2}\)/\(c^2\))。斯特拉飞船的接近速度为 2 V/(1 +

Exercise 1.12 Suppose that Terra sends regular time练习1.12 假设Terra发送固定时间

signals towards Astra and Stella at one-second intervals. Write down expressions for the frequency at which Astra receives the signals on the outward and return legs of her journey.每隔一秒向阿斯特拉和斯特拉发出信号。写下阿斯特拉在去程和回程旅程中接收信号的频率表达式。

Summary of Chapter 1第 1 章总结

1. Basic terms in the vocabulary of relativity include:1. 相对论词汇中的基本术语包括：

event, frame of reference, inertial frame and observer.事件、参考系、惯性系和观察者。

2. A theory of relativity concerns the relationships between2. 相对论关注的是之间的关系

observations made by observers in a specified state of relative motion. Special relativity is essentially restricted to inertial observers in uniform relative motion.观察者在特定的相对运动状态下所做的观察。狭义相对论本质上仅限于匀速相对运动的惯性观察者。

3. Einstein based special relativity on two postulates:3. 爱因斯坦将狭义相对论建立在两个假设之上：

the principle of relativity (that the laws of physics can be written in the same form in all inertial frames) and the principle of the constancy of the speed of light (that all inertial observers agree that light travels through empty space with the same fixed speed, c, in all directions).相对论原理(物理定律可以在所有惯性系中以相同的形式写出)和光速恒定原理(所有惯性观察者都同意光以相同的固定速度 c 在各个方向穿过真空)。

4. Given two inertial frames S and \(S'\) in standard configuration,4. 给定标准配置的两个惯性系S和\(S'\)，

the coordinates of an event observed in frame S are related to the coordinates of the same event observed in frame \(S'\) by the Lorentz transformations通过洛伦兹变换，在坐标系 S 中观察到的事件的坐标与在坐标系 \(S'\) 中观察到的同一事件的坐标相关

where the Lorentz factor is其中洛伦兹因子是

These transformations may also be represented by matrices,这些变换也可以用矩阵表示，

or as a set of summations或作为一组求和

5. The inverse Lorentz transformations may be written5.洛伦兹逆变换可以写成

as作为

6. Similar equations describe the transformation of intervals, \(\Delta t\), \(\Delta x\), etc.,6.类似的方程描述了区间、\(\Delta t\)、\(\Delta x\)等的变换，

between the two frames.两个框架之间。

7. The consequences of special relativity, deduced by considering the7. 狭义相对论的后果，通过考虑

transformation of events and intervals, include the following.事件和间隔的转换，包括以下内容。

(a) Time dilation:(a) 时间膨胀：

(b) Length contraction:(b) 长度收缩：

(c) The relativity of simultaneity.(c)同时性的相对性。

(c + V)/(c − V) (for an approaching source),(c + V)/(c − V)(对于接近源)，

(c − V)/(c + V) (for a receding source).(c − V)/(c + V)(对于后退源)。

(e) The velocity transformation:(e) 速度变换：

8. Four-dimensional Minkowski spacetime contains all possible events.8. 四维闵可夫斯基时空包含所有可能的事件。

9. Spacetime diagrams showing events as observed by a particular observer are9. 显示特定观察者观察到的事件的时空图是

a valuable tool that can provide pictorial insights into relativistic effects and the structure of Minkowski spacetime.这是一个有价值的工具，可以为相对论效应和闵可夫斯基时空结构提供图形化的见解。

10. Lightcones are particularly useful for understanding causal relationships10.光锥对于理解因果关系特别有用

between events in Minkowski spacetime.闵可夫斯基时空中的事件之间。

11. The invariant spacetime separation between two events has the form11. 两个事件之间的不变时空间隔具有以下形式

and may be positive (time-like), zero (light-like) or negative (space-like).并且可以是正值(类时间)、零(类光)或负值(类空间)。

12. The spacetime separation may be conveniently written as12. 时空分离可以方便地写为

where the \(\eta\) \(\mu\)\(\nu\) are the components of the Minkowski metric其中 \(\eta\) \(\mu\)\(\nu\) 是闵可夫斯基度规的组成部分

13. The proper time \(\Delta \tau\) between two time-like separated13. 两个类似时间间隔之间的正确时间\(\Delta \tau\)

events is given by事件由下式给出

This is the time that would be recorded on a clock that moves uniformly between the two events.这是记录在两个事件之间均匀移动的时钟上的时间。

14. The proper time between two events is an invariant14. 两个事件之间的固有时间是一个不变量

under Lorentz transformations.在洛伦兹变换下。

Chapter 2 Special relativity and physical laws第二章狭义相对论和物理定律

Introduction介绍

Physical laws are usually expressed mathematically, as equations. They are used by physicists to summarize their findings regarding the basic principles that govern the Universe. From the late 1600s to the mid-1800s, Newton’s laws and the Galilean relativity underpinning them were believed to be the fundamental rules. The precision engineering of the nineteenth century and the clock-like regularity of the Solar System all seemed to be consistent with this view.物理定律通常以数学形式表达为方程。物理学家用它们来总结他们关于支配宇宙的基本原理的发现。从 1600 年代末到 1800 年代中期，牛顿定律和支撑牛顿定律的伽利略相对论被认为是基本规则。十九世纪的精密工程和太阳系的时钟般的规律性似乎都与这一观点相一致。

However, as we have already seen, the investigation of electricity and magnetism, and their unification with optics through Maxwell’s demonstration of the electromagnetic nature of light, exposed a new conflict between fundamental laws. Lorentz and others worked on this problem but it was Einstein who recognized most clearly and completely that its essence was in a conflict between the invariance of the speed of light in a vacuum and the requirement of Galilean relativity that observers in relative motion should disagree about the speed of light. Einstein’s response was to extend the principle of relativity from the laws of mechanics to all the laws of physics, including specifically the constancy of the speed of light, and to accept as a consequence the need for a new theory of relativity based on the Lorentz transformations rather than Galilean transformations.然而，正如我们已经看到的，对电和磁的研究，以及通过麦克斯韦对光的电磁性质的论证，它们与光学的统一，暴露了基本定律之间的新冲突。洛伦兹和其他人致力于研究这个问题，但爱因斯坦最清楚、最彻底地认识到，其本质是真空中光速不变性与伽利略相对论的要求(即相对运动的观察者对光速的看法应该不一致)之间的冲突。爱因斯坦的回应是将相对论原理从力学定律扩展到所有物理定律，特别是光速恒定性，并因此接受基于洛伦兹变换而不是伽利略变换的新相对论的需要。

The requirement of special relativity, that physical laws should take the same form in all inertial frames, is highly restrictive. It prevents many candidates from being accepted as genuine physical laws. The principle of relativity cannot tell us which proposed laws are correct — that must be done by experiment — but it can show up those that are not acceptable in principle. When the coordinates used in two different frames are related by the Lorentz transformations, it is soon seen that the laws of Newtonian mechanics do not take the same form in all inertial frames. So an immediate implication of special relativity is the need for an extensive rewriting of the laws of mechanics. The new laws must be consistent with the well-established successes of Newtonian mechanics, but they must also show the invariance under Lorentz transformations required by the principle of relativity. In this chapter we shall consider those new laws of mechanics and see the extent to which Newtonian concepts had to be modified or replaced. We shall then go on to see what special relativity has to say about the laws of electricity and magnetism.狭义相对论的要求，即物理定律在所有惯性系中应采取相同的形式，是高度限制性的。它阻止了许多候选人被接受为真正的物理定律。相对性原理不能告诉我们哪些提出的定律是正确的——这必须通过实验来完成——但它可以显示那些原则上不可接受的定律。当两个不同框架中使用的坐标通过洛伦兹变换相关联时，很快就会发现牛顿力学定律在所有惯性系中并不采用相同的形式。因此，狭义相对论的直接含义是需要对力学定律进行广泛的重写。新定律必须与牛顿力学的既定成功相一致，但它们也必须显示相对论原理所需的洛伦兹变换下的不变性。在本章中，我们将考虑这些新的力学定律，并了解牛顿概念必须修改或替换的程度。然后我们将继续看看狭义相对论对电和磁定律的解释。

The discussion of physical laws in this chapter will introduce some important mathematical entities that may be new to you. These entities, called four-vectors and four-tensors, are of particular relevance to special relativity but they set the scene for the introduction of more general tensors in the later chapters that deal with general relativity. Pay special attention to these four-vectors and four-tensors. Appreciating their role in the formulation of physical laws that are consistent with special relativity is at least as important as learning about any specific feature of those laws.本章中对物理定律的讨论将介绍一些对您来说可能是新的重要数学实体。这些实体被称为四向量和四张量，与狭义相对论特别相关，但它们为后面讨论广义相对论的章节中引入更一般的张量奠定了基础。特别注意这四矢量量量和四个张量。理解它们在制定符合狭义相对论的物理定律中的作用至少与了解这些定律的任何具体特征一样重要。

2.1 Invariants and physical laws2.1 不变量和物理定律

2.1.1 The invariance of physical quantities2.1.1 物理量的不变性

Central to the formulation of physical laws in special relativity are invariant quantities or invariants for short. You have already met a number of these invariants: most obviously, the speed of light in a vacuum, but also the spacetime separation between events (\(\Delta s\)) 2 = (c \(\Delta t\)) 2 − (\(\Delta x\)) 2 − (\(\Delta y\)) 2 − (\(\Delta z\)) 2 and, in the case of time-like separated events, the closely related proper time interval \(\Delta \tau\) given by (\(\Delta \tau\)) 2 = (\(\Delta s\)) 2/\(c^2\).狭义相对论中物理定律表述的核心是不变量或简称不变量。您已经遇到了许多这样的不变量：最明显的是真空中的光速，还有事件之间的时空间隔 (\(\Delta s\)) 2 = (c \(\Delta t\)) 2 − (\(\Delta x\)) 2 − (\(\Delta y\)) 2 − (\(\Delta z\)) 2 以及在类似时间的分离事件的情况下，紧密相关的本征时间间隔 \(\Delta \tau\) 由 (\(\Delta \tau\)) 2 = (\(\Delta s\)) 给出2/\(c^2\)。

An alternative way of defining the proper time between two events is as the time between those events measured in a frame in which the two events occur at the same spatial position. (The fact that the events are time-like separated guarantees that such a frame exists.) This is an interesting definition since it uses a measurement made in one inertial frame to define a quantity that can then be used in all inertial frames. This approach to defining invariants is quite common. For example, we can and will say that the electric charge of a particle is the charge that it has when measured in the frame in which the particle is at rest. The charge is then defined in an invariant way, even though the prescription for measuring it involves a particular frame — the rest frame of the particle.定义两个事件之间的固有时间的另一种方法是在两个事件发生在同一空间位置的帧中测量的这些事件之间的时间。 (事件是类似时间的分离这一事实保证了这样一个框架的存在。)这是一个有趣的定义，因为它使用在一个惯性系中进行的测量来定义一个可以在所有惯性系中使用的量。这种定义不变量的方法很常见。例如，我们可以并且会说，粒子的电荷是在粒子静止的框架中测量时所具有的电荷。然后以不变的方式定义电荷，即使测量它的规定涉及特定的框架——粒子的其余框架。

A similar approach can be used to provide an invariant value for the mass of a particle. In keeping with the common practice of particle physicists, we shall say that the mass of a particle is the mass that would be measured in a frame in which the particle is at rest. This provides a mass that all observers can agree about. Some authors refer to this quantity as the rest mass of the particle, but we have no need to do so here since this is the only sense in which we shall use the term mass in this chapter. Incidentally, if you have studied relativity before, you may have encountered the idea of a relativistic mass that increases with the speed of the particle. This is based on a quite different definition of mass that will not be used in this book. The masses that we shall refer to are defined invariantly and will never depend on speed. Other invariant quantities — some of them very important — will be introduced later, but for the moment here is a summary of what we have said about invariants.可以使用类似的方法来提供粒子质量的不变值。根据粒子物理学家的惯例，我们应该说粒子的质量是在粒子静止的框架中测量的质量。这提供了所有观察者都同意的质量。一些作者将这个量称为粒子的剩余质量，但我们在这里没有必要这样做，因为这是我们在本章中使用术语“质量”的唯一含义。顺便说一句，如果您以前研究过相对论，您可能遇到过相对论质量随粒子速度增加而增加的想法。这是基于一个完全不同的质量定义，本书中不会使用该定义。我们所提到的质量是恒定定义的，并且永远不会依赖于速度。其他不变量(其中一些非常重要)将在稍后介绍，但目前这里是我们关于不变量所说的内容的总结。

Invariants不变量

An invariant is a quantity that has the same value in all inertial frames.不变量是在所有惯性系中具有相同值的量。

Invariant quantities include:不变量包括：

• • the speed of light in a vacuum, c• 真空中的光速，c
• • the spacetime separation (\(\Delta s\)) 2 = (c \(\Delta t\)) 2 − (Δ• 时空间隔 (\(\Delta s\)) 2 = (c \(\Delta t\)) 2 − (Δ
• • the proper time (\(\Delta \tau\)) 2 = (\(\Delta s\)) 2/\(c^2\) between time-like• 类时间之间的固有时间 (\(\Delta \tau\)) 2 = (\(\Delta s\)) 2/\(c^2\)

separated events分开的事件

• • the charge of a particle, q• 粒子的电荷，q
• • the mass of a particle, m.• 粒子的质量，m。

2.1.2 The invariance of physical laws2.1.2 物理定律的不变性

The requirement that the laws of physics should take the same form in all inertial frames involves extending the idea of invariance from invariance of a quantity to invariance of the form of an equation. The easiest way to appreciate this is by means of an example so, although it is mainly of historical interest, we shall now demonstrate the form invariance of Newton’s laws of motion under the Galilean coordinate transformation.物理定律在所有惯性系中应采用相同形式的要求涉及将不变性的概念从量的不变性扩展到方程形式的不变性。理解这一点的最简单方法是通过一个例子，尽管它主要具有历史意义，但我们现在将证明牛顿运动定律在伽利略坐标变换下的形式不变性。

Newton’s laws of motion can be stated as follows.牛顿运动定律可以表述如下。

1. A body maintains a constant velocity unless acted upon by an unbalanced1. 除非受到不平衡力的作用，否则物体保持恒定速度

external force.外力。

2. A body acted upon by an unbalanced force accelerates in the direction of2. 物体受到不平衡力的作用时，沿下列方向加速：

that force at a rate that is proportional to the force and inversely proportional to the body’s mass.该力的速率与力成正比，与身体质量成反比。

3. When body A exerts a force on body B, body B exerts a force on body A3.当A体对B体施加力时，B体也对A体施加力

that has the same magnitude but acts in the opposite direction. (This law is often stated as: to every action there is an equal and opposite reaction.)其大小相同，但作用方向相反。 (这条定律通常被表述为：每一个动作都会有一个相等且相反的反应。)

The first law is really telling us that in order to use the other laws, we should make sure that we observe from an inertial frame of reference. So we don’t need to give any further thought to this law as long as we restrict ourselves to inertial frames. The third law also presents no difficulty. Provided that oppositely directed forces of equal magnitude transform in the same way in Galilean relativity, there will not be any problem about agreeing on the form of the third law. This is true even for forces that act at a distance, such as the gravitational force acting on a person due to the Earth and the reaction to that force that acts simultaneously at the Earth’s centre of mass.第一定律实际上告诉我们，为了使用其他定律，我们应该确保我们从惯性参考系进行观察。因此，只要我们将自己限制在惯性系中，就无需进一步考虑该定律。第三定律也没有什么困难。假设伽利略相对论中方向相反、大小相等的力以同样的方式变换，那么就第三定律的形式达成一致就不会有任何问题。即使对于远距离作用的力也是如此，例如由于地球而作用在人身上的引力以及同时作用在地球质心的该力的反作用力。

The real challenge comes with Newton’s second law of motion. Let’s start by writing the second law as an equation真正的挑战来自牛顿第二运动定律。让我们首先将第二定律写成方程

where f is the applied force, m is the mass of the body, and a is its acceleration. If we take this to be the form of Newton’s second law in some particular inertial frame S with Cartesian coordinate axes x, y and z, we can relate the acceleration to the coordinates of the body in frame S by writing其中 f 是施加的力，m 是物体的质量，a 是物体的加速度。如果我们将其视为具有笛卡尔坐标轴 x、y 和 z 的特定惯性系 S 中牛顿第二定律的形式，我们可以将加速度与 S 系中物体的坐标联系起来：

Now suppose that we have a second frame of reference \(S'\) in standard configuration with S, so that the coordinates in the two frames are related by the Galilean transformations现在假设我们有第二个参考系 \(S'\)，其标准配置为 S，因此两个参考系中的坐标通过伽利略变换相关

Differentiating the expressions for the position coordinates twice with respect to \(t'\), and noting that this is equivalent to differentiating with respect to t (since \(t'\) = t), we see that对位置坐标的表达式关于 \(t'\) 进行两次微分，并注意这相当于对 t 进行微分(因为 \(t'\) = t)，我们看到

Mass is certainly an invariant in Galilean relativity, so, under a Galilean transformation from frame S to frame \(S'\), the right-hand side of Equation 2.2 becomes质量在伽利略相对论中当然是一个不变量，因此，在从 S 系到 \(S'\) 系的伽利略变换下，方​​程 2.2 的右侧变为

where the quantity \(a'\) has been introduced to emphasize the form-invariance of the right-hand side of Newton’s second law under a Galilean transformation. This is a promising start, but what about the left-hand side: how does the force f transform under a Galilean transformation? To answer that question, we need to know how the force depends on the coordinates.其中引入量 \(a'\) 是为了强调伽利略变换下牛顿第二定律右侧的形式不变性。这是一个充满希望的开始，但左边呢：力 f 在伽利略变换下如何变换？要回答这个问题，我们需要知道力如何取决于坐标。

For the sake of definiteness, let’s consider the case in which a body of mass m at position r = (x, y, z) is acted upon by a gravitational force due to a body of mass M at position R = (X, Y, Z) (see Figure 2.1). According to Newton’s law of universal gravitation, in frame S the force will be为了明确起见，我们考虑这样一种情况，即位置 r = (x, y, z) 处的质量体 m 受到位置 R = (X, Y, Z) 处质量体 M 的引力作用(见图 2.1)。根据牛顿万有引力定律，在 S 系中，力为

where G is Newton’s gravitational constant (an invariant constant with the value其中 G 是牛顿万有引力常数(一个不变常数，其值为

\(6.673\times10^{-11}\) N \(m^{2}\) kg − 2), the distance d is the magnitude\(6.673\times10^{-11}\) N \(m^{2}\) kg − 2)，距离 d 为震级

of the displacement vector d = r − R from the body of mass M to the body of mass m, and d E is a unit vector in the direction of d.从质量体 M 到质量体 m 的位移矢量 d = r − R，d E 是 d 方向的单位矢量。

\(S'\), the position vectors and \(R'\) ≡ (\(X'\), \(Y'\), \(Z'\)) = (X − V t, Y, Z), but the displacement between the bodies will be \(d'\) = \(r'\) − \(R'\) = (x − X, y − Y, z − Z), which is identical to the displacement d in frame S. It follows that the magnitude of the displacement \(d'\) and the unit vector d \(E'\) in the direction of the displacement will also retain their old values. Since masses are invariant in Galilean relativity, we thus see that Newton’s law of universal gravitation takes the same form in S and \(S'\). Consequently, we can conclude that, at least in the case of gravitational forces, Newton’s second law of motion, f = m a, also takes the same form in frames S and \(S'\), and by implication in all inertial frames. All we have to do to find the form of the law in frame \(S'\) is to add primes to all the old quantities, remembering that in the case of invariants the primes will be irrelevant since the primed quantities will have the same values as the unprimed quantities.\(S'\)，位置向量和 \(R'\) == (\(X'\), \(Y'\), \(Z'\)) = (X − V t, Y, Z)，但物体之间的位移将为 \(d'\) = \(r'\) − \(R'\) = (x − X, y − Y, z − Z)，这与坐标系 S 中的位移 d。 由此可见，位移 \(d'\) 的大小和位移方向上的单位向量 d \(E'\) 也将保留其旧值。由于质量在伽利略相对论中是不变的，因此我们看到牛顿万有引力定律在 S 和 \(S'\) 中具有相同的形式。因此，我们可以得出结论，至少在引力的情况下，牛顿第二运动定律 f = m a 在框架 S 和 \(S'\) 中也采取相同的形式，并且暗示在所有惯性系中。要在 \(S'\) 框架中找到定律的形式，我们所要做的就是将素数添加到所有旧量中，记住，在不变量的情况下，素数将是无关的，因为带素数的量将与未带素数的量具有相同的值。

An equation that is form-invariant under a given coordinate transformation is sometimes said to be covariant under that transformation. In the particular case that we have been considering, not only have we shown that Newton’s second law is covariant under the Galilean transformation, we have also concluded that the forces, masses and accelerations will have the same values in all inertial frames. So in this case, in addition to establishing the covariance of the equations, we have also shown the invariance of the quantities involved. Later in this chapter you will meet examples of physical laws that are covariant under a transformation but where the quantities involved are certainly not invariant.在给定坐标变换下形式不变的方程有时被称为在该变换下是协变的。在我们一直在考虑的特定情况下，我们不仅证明了牛顿第二定律在伽利略变换下是协变的，而且还得出了力、质量和加速度在所有惯性系中具有相同值的结论。因此，在这种情况下，除了建立方程的协方差之外，我们还证明了所涉及的量的不变性。在本章后面，您将遇到一些物理定律的示例，这些物理定律在变换下是协变的，但所涉及的数量肯定不是不变的。

The argument that we have already applied to Newton’s second law in the case of gravitational forces can be extended to any force that depends only on a combination of displacements and invariants. Such an extension would include Hooke’s law (for the force produced by the stretching of a spring) and even Coulomb’s law of electrostatic forces. However, the argument cannot be extended to all conceivable forces. It does not, for example, work for electromagnetic forces that depend on the velocity of a charged particle. Of course, this failure is not a great concern to us since we have already seen that it was problems arising from electromagnetism and light that persuaded Einstein to reject Galilean relativity in favour of special relativity, even at the price of having to accept new laws of mechanics.我们已经在引力情况下应用于牛顿第二定律的论点可以扩展到任何仅依赖于位移和不变量组合的力。这样的扩展将包括胡克定律(针对弹簧拉伸产生的力)，甚至库仑静电力定律。然而，这一论点不能扩展到所有可以想象的力量。例如，它不适用于取决于带电粒子速度的电磁力。当然，这种失败并不是我们所关心的，因为我们已经看到，正是电磁学和光引起的问题说服了爱因斯坦拒绝伽利略相对论而支持狭义相对论，甚至不惜以接受新的力学定律为代价。

So, now that the idea of covariance or form-invariance has been introduced in the relatively simple context of Galilean relativity, let us return to special relativity and go in search of laws of mechanics that are covariant under the Lorentz transformations.因此，现在协变或形式不变的概念已经在伽利略相对论相对简单的背景下引入，让我们回到狭义相对论并寻找洛伦兹变换下协变的力学定律。

2.2 The laws of mechanics2.2 力学定律

2.2.1 Relativistic momentum2.2.1 相对论动量

The best place to start the reformulation of mechanics is with the concept of momentum. This quantity plays a crucial role in the analysis of high-speed collisions between fundamental particles, one of the main areas where relativistic mechanics (i.e. Lorentz-covariant mechanics) is routinely used. Relativistic mechanics will be essential to the analysis of the high-energy proton–proton collisions in the Large Hadron Collider (Figure 2.2 overleaf) at CERN, near Geneva.重新表述力学的最佳起点是动量概念。这个量在基本粒子之间的高速碰撞分析中起着至关重要的作用，这是相对论力学(即洛伦兹协变力学)常规使用的主要领域之一。相对论力学对于分析日内瓦附近欧洲核子研究中心大型强子对撞机(背页图 2.2)中的高能质子对撞机至关重要。

In Newtonian mechanics, the momentum of a particle of mass m travelling with velocity v is given by p Newtonian = m v. The importance of momentum comes mainly from the observation that, provided that no external forces act on a system,在牛顿力学中，质量为 m 以速度 v 运动的粒子的动量由 p Newtonian = m v 给出。动量的重要性主要来自于观察，假设没有外力作用在系统上，

cavities and gigantic空洞和巨大的

This means, travelling with some travelling with initial v A and v B of those two这意味着，与一些以这两个的初始 v A 和 v B 一起旅行

transforming velocities in the Newtonian way raises doubts about the Detailed calculations momentum is conserved shows that it cannot be should seek a new definition of momentum, sometimes called relativistic momentum, that will transform simply under Lorentz transformations and will provide a conservation law that is Lorentz-covariant. Of course, in formulating a new definition of momentum, we should not forget that physicists spent many years believing that experiments supported the conservation of Newtonian momentum — we should also aim to account for that.以牛顿方式变换速度引起了对动量守恒的详细计算的怀疑，表明不能寻求新的动量定义，有时称为相对论动量，它将在洛伦兹变换下简单地变换，并提供洛伦兹协变的守恒定律。当然，在制定动量的新定义时，我们不应忘记物理学家多年来相信实验支持牛顿动量守恒——我们也应该致力于解释这一点。

Consider a particle of mass m travelling with uniform velocity v between two (\(\Delta t\), \(\Delta x\), \(\Delta y\), \(\Delta z\)). events, labelled 1 and 2, separated by the coordinate intervals What makes the Newtonian momentum of such a particle transform in a complicated way is its direct relationship to the particle’s velocity:考虑质量为 m 的粒子以匀速 v 在两个粒子 (\(\Delta t\)、\(\Delta x\)、\(\Delta y\)、\(\Delta z\)) 之间移动。事件，标记为 1 和 2，由坐标间隔分隔。使此类粒子的牛顿动量以复杂方式变换的原因在于它与粒子速度的直接关系：

\(\Delta t\) transform in This involves ratios such as \(\Delta x\)/\(\Delta t\) where both \(\Delta x\) and moderately complicated ways. Momentum would transform far more simply if all references to the time between the two events, \(\Delta t\), were replaced by references to the proper time between the events, \(\Delta \tau\), which is an invariant and therefore transforms very simply. This suggests that a simple definition of the relativistic momentum of the particle would be\(\Delta t\) 变换这涉及诸如 \(\Delta x\)/\(\Delta t\) 之类的比率，其中 \(\Delta x\) 和中等复杂的方式。如果对两个事件之间的时间 \(\Delta t\) 的所有引用都替换为对事件之间的正确时间 \(\Delta \tau\) 的引用(\(\Delta \tau\) 是一个不变量，因此转换非常简单)，那么动量的转换就会简单得多。这表明粒子相对论动量的简单定义是

Since the particle mass m and the proper time interval \(\Delta \tau\) are both invariants, relativistic momentum defined like this will transform in the same way as the displacement vector (\(\Delta x\), \(\Delta y\), \(\Delta z\)). Moreover, it follows from our discussion of proper time in Chapter 1 that since a particle travelling with speed v is present at both event 1 and event 2, the time between those events, \(\Delta t\), is related to the proper time between them by \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\), so we can rewrite the definition of relativistic momentum as由于粒子质量 m 和本征时间间隔 \(\Delta \tau\) 都是不变量，因此这样定义的相对论动量将以与位移矢量 (\(\Delta x\), \(\Delta y\), \(\Delta z\)) 相同的方式变换。此外，根据我们在第一章中对本征时间的讨论，由于事件 1 和事件 2 中都存在以速度 v 行进的粒子，因此这些事件之间的时间 \(\Delta t\) 与它们之间的本征时间相关，关系为 \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\)，因此我们可以将相对论动量的定义重写为

We now have a clear definition of relativistic momentum that is guaranteed to transform simply between different inertial frames. However, several issues remain to be resolved before we can accept it. First, does it lead to a Lorentz-covariant conservation law, so that the observed conservation of momentum in one inertial frame implies the conservation of momentum in all inertial frames? Second, is such a conservation law correct: is momentum defined in this new way really conserved in any inertial frame? (Remember, covariance establishes the acceptability of a law in principle, but only experiment can establish its truth in practice.) Third, how does this newly defined relativistic momentum relate to Newtonian momentum? Let’s deal with the last of these questions first.我们现在对相对论动量有了一个明确的定义，保证它可以在不同的惯性系之间简单地转换。然而，在我们接受之前，还有几个问题需要解决。首先，它是否会导致洛伦兹协变守恒定律，因此在一个惯性系中观察到的动量守恒意味着所有惯性系中的动量守恒？其次，这样的守恒定律是否正确：以这种新方式定义的动量在任何惯性系中真的守恒吗？ (请记住，协变原则上确立了定律的可接受性，但只有实验才能在实践中确立其真理。)第三，这种新定义的相对论动量与牛顿动量有何关系？让我们先解决最后一个问题。

The relativistic momentum p = mγ (v) v differs from Newtonian momentum only by a Lorentz factor \(\gamma(v)\). This means that at speeds that are small compared with the speed of light, where \(\gamma(v)\) ≈ 1, the two will be almost indistinguishable and all the apparent successes of Newtonian momentum conservation can be recovered.相对论动量 p = mγ (v) v 与牛顿动量的区别仅在于洛伦兹因子 \(\gamma(v)\)。这意味着，在与光速相比较小的速度下，其中 \(\gamma(v)\) ≈ 1，两者几乎无法区分，并且可以恢复牛顿动量守恒的所有明显成功。

As far as the covariance of relativistic momentum conservation is concerned, the question is this: if in some frame S就相对论动量守恒的协方差而言，问题是：如果在某个坐标系 S

will the velocity transformations also show that in some other inertial frame \(S'\)速度变换是否也表明在其他一些惯性系中 \(S'\)

Note that there are no primes on any of the masses in this last equation — that’s because they are invariant.请注意，最后一个方程中的任何质量上都没有素数 - 这是因为它们是不变的。

We could perform a detailed calculation to show that the law of relativistic momentum conservation is covariant under Lorentz transformations, but it’s really not necessary. There is a much neater way of reaching the same conclusion based on the fact that the relativistic momentum (\(p_{x}\), \(p_{y}\), \(p_{z}\)) transforms in the same way as the displacement vector (\(\Delta x\), \(\Delta y\), \(\Delta z\)). Suppose that we let the initial momenta in frame S be p A and p B, and let the final momenta be p A and p B. Then relativistic momentum conservation implies that p A + p B = p A + p B, or, after a slight rearrangement,我们可以通过详细的计算来证明相对论动量守恒定律在洛伦兹变换下是协变的，但这其实没有必要。基于相对论动量 (\(p_{x}\), \(p_{y}\), \(p_{z}\)) 以与位移矢量 (\(\Delta x\), \(\Delta y\), \(\Delta z\)) 相同的方式变换这一事实，有一种更简洁的方法可以得出相同的结论。假设我们令 S 系中的初始动量为 p A 和 p B，最终动量为 p A 和 p B。则相对论动量守恒意味着 p A + p B = p A + p B，或者稍微重新排列后，

Now, this equation can be represented geometrically as in Figure 2.4. With the arrows corresponding to the final momenta reversed in direction, the four arrows representing the individual momenta in frame S form a closed figure when drawn head to tail. Under a Lorentz transformation to some other frame, all of these momenta may change, but since they transform like displacement vectors, it will still be the case, even after transformation, that they will form a closed figure. Hence we can be sure that the transformed momenta will obey现在，这个方程可以用几何表示，如图 2.4 所示。由于最终动量对应的箭头方向相反，S 系中代表各个动量的四个箭头从头到尾绘制时形成一个闭合图形。在洛伦兹变换到其他坐标系下，所有这些动量都可能发生变化，但由于它们像位移矢量一样变换，因此即使在变换之后，它们仍然会形成一个闭合图形。因此我们可以确定变换后的动量将服从

and consequently \(p'_{A}\) + \(p'_{B}\) = \(p'_{A}\) + \(p'_{B}\). Thus we see, in this case at least, that if relativistic momentum is conserved in one inertial frame, then it will be conserved in all inertial frames. This geometric argument can be extended to as many colliding particles as we want, so the argument shows that relativistic momentum conservation is a Lorentz-covariant result.因此，\(p'_{A}\) + \(p'_{B}\) = \(p'_{A}\) + \(p'_{B}\)。因此，至少在这种情况下，我们看到，如果相对论动量在一个惯性系中守恒，那么它在所有惯性系中也将守恒。这个几何论证可以扩展到我们想要的任意数量的碰撞粒子，因此该论证表明相对论动量守恒是洛伦兹协变的结果。

• ● Why can’t this same geometric argument be used to● 为什么不能使用相同的几何参数

show that Newtonian momentum, if conserved in some inertial frame S, will also be conserved in all other inertial frames, even under Lorentz transformations?表明牛顿动量如果在某些惯性系 S 中守恒，那么即使在洛伦兹变换下，在所有其他惯性系中也将守恒？

❍ This is because Newtonian momentum does not transform in the same way as a displacement vector under a Lorentz transformation. Even if the Newtonian momentum vectors formed a closed figure in frame S, the complicated transformation law of Newtonian momentum would ensure that they did not form a closed figure in all other inertial frames.❍ 这是因为牛顿动量在洛伦兹变换下的变换方式与位移矢量不同。即使牛顿动量矢量在S系中形成闭图形，牛顿动量复杂的变换规律也将确保它们在所有其他惯性系中不会形成闭图形。

Now the only remaining question is: ‘does nature really make use of this possibility?’ Here experiment is the arbiter, and the analysis of an enormous number of high-speed particle collisions clearly indicates that nature does so. It is relativistic momentum that is found to be conserved in nature. So we can conclude the following.现在唯一剩下的问题是：“大自然真的利用了这种可能性吗？”这里实验是仲裁者，对大量高速粒子碰撞的分析清楚地表明大自然确实这样做了。人们发现，相对论动量在自然界中是守恒的。所以我们可以得出以下结论。

Relativistic momentum相对论动量

In Lorentz-covariant mechanics, the relativistic momentum of a particle of mass m moving with velocity v is defined as在洛伦兹协变力学中，质量为 m 以速度 v 运动的粒子的相对论动量定义为

The total relativistic momentum of a system is conserved in the absence of external forces.在没有外力的情况下，系统的总相对论动量是守恒的。

Exercise 2.1 An electron of mass m = 9.11 × 10 −练习 2.1 质量为 m = 9.11 × 10 − 的电子

31 kg has speed 4 c/5. What is the magnitude of its (relativistic) momentum?31公斤的速度为4 c/5。它的(相对论)动量有多大？

2.2.2 Relativistic kinetic energy2.2.2 相对论动能

Another quantity of importance in mechanics is kinetic energy. As in the case of momentum, special relativity demands that we modify the definition of kinetic energy before it can take its proper place in a Lorentz-covariant formulation of mechanics.力学中另一个重要的量是动能。与动量的情况一样，狭义相对论要求我们修改动能的定义，然后才能在力学的洛伦兹协变公式中占据适当的位置。

In Newtonian mechanics, the kinetic energy of a particle travelling with speed v can be found from the work W done in accelerating that particle from rest to its final speed v. If we consider the case of a particle with speed u accelerated along the x -axis by a force of magnitude f, we can write the kinetic energy as在牛顿力学中，以速度 v 行进的粒子的动能可以通过将该粒子从静止加速到最终速度 v 所做的功 W 求得。如果我们考虑速度为 u 的粒子被大小为 f 的力沿 x 轴加速的情况，我们可以将动能写为

In Newtonian mechanics, the applied force is the same as the rate of change of momentum, f = ma = m d v/d t = d p/d t, so在牛顿力学中，施加的力与动量的变化率相同，f = ma = m d v/d t = d p/d t，因此

The integral can be rewritten in a much more useful form by changing integration variables and using the chain rule:通过更改积分变量并使用链式法则，可以将积分重写为更有用的形式：

In this way a Newtonian expression for kinetic energy that initially involved distance and force can be re-expressed in terms of speed u and momentum magnitude p. This latter expression can be taken over to special relativity, where we already know the relationship between speed and the magnitude of momentum.这样，最初涉及距离和力的牛顿动能表达式可以用速度 u 和动量大小 p 重新表达。后一种表达可以采用狭义相对论，我们已经知道速度和动量大小之间的关系。

So, in special relativity, a reasonable starting point from which to define the relativistic kinetic energy of a particle of mass m moving with speed v is因此，在狭义相对论中，定义质量为 m、速度为 v 的粒子的相对论动能的合理起点是

This integral can be evaluated using the technique of integration by parts:该积分可以使用分部积分技术来计算：

The remaining integral can be performed by inspection, giving剩余积分可以通过检查进行，给出

A compact final result can be found by putting both terms over a common denominator:通过将两项放在一个公分母上可以找到紧凑的最终结果：

Thus the suggested expression for the relativistic kinetic energy of a particle of mass m moving with speed v is因此，质量为 m 以速度 v 运动的粒子的相对论动能的建议表达式为

There is no general principle of conservation of kinetic energy for us to consider in this case, but in Newtonian physics, kinetic energy is conserved in elastic collisions. In an elastic collision, the particles do not change their number, state or nature, so what goes in is also what comes out. As far as covariance under在这种情况下，我们没有考虑动能守恒的一般原理，但在牛顿物理学中，弹性碰撞中动能守恒。在弹性碰撞中，粒子的数量、状态或性质不会改变，所以进去的也是出来的。就协方差而言

Lorentz transformations is concerned, it is possible to show that in the case of elastic collisions, the proposed expression for relativistic kinetic energy does ensure that an elastic collision in one inertial frame will also be elastic in all other inertial frames.就洛伦兹变换而言，可以证明，在弹性碰撞的情况下，所提出的相对论动能表达式确实确保了一个惯性系中的弹性碰撞在所有其他惯性系中也将是弹性的。

How does the relativistic kinetic energy relate to the Newtonian kinetic energy? At first sight the relationship is not at all obvious, but it soon becomes clear if we factor, \(\gamma(v)\), obtained use the following mathematical expansion of the Lorentz via Taylor’s theorem or the binomial expansion:相对论动能与牛顿动能有何关系？乍一看，这种关系并不明显，但如果我们通过泰勒定理或二项式展开使用洛伦兹的以下数学展开式对 \(\gamma(v)\) 进行因式分解，很快就会变得清晰：

The expansion continues with higher orders of \(v^{2}\)/\(c^2\). In Newtonian physics, the speed v will generally be small compared with the speed of light c, so these expression for \(\gamma(v)\) higher-order terms can be ignored. Substituting the truncated into Equation 2.19 gives随着 \(v^{2}\)/\(c^2\) 订单的增加，扩展仍在继续。在牛顿物理学中，速度 v 与光速 c 相比通常会很小，因此 \(\gamma(v)\) 高阶项的这些表达式可以忽略。将截断代入公式 2.19 得出

So the Newtonian expression for kinetic energy emerges as a low-speed approximation to the relativistic expression. All the low-speed experiments that support the Newtonian expression will also support the more general expression of relativistic mechanics.因此，牛顿动能表达式是相对论表达式的低速近似。所有支持牛顿表达式的低速实验也将支持相对论力学的更一般的表达式。

Henceforth we shall adopt the proposed definition, so we can say the following.今后我们将采用提议的定义，因此我们可以说以下内容。

Relativistic kinetic energy相对论动能

In Lorentz-covariant mechanics, the relativistic kinetic energy of a particle of mass m moving with speed v is在洛伦兹协变力学中，质量为 m 以速度 v 运动的粒子的相对论动能为

Exercise 2.2 Compute the kinetic energy of a muon练习2.2 计算\(\mu\)子的动能

(mass m = \(1.88\times10^{-28}\) kg) travelling with speed 9 c/10.(质量 m = \(1.88\times10^{-28}\) kg)以 9 c/10 的速度行驶。

2.2.3 Total relativistic energy and mass2.2.3 总相对论能量和质量

energy活力

In the 1905 paper in which Einstein introduced the special theory of relativity, he considered the acceleration of an electron and arrived at the expressions for momentum and kinetic energy that have been introduced in this chapter. However, our next topic is one that was not considered in that first paper. It concerns the best known result of special relativity, E = \(mc^2\), and the ‘equivalence’ between mass and energy that it is usually said to indicate. It was first indicated in a three-page paper (‘Does the inertia of a body depend upon its energy content?’) published a few months after the first paper, and then more fully developed in later publications.在 1905 年爱因斯坦介绍狭义相对论的论文中，他考虑了电子的加速，并得出了本章介绍的动量和动能的表达式。然而，我们的下一个主题是第一篇论文中未考虑的主题。它涉及狭义相对论最著名的结果 E = \(mc^2\)，以及通常所说的质量和能量之间的“等价性”。它首先在一篇三页纸的论文(“物体的惯性取决于其能量含量吗？”)中指出，该论文在第一篇论文发表几个月后发表，然后在后来的出版物中得到了更充分的发展。

The crucial result is already suggested by the expression for relativistic kinetic energy E = (\(\gamma(v)\) − 1) \(mc^2\), which can be rewritten as相对论动能 E = (\(\gamma(v)\) − 1) \(mc^2\) 的表达式已经表明了关键结果，该表达式可以重写为

This is now interpreted as showing that in an inertial frame S, where a particle of mass m has speed v, that particle will have a total relativistic energy E = \(\gamma(v)\) \(mc^2\) that is the sum of a relativistic kinetic energy \(E_K\) and a mass energy \(E_0\) = \(mc^2\). As a mere rearrangement and renaming of terms, this is a harmless exercise. The revolutionary step is in the proposal that in relativistic mechanics generally, and high-speed particle collisions in particular, it is the total relativistic energy that is conserved. In high-speed collisions, neither kinetic energy nor mass energy will necessarily be conserved, but their sum, represented by the total relativistic energy E, will be.现在这被解释为表明，在惯性系 S 中，质量为 m 的粒子具有速度 v，该粒子将具有总相对论能量 E = γ (v) mc 2，即相对论动能 \(E_K\) 和质量能量 \(E_0\) = mc 2 之和。作为仅仅是术语的重新排列和重命名，这是一种无害的练习。革命性的一步是提出，在一般的相对论力学中，特别是在高速粒子碰撞中，总的相对论能量是守恒的。在高速碰撞中，动能和质能都不一定守恒，但它们的总和(以相对论总能量 E 表示)将会守恒。

The startling possibility opened up by this suggestion is that in high-speed collisions, particles with mass may be created at the expense of relativistic kinetic energy. It is also possible for some or all of the particles involved in a collision to be annihilated, releasing mass energy that may emerge from the collision either as the mass energy of particles created in the collision or as a contribution to the kinetic energy of all the particles that emerge, or both. This takes relativistic mechanics into an important domain that was completely unexplored by Newtonian mechanics.这一建议所揭示的令人震惊的可能性是，在高速碰撞中，具有质量的粒子可能会以相对论动能为代价而产生。参与碰撞的部分或全部粒子也有可能被湮灭，释放出碰撞中产生的质量能，或者作为碰撞中产生的粒子的质量能，或者作为对所有出现的粒子的动能的贡献，或者两者兼而有之。这将相对论力学带入了牛顿力学完全未探索过的重要领域。

It’s worth noting that the relationship between mass and energy represented by the formula \(E_0\) = \(mc^2\) is not limited to high-speed particle collisions. The initial arguments in favour of such a relationship were based on considerations of the emission of radiation from a body, and it has often been stressed that in the case of a composite body, such as a piece of metal, the simple act of heating it so as to raise its temperature will increase its internal energy and thereby increase its mass. Note that this has nothing to do with the speed of the body; it is a change in the invariant mass that we are discussing.值得注意的是，\(E_0\) = mc 2 公式所表示的质量与能量的关系并不局限于高速粒子碰撞。支持这种关系的最初论据是基于对物体辐射发射的考虑，并且经常强调，在复合物体(例如一块金属)的情况下，加热它以提高其温度的简单行为将增加其内能，从而增加其质量。请注意，这与身体的速度无关；我们正在讨论的是不变质量的变化。

When Einstein first proposed the equivalence of mass and energy, he suggested that it might account for the energy associated with radioactive decay. This is now known to be the case. \(E_0\) = \(mc^2\) plays a vital role in explaining many nuclear phenomena, and particle creation (see Figure 2.5) is the basis of much of the work carried out in particle physics laboratories. Ironically, Einstein’s famous relation has also become indissolubly linked with the awesome energy release of nuclear weapons (Figure 2.6) despite Einstein’s many pronouncements on the need for world peace.当爱因斯坦首次提出质量和能量的等价性时，他认为这可以解释与放射性衰变相关的能量。现在已知情况确实如此。 \(E_0\) = mc 2 在解释许多核现象中起着至关重要的作用，而粒子的产生(见图 2.5)是粒子物理实验室中进行的许多工作的基础。具有讽刺意味的是，尽管爱因斯坦多次宣称需要世界和平，但爱因斯坦著名的关系也与核武器释放的惊人能量有着不可分割的联系(图2.6)。

Total relativistic energy and mass energy总相对论能量和质量能量

In Lorentz-covariant mechanics, the total relativistic energy E and the mass energy 0 (sometimes called the rest energy) of a particle of mass with speed are given by在洛伦兹协变力学中，具有速度的质量粒子的总相对论能量 E 和质量能量 0(有时称为静止能量)由下式给出

Exercise 2.3 The proton has mass m = 1.67 × 10练习 2.3 质子的质量 m = 1.67 × 10

− 27 kg. Compute the total 3 c/5. (relativistic) energy of a proton moving with speed v =- 27 公斤。计算总计 3 c/5。以速度 v 运动的质子的(相对论)能量 =

Exercise 2.4 At what speed is the total energy of a练习2.4 物体速度为多少时，其总能量为

particle twice the mass energy?粒子质量能量的两倍？

Exercise 2.5 (a) In a nuclear fission of uranium- 235练习2.5(a)在铀的核裂变中- 235

caused by the absorption of a neutron, nuclei of krypton and barium are produced, and three neutrons are emitted. The difference in the total mass of the particles present at the start of the process and those present at the end is \(3.08\times10^{-28}\) kg. What is the energy (1 eV = \(1.60\times10^{-19}\) J)? released in this process, in both joules and electronvolts由于吸收一个中子，产生氪和钡的原子核，并发射三个中子。过程开始时和结束时存在的颗粒总质量之差为 \(3.08\times10^{-28}\) kg。能量是多少(1 eV = \(1.60\times10^{-19}\) J)？在此过程中释放的能量，单位为焦耳和电子伏特

(b) Given that the binding energy of a hydrogen atom is 13.6 eV, what is the difference between the mass of a hydrogen atom and the masses of its constituent electron and proton?(b) 假设氢原子的结合能为 13.6 eV，氢原子的质量与其组成电子和质子的质量有什么区别？

2.2.4 Four-momentum2.2.4 四动量

In Chapter 1 you were briefly introduced to the four-position, that is, the four-component object在第一章中，您简要介绍了四位置，即四分量对象

which usefully combined space and time coordinates while using ct rather than t to ensure that they could all be expressed in units of distance. Often when writing \([x^{\mu}]\), it is convenient to write r instead of the three space components x, y and z, so we can write它有效地结合了空间和时间坐标，同时使用 ct 而不是 t 以确保它们都可以用距离单位表示。通常在写\([x^{\mu}]\)时，写r代替三个空间分量x、y和z会更方便，所以我们可以写

Now suppose that the four-position \([x^{\mu}]\) describes the events on the spacetime pathway (i.e. the world-line) of a particle of mass m. We can imagine that the particle carries a clock with it that records the proper time \(\tau\) between successive events as it moves along its world-line. We can then regard each component x \(\mu\) of the particle’s four-position \([x^{\mu}]\) as a function of proper time \(\tau\). Differentiating each of those components x \(\mu\) with respect to \(\tau\) gives us four so-called proper derivatives d x \(\mu\)/d \(\tau\) that we can use as the components of another four-component entity that we shall denote \([U^{\mu}]\). Thus现在假设四位 \([x^{\mu}]\) 描述了质量为 m 的粒子的时空路径(即世界线)上的事件。我们可以想象粒子携带一个时钟，记录它沿着其世界线移动时连续事件之间的固有时间 \(\tau\)。然后，我们可以将粒子四位置 \([x^{\mu}]\) 的每个分量 x \(\mu\) 视为本征时间 \(\tau\) 的函数。将每个分量 x \(\mu\) 相对于 \(\tau\) 求导，得到四个所谓的真导数 d x \(\mu\)/d \(\tau\)，我们可以将其用作另一个四分量实体的分量，我们将其表示为 \([U^{\mu}]\)。因此

The derivatives d x/d \(\tau\), d y/d \(\tau\) and d z/d \(\tau\) that appear on the right can be regarded \(\Delta z\)/\(\Delta \tau\) that we as infinitesimal limits of the ratios \(\Delta x\)/\(\Delta \tau\), \(\Delta y\)/\(\Delta \tau\) and considered earlier when introducing relativistic momentum. On that earlier occasion, the relation \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\) was used to relate those ratios to the scaled velocity components \(\gamma(v)\) v, \(\gamma(v)\) v and \(\gamma(v)\) v. Doing the same here, and also noting that d t/d \(\tau\) is the limit of \(\Delta t\)/\(\Delta \tau =\)\(\gamma(v)\), we can write右侧出现的导数 d x/d \(\tau\)、d y/d \(\tau\) 和 d z/d \(\tau\) 可以视为 \(\Delta z\)/\(\Delta \tau\)，我们将其视为比率 \(\Delta x\)/\(\Delta \tau\)、\(\Delta y\)/\(\Delta \tau\) 的无穷小极限，并在前面介绍时考虑过相对论动量。在那之前的情况下，关系 \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\) 用于将这些比率与缩放速度分量 \(\gamma(v)\) v、\(\gamma(v)\) v 和 \(\gamma(v)\) v 联系起来。在这里做同样的事情，并且还注意到 d t/d \(\tau\) 是\(\Delta t\)/\(\Delta \tau =\)\(\gamma(v)\)，我们可以写

Since \([U^{\mu}]\) is the This quantity is called the four-velocity of the particle. \([U^{\mu}]\) behaves derivative of \([x^{\mu}]\) with respect to the invariant \(\tau\), the four-velocity just as the four-position does under Lorentz transformations.由于 \([U^{\mu}]\) 是该量称为粒子的四速度。 \([U^{\mu}]\) 表现为 \([x^{\mu}]\) 相对于不变 \(\tau\) 的导数，四速度就像洛伦兹变换下的四位置一样。

The four-velocity has an interesting property that becomes apparent when \([U^{\mu}]\) is combined with the Minkowski metric [\(\eta\) \(\mu\)\(\nu\)] that, was introduced in Chapter 1. In \(\eta\) \(\Delta x\) \(\mu\) \(\Delta x\) \(\nu\). Now we that earlier case, we met the invariant (\(\Delta s\)) 2 = 3 can see that四速度有一个有趣的属性，当 \([U^{\mu}]\) 与第 1 章中介绍的闵可夫斯基度规 [\(\eta\) \(\mu\)\(\nu\)] 相结合时，这个属性就变得很明显。现在我们在前面的情况下，遇到了不变式 (\(\Delta s\)) 2 = 3 可以看到

But since但自从

it is clear that the original sum has the invariant value显然原和具有不变值

Multiplying the four-velocity \([U^{\mu}]\) by the invariant mass m gives a related four-component entity called the four-momentum:将四速度 \([U^{\mu}]\) 乘以不变质量 m 得到一个相关的四分量实体，称为四动量：

All the terms on the right should already be familiar. The first is the total relativistic energy divided by the speed of light, E/c; the other three are the components of the relativistic momentum p, so we can write右边的所有术语应该已经很熟悉了。第一个是总相对论能量除以光速，E/c；另外三个是相对论动量 p 的分量，所以我们可以写成

It is clear that the four-momentum contains all the information about the relativistic energy and relativistic momentum of any particle.显然，四动量包含了任何粒子的相对论能量和相对论动量的所有信息。

The crucial point about all this is that under a Lorentz transformation from one inertial frame to another (S to \(S'\), say), the four-momentum \([P^{\mu}]\) transforms in exactly the same way as the four-position.所有这一切的关键点是，在从一个惯性系到另一个惯性系(例如 S 到 \(S'\))的洛伦兹变换下，四动量 \([P^{\mu}]\) 的变换方式与四位置完全相同。

• ● Why must the four-momentum transform in the same way as the为什么四动量必须以与四动量相同的方式转变？

four-position? ❍ Because \(\tau\) is an invariant, the four-velocity \([U^{\mu}]\) = \(\left[\frac{dx^\mu}{d\tau}\right]\) will transform in the same way as the four-position \([x^{\mu}]\). Since m is also invariant, it follows that the four-momentum \([P^{\mu}]\) = \([mU^{\mu}]\) must also transform like \([x^{\mu}]\).四位？ ❍ 因为 \(\tau\) 是不变量，所以四速度 \([U^{\mu}]\) = \(\left[\frac{dx^\mu}{d\tau}\right]\) 将以与四位置 \([x^{\mu}]\) 相同的方式进行变换。由于 m 也是不变的，因此四动量 \([P^{\mu}]\) = \([mU^{\mu}]\) 也必须像 \([x^{\mu}]\) 一样变换。

As a result of the simple behaviour of \([P^{\mu}]\) under a Lorentz transformation, we can say that if the frames S and \(S'\) are in standard configuration, then a particle of mass m with velocity v, that has relativistic energy E and relativistic momentum p in inertial frame S, will be found to have energy \(E'\) and由于 \([P^{\mu}]\) 在洛伦兹变换下的简单行为，我们可以说，如果框架 S 和 \(S'\) 处于标准配置，则质量为 m 且速度为 v 的粒子，在惯性系 S 中具有相对论能量 E 和相对论动量 p，将发现具有能量 \(E'\) 和

Note that the particle speeds v and \(v'\) that help to determine the energy and momentum in S and \(S'\) are quite distinct from V, which represents the speed of frame \(S'\) as measured in frame S. For a particle travelling along the x -axis, so that v = (v, 0, 0), the relation between v, \(v'\) and V follows from the velocity V)/(1 − vV/\(c^2\)). transformation of Chapter 1, and is given by \(v'\) = (v −请注意，有助于确定 S 和 \(S'\) 中的能量和动量的粒子速度 v 和 \(v'\) 与 V 完全不同，V 代表在坐标系 S 中测量的坐标系 \(S'\) 的速度。对于沿 x 轴运动的粒子，使得 v = (v, 0, 0)，v、\(v'\) 和 V 之间的关系由速度 V)/(1 − vV/\(c^2\))。第 1 章的变换，由下式给出： \(v'\) = (v −

As was the case with the four-position, the transformation rule for four-momentum can be written in a number of equivalent ways using the Lorentz transformation matrix [\(\Lambda\) \(\mu\) \(\nu\)]. In terms of matrices,与四位置的情况一样，四动量的变换规则可以使用洛伦兹变换矩阵 [\(\Lambda\) \(\mu\) \(\nu\)] 以多种等效方式编写。就矩阵而言，

which may be represented more compactly as可以更紧凑地表示为

Alternatively, we can represent the transformation using components and summations:或者，我们可以使用分量和求和来表示变换：

The fact that the four-momentum transforms in exactly the same way as the four-position says something quite profound about energy and momentum. Under Lorentz transformation, the energy and momentum components intertwine, and can be thought of as aspects of a single quantity, just as space and time are unified into spacetime. That which is energy to one observer is a mix of energy and momentum to another.事实上，四动量的变换方式与四位置的变换方式完全相同，这一事实说明了有关能量和动量的深刻含义。在洛伦兹变换下，能量和动量分量交织在一起，可以被视为单个量的各个方面，就像空间和时间统一为时空一样。对于一个观察者来说是能量，对于另一个观察者来说则是能量和动量的混合体。

Exercise 2.6 In frame of reference S, an √ electron moving练习 2.6 在参考系 S 中，一个 √ 电子移动

along the x -axis has energy 3 m e \(c^2\) and momentum magnitude 8 m e c. Use the transformations of energy and momentum to find the energy and momentum magnitude observed in x -direction. frame \(S'\) moving with speed 4 c/5 relative to S in the positive沿 x 轴的能量为 3 m e \(c^2\)，动量大小为 8 m e c。使用能量和动量的变换来查找在 x 方向上观察到的能量和动量大小。坐标系 \(S'\) 相对于 S 正向以 4 c/5 的速度移动

2.2.5 The energy–momentum relation2.2.5 能量-动量关系

It was shown in Equation 2.31 that公式 2.31 表明

Since \([P^{\mu}]\) is defined by \([P^{\mu}]\) = m \([U^{\mu}]\), it follows that由于 \([P^{\mu}]\) 由 \([P^{\mu}]\) = m \([U^{\mu}]\) 定义，因此

But using the Minkowski metric we also know that但使用闵可夫斯基度规我们也知道

Consequently, we can say that因此，我们可以说

So, regardless of the frame of reference of an observer, the difference between the squared total energy and the squared momentum magnitude multiplied by the speed of light squared is proportional to the squared invariant mass. This extremely useful relationship is often called the energy–momentum relation and is usually written as follows.因此，无论观察者的参考系如何，总能量平方与动量大小平方乘以光速平方之间的差值与不变质量平方成正比。这种非常有用的关系通常称为能量-动量关系，通常写如下。

Energy–momentum relation能量-动量关系

Taking the positive square root, we see that取正平方根，我们看到

A plot of this relation can be seen in Figure 2.7. Apart from the presence of the distinctly non-Newtonian rest energy \(E_0\) = \(mc^2\), the behaviour at low speeds (when p is close to zero) is what would be expected in Newtonian mechanics, with (kinetic) energy increasing in proportion to \(p^{2}\). However, as the momentum magnitude increases, the total energy becomes more and more nearly proportional to the momentum magnitude, as special relativity requires.这种关系的图如图 2.7 所示。除了存在明显的非牛顿静止能量 \(E_0\) = mc 2 之外，低速时的行为(当 p 接近于零时)也是牛顿力学所期望的，(动能)能量与 p 2 成比例增加。然而，随着动量大小的增加，总能量变得越来越接近与动量大小成正比，正如狭义相对论所要求的那样。

• ● The electron has mass m e = \(9.11\times10^{-31}\) kg. What is the energy of an电子的质量为 m e = \(9.11\times10^{-31}\) kg。一个物体的能量是多少

electron that has a momentum of magnitude \(1.00\times10^{-22}\) \(\mathrm{kg\,m\,s^{-1}}\)? ❍ Making the substitutions \(m^{2}\) \(c^4\) = \(6.72\times10^{-27}\) J and \(c^2\) \(p^{2}\) = \(9.00\times10^{-28}\) J, the energy–momentum √ relation shows that the energy is E = \(6.72\times10^{-27}\) + \(9.00\times10^{-28}\) J = \(8.73\times10^{-14}\) J.动量大小为 \(1.00\times10^{-22}\) \(\mathrm{kg\,m\,s^{-1}}\) 的电子？ ❍ 将 m 2 \(c^4\) = \(6.72\times10^{-27}\) J 和 \(c^2\) p 2 = \(9.00\times10^{-28}\) J 代入，则能量-动量 √ 关系表明，能量为 E = \(6.72\times10^{-27}\) + \(9.00\times10^{-28}\) J = \(8.73\times10^{-14}\) J。

The energy–momentum relation has an important consequence with no analogue in Newtonian mechanics. A symmetry principle known as gauge-invariance, which is of great importance in particle physics, demands that the photon, which is often described as the ‘particle of light’, should be strictly massless with m = 0. It follows from the energy–momentum relation that for a photon, or any other massless particle,能量-动量关系具有牛顿力学中无可比拟的重要结论。被称为规范不变性的对称原理在粒子物理学中非常重要，它要求通常被描述为“光粒子”的光子应该严格无质量，m = 0。从能量-动量关系可以得出，对于光子或任何其他无质量粒子，

The photon carries energy, so even though it has no mass, it does have a momentum. This clearly shows the non-Newtonian nature of relativistic momentum.光子携带能量，因此即使它没有质量，它也有动量。这清楚地表明了相对论动量的非牛顿性质。

It has been suggested that the momentum of the photon could be harnessed to make solar sails, a kind of propulsion system for spacecraft. A depiction of a solar-sail craft is shown in Figure 2.8.有人建议可以利用光子的动量来制造太阳帆，一种航天器的推进系统。太阳帆飞船的描述如图 2.8 所示。

h = \(6.63\times10^{-34}\) \(\mathrm{J\,s}\) ish = \(6.63\times10^{-34}\) \(\mathrm{J\,s}\) 是

Exercise 2.7 (a) The energy of a photon is hf, where练习2.7 (a) 光子的能量是hf，其中

Planck’s constant and f is the frequency of the photon. What is the magnitude of the momentum of a single photon belonging to a monochromatic beam of light with frequency \(5.00\times10^{14}\) Hz?普朗克常数，f 是光子的频率。属于频率为 \(5.00\times10^{14}\) Hz 的单色光束的单个光子的动量大小是多少？

(b) At what rate must such photons be absorbed by a solar sail if they are to cause a steady force of magnitude 10 N on the sail?(b) 如果这些光子要在太阳帆上产生 10 N 量级的稳定力，那么它们必须以什么速率被太阳帆吸收？

Exercise 2.8 You are told by a scientist of ill repute练习 2.8 一位声名狼藉的科学家告诉你

that a ficteron particle of mass m f has been measured to have energy E f = 3 m f \(c^2\) and momentum of magnitude p f = 7 m f c. Are those values consistent with special relativity?质量为 m f 的菲克特粒子已被测量为具有能量 E f = 3 m f \(c^2\) 和动量大小 p f = 7 m f c。这些值符合狭义相对论吗？

2.2.6 The conservation of energy and momentum2.2.6 能量和动量守恒

Now that we know how the four-momentum transforms under a Lorentz transformation, it is easy to demonstrate the Lorentz covariance of the conservation laws of relativistic energy and momentum.现在我们知道了四动量在洛伦兹变换下是如何变换的，就很容易证明相对论能量和动量守恒定律的洛伦兹协变性了。

particles emerge. Let Imagine a collision in which N particles collide, and N incident particle i have mass m i and an incident four-momentum (i = 1, 2, 3,..., N), and remember that some of the masses may be zero. Similarly, let the particles that emerge from the collision have masses m j and four-momenta \([P^{\nu}]\) (j = 1, 2, 3,..., N). Note that the index representing the particle has been placed in parentheses to avoid confusing it with the index that denotes a particular component of the four-momentum. The conservation of energy and momentum in an inertial frame S is represented by the relation粒子出现。想象一次碰撞，其中 N 个粒子发生碰撞，N 个入射粒子 i 具有质量 m i 和入射四动量 (i = 1, 2, 3,..., N)，并记住某些质量可能为零。类似地，让碰撞中出现的粒子具有质量 m j 和四动量 \([P^{\nu}]\) (j = 1, 2, 3,..., N)。请注意，代表粒子的索引已放在括号中，以避免将其与表示四动量特定分量的索引混淆。惯性系 S 中的能量和动量守恒由以下关系表示

Note that \(\nu\) is a free index in this expression, so this one line really represents four different equations, one for each possible value of \(\nu\).请注意，\(\nu\) 在此表达式中是一个自由索引，因此这一行实际上代表四个不同的方程，每个方程对应 \(\nu\) 的每个可能值。

What will be the energy and momentum involved in this collision as observed by some other inertial observer who uses frame \(S'\)? Performing the same Lorentz transformation on each side of the equation, we see that使用 \(S'\) 系的其他惯性观察者观察到的这次碰撞涉及的能量和动量是多少？在方程的每一边执行相同的洛伦兹变换，我们看到

Since the transformation law of an individual four-momentum takes the form [\(P'\) \(\mu\)] = [\(\Lambda\) \(\mu\)] \([P^{\nu}]\), we know that each individual four-momentum in the sum will transform in the same way under the Lorentz transformation to frame \(S'\). Consequently, transforming both sides of the conservation law, we get由于单个四动量的变换定律采用 [\(P'\) \(\mu\)] = [\(\Lambda\) \(\mu\)] \([P^{\nu}]\) 的形式，我们知道在洛伦兹变换下，总和中的每个单独的四动量将以相同的方式变换到坐标系 \(S'\)。因此，变换守恒定律两边，我们得到

Apart from an irrelevant switch in the symbol used to represent the free index, from \(\nu\) to \(\mu\), the only difference between the conservation law in frame S and that in frame \(S'\) is the addition of some primes.除了用于表示自由索引的符号中不相关的转换(从 \(\nu\) 到 \(\mu\))之外，框架 S 中的守恒定律与框架 \(S'\) 中的守恒定律之间的唯一区别是添加了一些素数。

The lesson is clear: by expressing the conservation laws of relativistic energy and relativistic momentum in terms of four-momenta that transform simply under Lorentz transformations, it has become obvious that the conservation laws can be written in the same form in all inertial frames without any need to carry out complicated transformations of E and p. In such situations we say that the law is manifestly covariant. This is only a first glimpse of manifest covariance. We shall have much more to say on the subject later.教训很清楚：通过用洛伦兹变换下简单变换的四动量来表达相对论能量和相对论动量守恒定律，很明显，守恒定律可以在所有惯性系中以相同的形式写出，而不需要对 E 和 p 进行复杂的变换。在这种情况下，我们说该定律是明显协变的。这只是显性协方差的初步体现。稍后我们将就这个主题有更多的内容要说。

2.2.7 Four-force2.2.7 四力

The last major mechanics concept that we shall discuss is that of force. Recalling that in Newtonian particle mechanics, force may be defined by the rate of change of momentum, and taking the introduction of the four-velocity as a guide, a natural way to introduce a four-force in relativistic mechanics is via the manifestly covariant relation我们要讨论的最后一个主要力学概念是力的概念。回想一下，在牛顿粒子力学中，力可以通过动量的变化率来定义，并以四速度的引入为指导，在相对论力学中引入四力的自然方法是通过明显的协变关系

Note that the differentiation is with respect to the invariant proper time \(\tau\). To make the link with Newtonian mechanics as close as possible, we identify the spatial components of the four-force with \(\gamma(v)\) f, where f is a ‘conventional’ force vector: f = (f x, f y, f z). (This is similar to our identification of the scaled velocity \(\gamma(v)\) v with the spatial components of \([U^{\mu}]\).) Making the usual identification \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\), and taking the limit as \(\Delta \tau\) tends to zero, gives请注意，微分是相对于不变的固有时间 \(\tau\) 而言的。为了使与牛顿力学的联系尽可能紧密，我们用 \(\gamma(v)\) f 来识别四力的空间分量，其中 f 是“传统”力矢量：f = (f x, f y, f z)。(这类似于我们用 \([U^{\mu}]\) 的空间分量来识别缩放速度 \(\gamma(v)\) v。)进行通常的识别 \(\Delta t =\)\(\gamma(v)\) \(\Delta \tau\)，并以 \(\Delta \tau\) 趋于零为极限，得到

and we can then identify d E/d t, the rate of change of total energy, with the rate at which the force f performs work, which is given by the scalar product f · v. So we have然后我们可以将总能量的变化率 d E/d t 与力 f 做功的速率(由标量积 f · v 给出)确定。

It’s tempting to think that the ‘conventional’ force vector f must be the Newtonian force, but things are not quite so simple. Having defined the four-force as the derivative of the four-momentum with respect to the proper time, we can be sure that under a Lorentz transformation, the four-force will transform in a simple way, similar to that of the four-momentum and four-position. For the usual case of frames S and \(S'\) in standard configuration, that means人们很容易认为“传统”力矢量 f 一定是牛顿力，但事情并不那么简单。将四力定义为四动量对原时间的导数，我们可以肯定，在洛伦兹变换下，四力会以简单的方式进行变换，类似于四动量和四位置的变换。对于标准配置中框架 S 和 \(S'\) 的常见情况，这意味着

This will automatically determine the way in which the force vector f must transform. It turns out that the electromagnetic Lorentz force that we consider in the next section does transform in just the required way, but the Newtonian gravitational force (which was shown to be form-invariant under the Galilean transformation in an earlier section) does not obey the required transformation. This means that it will be relatively simple to extend the ideas that we have been developing in this section to include electromagnetic forces, but we shall not be able to treat the Newtonian gravitational force as part of a four-force. In fact, we shall have to develop an entirely new theory of gravitation that will take us beyond special relativity and in which force will have almost no part to play at all. This is the role of general relativity.这将自动确定力矢量 f 必须变换的方式。事实证明，我们在下一节中考虑的电磁洛伦兹力确实以所需的方式进行变换，但牛顿引力(在前面的伽利略变换下被证明是形式不变的)并不遵守所需的变换。这意味着将我们在本节中发展的思想扩展到包括电磁力将相对简单，但我们不能将牛顿引力视为四力的一部分。事实上，我们必须发展一种全新的引力理论，它将带我们超越狭义相对论，并且力在其中几乎不起作用。这就是广义相对论的作用。

Exercise 2.9 Given frames S and \(S'\) in standard configuration练习 2.9 给定标准配置中的帧 S 和 \(S'\)

with relative speed V, write down the expressions that relate the component of the three-force \(f'\) measured in frame \(S'\) to the components of the same three-force f that would be measured in frame S.相对速度 V，写出将在坐标系 \(S'\) 中测量的三力 \(f'\) 的分量与在坐标系 S 中测量的相同三力 f 的分量联系起来的表达式。

2.2.8 Four-vectors2.2.8 四向量

You will have gathered by now that among the most important quantities in Lorentz-covariant mechanics are several four-component entities, including:到目前为止，您已经了解到洛伦兹协变力学中最重要的量是几个四分量实体，包括：

• • the four-position \([x^{\mu}]\) = (ct, r)• 四位\([x^{\mu}]\) = (ct, r)
• • the four-velocity \([U^{\mu}]\) = (γc, γ v)• 四速度 \([U^{\mu}]\) = (γc, γ v)
• • the four-momentum \([P^{\mu}]\) = (E/c, p)• 四动量 \([P^{\mu}]\) = (E/c, p)
• • the four-force \([F^{\mu}]\) = (γ f · v/c, γ f).• 四力\([F^{\mu}]\) = (γ f · v/c, γ f)。

= (c \(\Delta t\), \(\Delta r\)). (The To this list we may add the four-displacement [\(\Delta x\) \(\mu\)] four-position is really a special case of the four-displacement in which the coordinate intervals are measured from the origin.) These quantities are all examples of a general class of four-component entities called contravariant four-vectors.= (c \(\Delta t\), \(\Delta r\))。 (在此列表中，我们可以添加四位移 [\(\Delta x\) \(\mu\)] 四位置实际上是四位移的特例，其中坐标间隔是从原点测量的。)这些量都是称为逆变四向量的一般类四分量实体的示例。

Four-vectors will play an important role in the next section, so we shall take this opportunity to introduce them properly and explain their mathematical properties. The defining characteristic that distinguishes the four-vectors introduced so far from other four-component objects is the way that they behave under a Lorentz transformation.四向量将在下一节中发挥重要作用，因此我们将借此机会正确介绍它们并解释它们的数学性质。迄今为止引入的四向量与其他四分量对象的区别特征是它们在洛伦兹变换下的行为方式。

Given two inertial frames S and \(S'\) in standard configuration, the components A \(\mu\) of a contravariant four-vector \([A^{\mu}]\) ≡ (A 0, A 1, A 2, A 3) transform according to给定标准配置中的两个惯性系 S 和 \(S'\)，逆变四向量 \([A^{\mu}]\) ≡ (A 0, A 1, A 2, A 3) 的分量 A \(\mu\) 根据以下变换：

which may be written more compactly in terms of matrices or as a summation:可以用矩阵或求和的形式写得更紧凑：

To this extent, all contravariant four-vectors behave like four-displacements. However, not all four-component objects are four-vectors, nor, as you are about to see, are contravariant four-vectors the only kind of four-vectors.就这个程度而言，所有逆变四向量的行为都类似于四位移。然而，并非所有四分量对象都是四向量，正如您将要看到的，逆变四向量也不是唯一的四向量。

Suppose that \(\phi\) is some scalar function of \(x^0\), \(x^1\), \(x_{2}\) and \(x^{3}\) that is invariant under Lorentz transformations, so \(\phi'\) (\(x'_{0}\), \(x'_{1}\), \(x'_{2}\), \(x'_{3}\)) = \(\phi\) (\(x^0\), \(x^1\), \(x_{2}\), \(x^{3}\)). Consider the behaviour of the derivative \(\partial\phi/\partial x^0\), which we shall denote \(B_0\). Under the usual Lorentz transformation from frame S to frame \(S'\), the function \(B_0\) will become some new function \(B'_{0}\), the form of which can be determined using the chain rule of partial differentiation:Suppose that \(\phi\) is some scalar function of \(x^0\), \(x^1\), \(x_{2}\) and \(x^{3}\) that is invariant under Lorentz transformations, so \(\phi'\) (\(x'_{0}\), \(x'_{1}\), \(x'_{2}\), \(x'_{3}\)) = \(\phi\) (\(x^0\), \(x^1\), \(x_{2}\), \(x^{3}\)).考虑导数 \(\partial\phi/\partial x^0\) 的行为，我们将其表示为 \(B_0\)。在通常的从坐标系 S 到坐标系 \(S'\) 的洛伦兹变换下，函数 \(B_0\) 会变成某个新函数 \(B'_{0}\)，其形式可以使用偏微分链式法则确定：

The partial derivatives ∂\(x^0\)/∂\(x'_{0}\), ∂\(x^1\)/∂\(x'_{0}\), ∂\(x_{2}\)/∂\(x'_{0}\) and ∂\(x^{3}\)/∂\(x'_{0}\) can each be easily determined from the inverse Lorentz transformations given in Chapter 1 as Equations 1.14–1.17, and turn out to be偏导数 ∂\(x^0\)/∂\(x'_{0}\)、∂\(x^1\)/∂\(x'_{0}\)、∂\(x_{2}\)/∂\(x'_{0}\) 和 ∂\(x^{3}\)/∂\(x'_{0}\) 都可以根据第 1 章中给出的洛伦兹逆变换轻松确定，如方程 1.14-1.17 所示，结果为

Substituting these results into Equation 2.62, and representing ∂\(\phi\)/∂x \(\mu\) by B \(\mu\), you can see that under a Lorentz transformation,将这些结果代入方程 2.62，并用 B \(\mu\) 表示 ∂\(\phi\)/∂x \(\mu\)，您可以看到在洛伦兹变换下，

Performing similar calculations for all the other partial derivatives of \(\phi\) leads to the following transformation rule for the four quantities B \(\mu\):对 \(\phi\) 的所有其他偏导数执行类似的计算，得出四个量 B \(\mu\) 的以下变换规则：

Now, this is very similar to an inverse Lorentz transformation. In fact, if we use the symbol [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] to represent the inverse Lorentz transformation matrix现在，这与洛伦兹逆变换非常相似。事实上，如果我们用符号[(\(\Lambda\) − 1) \(\mu\) \(\nu\)]来表示洛伦兹逆变换矩阵

entity \([B^{\mu}]\) in then we can write the transformation rule for the four-component terms of matrices or components:实体 \([B^{\mu}]\) 那么我们可以写出矩阵或分量的四分量项的变换规则：

Any four-component entity that obeys this transformation law is said to be a covariant four-vector. Note that contravariant four-vectors transform like four-positions or four-displacements and are indicated by a raised index as in \([A^{\mu}]\), while covariant four-vectors transform like derivatives of scalar functions and are indicated by a lowered index as in \([B^{\mu}]\).任何遵守此变换定律的四分量实体被称为协变四向量。请注意，逆变四向量变换类似于四位置或四位移，并由升高的索引表示，如 \([A^{\mu}]\) 中，而协变四向量变换类似于标量函数的导数，并由降低的索引表示，如 \([B^{\mu}]\) 中。

There are three important points to note concerning contravariant and covariant four-vectors.关于逆变和协变四向量，需要注意三个要点。

1 Raising and lowering four-vector indices1 四向量索引的升高和降低

For every contravariant four-vector, a corresponding covariant four-vector can be formed, and vice versa. This is achieved by using the Minkowski metric introduced in Chapter 1:假设 \(\phi\) 是 \(x^0\)、\(x^1\)、\(x_{2}\) 和 \(x^{3}\) 的某个标量函数，并且在洛伦兹变换下保持不变，即 \(\phi'\) (\(x'_{0}\), \(x'_{1}\), \(x'_{2}\), \(x'_{3}\)) = \(\phi\) (\(x^0\), \(x^1\), \(x_{2}\), \(x^{3}\))。考虑导数 \(\partial\phi/\partial x^0\) 的行为，我们将其记为 \(B_0\)。在通常的从坐标系 S 到坐标系 \(S'\) 的洛伦兹变换下，函数 \(B_0\) 会变成某个新函数 \(B'_{0}\)，其形式可以用偏微分链式法则确定：

If the four quantities A 0, A 1, A 2 and A 3 transform as a contravariant four-vector, then the four quantities defined by the sums如果四个量 A 0、A 1、A 2 和 A 3 变换为逆变四向量，则由和定义的四个量

will transform as a covariant four-vector. So the Minkowski metric can be used to lower the indices on four-vectors. Thanks to the very simple form of the Minkowski metric, it is easy to perform the necessary sums and to see that将变换为协变四向量。因此，闵可夫斯基度规可用于降低四向量的索引。由于闵可夫斯基度规的形式非常简单，很容易执行必要的求和并看到

if \([A^{\mu}]\) = (a, b, c, d), then [A] = (a, − b, −如果 \([A^{\mu}]\) = (a, b, c, d)，则 [A] = (a, − b, −

This means that starting from the contravariant four-vectors that have already been introduced, we can now introduce a set of covariant counterparts simply by reversing the signs of the spatial components. This gives:这意味着从已经引入的逆变四向量开始，我们现在可以简单地通过反转空间分量的符号来引入一组协变对应项。这给出：

• • the covariant four-displacement [\(\Delta x\)] = (ct, − \(\Delta r\))协变四位移 [\(\Delta x\)] = (ct, − \(\Delta r\))
• • the covariant four-velocity [U] = (γc, − γ v)• 协变四速度 [U] = (γc, − γ v)
• • the covariant four-momentum [P] = (E/c, − p)• 协变四动量 [P] = (E/c, − p)
• • the covariant four-force [F] = (γ f · v/c, − γ f).• 协变四力[F] = (γ f · v/c, − γ f)。

Furthermore, if we introduce a new 16 -component entity [\(\eta\) \(\mu\)\(\nu\)] with components \(\eta\) \(\mu\)\(\nu\) that can be identified from此外，如果我们引入一个新的 16 分量实体 [\(\eta\) \(\mu\)\(\nu\)]，其分量为 \(\eta\) \(\mu\)\(\nu\)，可以从

then we can use sums over those components to raise four-vector indices and convert covariant four-vectors into contravariant ones:然后我们可以使用这些分量的和来提高四向量索引并将协变四向量转换为逆变向量：

Incidentally, it’s worth noting for future reference that although [\(\eta\) \(\mu\)\(\nu\)] and [\(\eta\) \(\mu\)\(\nu\)] have identical components, the two quantities are actually inversely related, in the sense that顺便说一句，值得注意的是，尽管 [\(\eta\) \(\mu\)\(\nu\)] 和 [\(\eta\) \(\mu\)\(\nu\)] 具有相同的分量，但这两个量实际上是负相关的，即

where [δ \(\alpha\)] is represented by the 4 × 4 matrix其中 [δ \(\alpha\)] 由 4 × 4 矩阵表示

2 Forming invariants by contraction2 通过收缩形成不变量

The second point concerns invariants. We saw earlier that we could find invariants by considering sums of components such as第二点涉及不变量。我们之前看到，我们可以通过考虑分量之和来找到不变量，例如

But it can now be seen that such a sum actually involves the corresponding components of a contravariant four-vector and its covariant counterpart:但现在可以看出，这样的和实际上涉及逆变四向量及其协变对应项的相应分量：

Since the contravariant and covariant components transform in inversely related ways under a Lorentz transformation, it is really not surprising that this kind, of sum is invariant. Other examples, that you have already met include 3 P P \(\nu\) = \(m^{2}\) \(c^2\) and even 3 \(\Delta x\) \(\Delta x\) \(\nu\) = (\(\Delta s\)) 2.由于逆变和协变分量在洛伦兹变换下以逆相关的方式变换，因此这种总和是不变的并不奇怪。您已经遇到过的其他示例包括 3 P P \(\nu\) = \(m^{2}\) \(c^2\) 甚至 3 \(\Delta x\) \(\Delta x\) \(\nu\) = (\(\Delta s\)) 2。

It is very common to see expressions involving four-vectors in which a sum runs over one raised index and one lowered index. The process is often referred to as contraction, and is not limited to cases where the indices are on identical four-vectors. The contraction of \([A^{\mu}]\) with \([B^{\nu}]\), for example, would be the invariant quantity涉及四矢量量量的表达式很常见，其中求和在一个升高的索引和一个降低的索引上运行。该过程通常称为收缩，并且不限于索引位于相同的四向量上的情况。例如，\([A^{\mu}]\) 与 \([B^{\nu}]\) 的收缩将是不变量

The contraction of four-vectors is rather like the formation of a scalar product of ordinary (three-) vectors. Indeed, quantities that are invariant under Lorentz transformations are sometimes referred to as Lorentz scalars.四向量的收缩很像普通(三)向量的标量积的形成。事实上，洛伦兹变换下不变的量有时被称为洛伦兹标量。

3 Transformation under arbitrary Lorentz transformation3 任意洛伦兹变换下的变换

The third point concerns the generality of the definition of four-vectors. So far, when considering Lorentz transformations, we have always considered the case where the frames S and \(S'\) are in standard configuration, though we have emphasized that there is no real loss of generality in doing this. Nonetheless, now that we are using behaviour under Lorentz transformation as the defining characteristic of four-vectors, we should make it clear that the definition applies to arbitrary Lorentz transformations and not just those that describe standard configuration. We shall have more to say about this later. The box below summarizes what has already been said.第三点涉及四向量定义的一般性。到目前为止，在考虑洛伦兹变换时，我们总是考虑框架 S 和 \(S'\) 处于标准配置的情况，尽管我们强调这样做并没有真正损失一般性。尽管如此，既然我们使用洛伦兹变换下的行为作为四向量的定义特征，我们应该明确该定义适用于任意洛伦兹变换，而不仅仅是那些描述标准配置的变换。稍后我们将对此进行更多讨论。下面的方框总结了已经说过的内容。

Four-vectors and their transformation四向量及其变换

The behaviour of momentum, energy and force under Lorentz transformation is most easily described in terms of four-vectors. Important \([U^{\mu}]\) = (γc, γ v), contravariant four-vectors include the velocity four-vector the momentum four-vector \([P^{\mu}]\) = (E/c, p) and the force four-vector \([F^{\mu}]\) = (γ f · v/c, γ f).洛伦兹变换下动量、能量和力的行为最容易用四向量来描述。重要的是\([U^{\mu}]\) = (γc, γ v)，逆变四向量包括速度四向量、动量四向量\([P^{\mu}]\) = (E/c, p) 和力四向量\([F^{\mu}]\) = (γ f · v/c, γ f)。

\(\Lambda\) \(\mu\) x \(\nu\), a Under a Lorentz transformation in which \(x'\) \(\mu\) = 3 contravariant four-vector \([A^{\mu}]\) transforms in the same way as a four-displacement:\(\Lambda\) \(\mu\) x \(\nu\)， a 在洛伦兹变换下，其中 \(x'\) \(\mu\) = 3 个逆变四向量 \([A^{\mu}]\) 以与四位移相同的方式进行变换：

Under the same Lorentz transformation, a covariant transforms in the same way as a set of derivatives:在相同的洛伦兹变换下，协变的变换方式与一组导数相同：

where [(\(\Lambda\) − 1) \(\nu\)] is the matrix inverse of [\(\Lambda\) \(\mu\)].其中 [(\(\Lambda\) − 1) \(\nu\)] 是 [\(\Lambda\) \(\mu\)] 的矩阵逆。

Indices on four-vectors may be lowered or raised using the Minkowski \(\eta\) \(\mu\)\(\nu\) or the related quantity \(\eta\) \(\mu\)\(\nu\) (defined by requiring that可以使用闵可夫斯基 \(\eta\) \(\mu\)\(\nu\) 或相关量 \(\eta\) \(\mu\)\(\nu\) 来降低或提高四向量的指数(通过要求定义)

and和

Lorentz invariants may be formed by the process, of contraction (summing, over one raised and one lowered index) as in 3 \(\nu\) =0 U \(\nu\) U \(\nu\) = \(c^2\) and 3 P P \(\nu\) = \(m^{2}\) \(c^2\) and, more generally,洛伦兹不变量可以通过收缩过程(对一个升高的索引和一个降低的索引求和)形成，如 3 \(\nu\) =0 U \(\nu\) U \(\nu\) = \(c^2\) 和 3 P P \(\nu\) = \(m^{2}\) \(c^2\)，更一般地，

Four-vectors may be used to formulate the laws of mechanics in a manifestly Lorentz covariant way, as in the relation F \(\mu\) = d P \(\mu\)/d \(\tau\). However, the force described by Newton’s inverse square law of gravitation fails to transform in the required way, so Newtonian gravitation is inconsistent with special relativity and must be replaced by a different theory of gravitation.四向量可用于以明显洛伦兹协变的方式制定力学定律，如关系式 F \(\mu\) = d P \(\mu\)/d \(\tau\) 中所示。然而，牛顿引力平方反比定律所描述的力无法按要求的方式变换，因此牛顿引力与狭义相对论不一致，必须用不同的引力理论来取代。

Exercise 2.10 (cρ, J, J, J) is a contravariant four-vector that you will meet练习 2.10 (cρ, J, J, J) 是您将遇到的逆变四向量

in the next section. Even without knowing what the symbols represent, you should be able to write down the four equations that show how these quantities will transform under a Lorentz transformation. Do that for the case of frames S and S in standard configuration, then write down the four components of the counterpart covariant four-vector that will transform according to the corresponding inverse Lorentz transformation.在下一节中。即使不知道这些符号代表什么，您也应该能够写下四个方程，以显示这些量在洛伦兹变换下如何变换。对于标准配置中的帧 S 和 S 的情况执行此操作，然后写下将根据相应的洛伦兹逆变换进行变换的对应协变四向量的四个分量。

Exercise 2.11 If the four-vector given in the previous question is represented练习2.11 如果表示上一题给出的四向量

by \([J^{\mu}]\) = (cρ, J, J, J), explain why you should expect the quantity通过 \([J^{\mu}]\) = (cρ, J, J, J)，解释为什么您应该预期数量

3 \(\mu\) =0 J \(\mu\) J \(\mu\) to be invariant under a Lorentz transformation, but not the quantities 3 J J or 3 J \(\mu\) J \(\mu\).3 \(\mu\) =0 J \(\mu\) J \(\mu\) 在洛伦兹变换下不变，但数量 3 J J 或 3 J \(\mu\) J \(\mu\) 不变。

2.3 The laws of electromagnetism2.3 电磁定律

Turning to the laws of electromagnetism, the situation is rather different from that in mechanics. It turns out that the existing laws of electromagnetism are already consistent with special relativity. What is needed is a recasting of those laws so that the Lorentz covariance will be manifest. This involves identifying all the important electromagnetic quantities as components of four-vectors or other similar entities that behave simply under Lorentz transformations, and then expressing the laws of electromagnetism as relations between those entities. That is what we shall do in this section. To keep the discussion as simple as possible, we shall consider electromagnetism only in a vacuum.谈到电磁学定律，情况与力学中的情况截然不同。事实证明，现有的电磁学定律已经与狭义相对论相一致。需要的是重新制定这些定律，以使洛伦兹协方差变得明显。这涉及将所有重要的电磁量识别为四向量或其他类似实体的组成部分，这些实体在洛伦兹变换下表现简单，然后将电磁定律表达为这些实体之间的关系。这就是我们在本节中要做的事情。为了使讨论尽可能简单，我们将仅考虑真空中的电磁学。

2.3.1 The conservation of charge2.3.1 电荷守恒

One of the most fundamental laws of electromagnetism is the conservation of electric charge. Charge can be neither created nor destroyed. If particle physicists perform an experiment in which a positively-charged particle is produced, then an equal amount of negative charge must be produced at the same time. In less extreme circumstances, if the total amount of charge in some region changes, it must be because electric charge has been carried in or out of that region by electric currents. The law of electromagnetism that describes the conservation of electric charge is called the equation of continuity and is usually written as电磁学最基本的定律之一是电荷守恒定律。电荷既不能被创造也不能被消灭。如果粒子物理学家进行实验，产生带正电的粒子，那么必须同时产生等量的负电荷。在不太极端的情况下，如果某个区域的电荷总量发生变化，那一定是因为电荷已被电流带入或带出该区域。描述电荷守恒定律的电磁定律称为连续性方程，通常写为

where \(\rho\) represents the density of electric charge (measured in coulombs per cubic metre) and \(J_x\), \(J_y\) and \(J_z\) are the three components of a vector that represents the electric current density (measured in amperes per square metre). When carefully examined, it turns out that under a Lorentz transformation the charge density and the current density transform as the components of a contravariant four-vector \([J^{\mu}]\) ≡ (J 0, J 1, J 2, J 3) = (cρ, J, J, J), usually called the electric four-current, and the equation of continuity can be written as其中 \(\rho\) 表示电荷密度(以库仑/立方米为单位)，\(J_x\)、\(J_y\) 和 \(J_z\) 是表示电流密度(以安培/平方米为单位)的矢量的三个分量。仔细检查后发现，在洛伦兹变换下，电荷密度和电流密度变换为逆变四向量 [J \(\mu\)] ≡ (J 0, J 1, J 2, J 3) = (cρ, J, J, J) 的分量，通常称为电四电流，连续性方程可以写为

You will recall from our earlier discussion that derivatives transform like a covariant four-vector (the raised index in the denominator acts like a lowered index in the numerator). Consequently, the left-hand side of Equation 2.77 has the form of an invariant formed by contraction, and the right-hand side tells us that it is zero. The relationship is manifestly covariant — it is constructed from four-vectors, and there are no free indices on either side of the equation. So if experiment tells us — which it does — that the equation of continuity is true in some inertial frame S, then the theory of relativity tells us that it will also be true in any other inertial frame \(S'\). We now have our first law of manifestly Lorentz-covariant electromagnetism.您会记得我们之前的讨论中，导数像协变四向量一样进行变换(分母中升高的索引就像分子中降低的索引)。因此，方程 2.77 的左侧具有收缩形成的不变量的形式，右侧告诉我们它为零。这种关系显然是协变的——它是由四矢量量量构建的，并且等式两边都没有自由索引。因此，如果实验告诉我们(事实确实如此)，连续性方程在某个惯性系 S 中成立，那么相对论告诉我们，它在任何其他惯性系 \(S'\) 中也成立。我们现在有了明显洛伦兹协变电磁学的第一定律。

The covariant equation of continuity连续性协变方程

2.3.2 The Lorentz force law2.3.2 洛伦兹力定律

The electrostatic force on a particle of charge q at position r due to another particle of charge Q at position R is given by Coulomb’s law:由于位置 R 处的另一个电荷 Q 粒子对位置 r 处的电荷 q 粒子产生的静电力由库仑定律给出：

where ε 0 is the permittivity of free space (an invariant constant with the value其中 ε 0 是自由空间的介电常数(一个不变常数，其值为

\(8.854\times10^{-12}\) \(\mathrm{C^2\,m^{-2}\,N^{-1}}\)) and d = r − R is the displacement\(8.854\times10^{-12}\) \(\mathrm{C^2\,m^{-2}\,N^{-1}}\)) 且 d = r − R 为位移

vector from the particle of charge Q to the particle of charge q (see Figure从带电 Q 的粒子到带电 q 的粒子的向量(见图

2.9), so d is the2.9), 所以 d 是

distance between the two particles and d E is a unit vector in the direction of d.两个粒子之间的距离，d E 是 d 方向的单位向量。

Knowledge of this force is useful only in some highly specific cases. What is generally of much greater value is knowledge of the electric field E (r). This is a vector field, meaning that it is a function of position that assigns a vector E to each point r throughout some region. At any point r, the assigned vector E is the force per unit charge that would act on a test charge q placed at r:对这种力的了解仅在某些非常具体的情况下才有用。通常更有价值的是电场 E (r) 的知识。这是一个向量场，意味着它是位置的函数，将向量 E 分配给整个某个区域的每个点 r。在任意点 r，指定的矢量 E 是作用于放置在 r 处的测试电荷 q 的每单位电荷的力：

So, once the electric field throughout some region has been determined, the electrostatic force on any test charge q introduced at a point r can be predicted using因此，一旦确定了整个某个区域的电场，就可以使用以下公式来预测在点 r 引入的任何测试电荷 q 上的静电力：

A similar approach may be taken to magnetic forces. This case is somewhat more complicated because the magnetic force on a charged particle generally depends on the particle’s velocity as well as its position and charge. For example, the force on a charge q moving with velocity v through a point r that is at a perpendicular distance d from a long straight wire carrying a current I is given by对于磁力可以采取类似的方法。这种情况稍微复杂一些，因为带电粒子上的磁力通常取决于粒子的速度及其位置和电荷。例如，以速度 v 穿过点 r 的电荷 q 上的力，该点与载有电流 I 的长直导线的垂直距离 d 为

where \(\mu\) 0 is the permeability of free space (an invariant constant with the value 4 \(\pi\) × 10 − 7 N m \(\mathrm{A^{-1}}\)) and \(\theta\) E is a unit vector at right angles to the wire, as indicated in Figure 2.10 overleaf. Note that the symbol × in Equation 2.80 indicates a vector product, so directions are very important if it is to be correctly interpreted.其中 \(\mu\) 0 是自由空间的磁导率(恒定常数，值为 4 \(\pi\) × 10 − 7 N m \(\mathrm{A^{-1}}\))，\(\theta\) E 是与导线成直角的单位向量，如背面图 2.10 所示。请注意，公式 2.80 中的符号 × 表示矢量积，因此如果要正确解释方向，方向非常重要。

Once again, it is useful to have a more general prescription for the force, and this again involves the introduction of a vector field — in this case the magnetic field B (r), which is defined so that at the point r,再一次，对力有一个更通用的规定是有用的，这又涉及到矢量场的引入——在本例中是磁场 B (r)，其定义为在点 r，

Once the magnetic field has been determined throughout some region, the force on any test charge moving through that region can be predicted.一旦确定了某个区域的磁场，就可以预测移动穿过该区域的任何测试电荷所受的力。

Combining these descriptions of electric and magnetic forces, we see that in a region where there is both an electric field and a magnetic field, the electromagnetic force on a particle of charge q travelling with velocity v is given by the Lorentz force law结合这些对电力和磁力的描述，我们看到，在同时存在电场和磁场的区域中，以速度 v 运动的电荷粒子 q 上的电磁力由洛伦兹力定律给出

The role of the vector product can be seen by writing out the individual components of the Lorentz force,通过写出洛伦兹力的各个分量可以看出矢量积的作用，

which can also be written in matrix form as也可以写成矩阵形式

Our aim now is to find a way of rewriting the Lorentz force law in a manifestly covariant way. We should expect the final result to include four-vectors such as the four-force and the four-velocity, but the complexity of the above expressions suggests that something more will be required. The key extra ingredient is a new multi-component entity called the electromagnetic four-tensor or sometimes simply the field tensor. This can be denoted [F \(\mu\)\(\nu\)] and will have 16 components F \(\mu\)\(\nu\) that may be identified from the following:我们现在的目标是找到一种以明显协变的方式重写洛伦兹力定律的方法。我们应该期望最终结果包括四向量，例如四力和四速度，但上述表达式的复杂性表明还需要更多的东西。关键的额外成分是一种新的多分量实体，称为电磁四张量，有时简称为场张量。这可以表示为 [F \(\mu\)\(\nu\)] 并且将具有 16 个分量 F \(\mu\)\(\nu\)，可以通过以下方式识别：

It is unfortunate and potentially confusing that both the four-force and the field tensor are represented by an upper-case F, so we have used different typefaces for the two quantities. It may help that the field tensor will always have two indices while the four-force has only one. Nonetheless, you will need to take care not to confuse the two symbols.不幸的是，四力和场张量都由大写 F 表示，因此可能会造成混淆，因此我们对这两个量使用了不同的字体。场张量总是有两个索引，而四力只有一个索引，这可能会有所帮助。尽管如此，您需要注意不要混淆这两个符号。

Now, the truly remarkable thing about the electromagnetic four-tensor is that it behaves very simply under a Lorentz transformation. The positioning of the indices \(\mu\) and \(\nu\) in the raised contravariant location indicates the exact behaviour. If S and \(S'\) are two, inertial frames in standard configuration, with coordinates related by \(x'\) \(\mu\) = 3 \(\Lambda\) \(\mu\) x \(\nu\), then the field tensor components \(F'\) \(\mu\)\(\nu\) measured in frame \(S'\) will be related to those measured in frame S by现在，电磁四张量真正值得注意的是它在洛伦兹变换下的表现非常简单。索引 \(\mu\) 和 \(\nu\) 在凸起逆变位置的定位指示了确切的行为。如果 S 和 \(S'\) 是标准配置中的两个惯性坐标系，坐标由 \(x'\) \(\mu\) = 3 \(\Lambda\) \(\mu\) x \(\nu\) 相关，则在 \(S'\) 坐标系中测量的场张量分量 \(F'\) \(\mu\)\(\nu\) 将与在 S 坐标系中测量的场张量分量相关：

Given that the fully contravariant field tensor [F \(\mu\)\(\nu\)] does behave in this way, we can use the Minkowski metric to lower one of the indices, giving what is often referred to as the mixed version of the field tensor:鉴于完全逆变场张量 [F \(\mu\)\(\nu\)] 确实以这种方式表现，我们可以使用闵可夫斯基度规来降低其中一个索引，给出通常称为场张量的混合版本：

And then we can use the metric again to lower the remaining index, giving the fully covariant form:然后我们可以再次使用该度规来降低剩余索引，给出完全协变形式：

Performing the sums is tedious and needs care, but the process is straightforward and leads to the result进行求和是乏味且需要小心的，但过程很简单并且会得出结果

Once again we see that, superficially at least, all that the index lowering has achieved is the reversal of some signs; but the real significance is that the fully covariant form of the field tensor transforms differently from the fully contravariant form. Under a Lorentz transformation implemented by the transformation matrix [\(\Lambda\) \(\mu\) \(\nu\)], the fully covariant form transforms with the inverse Lorentz transformation matrix [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] just as a derivative did. Specifically,我们再次看到，至少从表面上看，指数下调所带来的只是一些迹象的逆转；但真正的意义在于，场张量的完全协变形式与完全逆变形式的变换不同。在由变换矩阵 [\(\Lambda\) \(\mu\) \(\nu\)] 实现的洛伦兹变换下，完全协变形式与逆洛伦兹变换矩阵 [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] 进行变换，就像导数一样。具体来说，

These transformations are of great interest in their own right, and their implications for the electric and magnetic fields will be discussed in the next subsection. For the moment, however, we shall simply note that the transformations involve products of elements of the Lorentz transformation matrix or its inverse, and concentrate on the implications of this for the Lorentz force law.这些变换本身就很有趣，它们对电场和磁场的影响将在下一小节中讨论。然而，目前我们只需注意变换涉及洛伦兹变换矩阵或其逆矩阵的元素的乘积，并集中讨论这对洛伦兹力定律的影响。

Now consider the following equation:现在考虑以下等式：

On the left is a contravariant four-force; on the right is the product of an invariant (q), a four-tensor with two contravariant indices (F \(\mu\)\(\nu\)) and a covariant four-vector (U \(\nu\)) — there is a contraction over one raised index and the lowered one. So the right-hand side has only one free index, and that is raised, just like the one free index on the left-hand side. The upshot of all this is that the equation is expressed entirely in terms of entities that transform in simple ways under a Lorentz transformation, and those entities are combined in such a way that both sides of the equation will transform in the same manner. In other words, the given equation is manifestly covariant under Lorentz transformation. (Incidentally, note that in the last sentence we are using ‘covariant’ in the sense of ‘form-invariant’, not in the sense of ‘transforming like a derivative’. It is unfortunate that the word is used in these two ways, but it is a customary practice.)左边是逆变四力；右边是一个不变量 (q)、一个具有两个逆变索引 (F \(\mu\)\(\nu\)) 的四张量和一个协变四向量 (U \(\nu\)) 的乘积 - 一个升高的索引和降低的索引存在收缩。因此，右侧只有一个空闲索引，并且该索引被提升，就像左侧的一个空闲索引一样。所有这一切的结果是，方程完全用在洛伦兹变换下以简单方式变换的实体来表达，并且这些实体以这样的方式组合，使得方程两边将以相同的方式变换。换句话说，给定的方程在洛伦兹变换下是明显协变的。(顺便说一句，请注意，在最后一句中，我们在“形式不变”的意义上使用“协变”，而不是“像导数一样变换”。不幸的是，这个词以这两种方式使用，但这是一种习惯做法。)

Of course, the real reason for our interest in Equation 2.90 is that it provides a covariant formulation of the Lorentz force law. You should convince yourself of this by actually performing the sum and checking the result, but the outcome can be more easily seen by interpreting the sum as the following matrix relationship (take the first index on any element to indicate the row):当然，我们对方程 2.90 感兴趣的真正原因是它提供了洛伦兹力定律的协变公式。您应该通过实际执行求和并检查结果来说服自己这一点，但是通过将求和解释为以下矩阵关系(采用任何元素上的第一个索引来指示行)可以更容易地看到结果：

Note that the v in this expression is the speed of the particle, the magnitude of the velocity v = (\(v_{x}\), \(v_{y}\), \(v_{z}\)). Also note that the negative signs in the right-hand column vector are there because it represents the covariant four-velocity. It is \(\gamma(v)\) on both sides, the clear from the matrix expression that, after cancelling a last three rows reproduce the component expressions of the Lorentz force law that were given earlier.注意，这个表达式中的 v 是粒子的速度，速度的大小 v = (\(v_{x}\), \(v_{y}\), \(v_{z}\))。另请注意，右侧列向量中的负号在那里，因为它代表协变四速度。两边都是 \(\gamma(v)\)，从矩阵表达式可以清楚地看出，取消最后三行后，再现了前面给出的洛伦兹力定律的分量表达式。

What about the first row in the matrix equation? That is the equation矩阵方程的第一行呢？这就是等式

It tells us the rate at which the Lorentz force does work and thereby increases the total energy. It does not contain any surprises, but it reflects the well-known fact that only the electric field is effective in doing work on the particle; this is because the magnetic part of the Lorentz force always acts at right angles to the particle’s velocity.它告诉我们洛伦兹力做功的速率，从而增加总能量。它并不包含任何令人惊讶的内容，但它反映了一个众所周知的事实：只有电场才能有效地对粒子做功；这是因为洛伦兹力的磁性部分始终与粒子的速度成直角。

So we now have a second law of Lorentz-covariant electromagnetism.所以我们现在有了洛伦兹协变电磁学第二定律。

The covariant Lorentz force law协变洛伦兹力定律

Exercise 2.12 The, Lorentz force law may also be, expressed covariantly using练习 2.12 洛伦兹力定律也可以用协变的方式表示

the equation F = q 3 F U \(\nu\), but not F = q 3 F U \(\nu\). Why does the former work, but not the latter?方程 F = q 3 F U \(\nu\)，但不是 F = q 3 F U \(\nu\)。为什么前者有效，而后者无效？

2.3.3 The transformation of electric and magnetic fields2.3.3 电场和磁场的变换

The ‘simple’ transformation law of the electromagnetic four-tensor is vital for the successful formulation of the Lorentz-covariant force law, but it is also of great interest in itself. In particular, it shows that electric and magnetic fields become mixed together in relativity, in a way that is not unlike the mixing of energy and momentum seen earlier. What is an electric field to an observer in frame S will be observed as a combination of electric and magnetic fields by an observer in frame \(S'\). In a relativistic universe, electric and magnetic phenomena are not completely separate. The existence of electric charge, combined with the requirements of special relativity, demands the existence of magnetism.电磁四张量的“简单”变换定律对于洛伦兹协变力定律的成功制定至关重要，但它本身也很有趣。特别是，它表明电场和磁场在相对论中混合在一起，其方式与之前看到的能量和动量的混合没有什么不同。对于 S 系中的观察者来说电场是什么，将被 S 系中的观察者观察到电场和磁场的组合。在相对论宇宙中，电现象和磁现象并不是完全分开的。电荷的存在，结合狭义相对论的要求，需要磁性的存在。

The transformation properties of electric and magnetic fields follow from the transformation properties of the field tensor. We already know that电场和磁场的变换性质源自场张量的变换性质。我们已经知道

and we know that in this fully contravariant case,我们知道，在这个完全逆变的情况下，

In the case where the Lorentz transformation matrix is the usual one, relating frames S and \(S'\) in standard configuration, the transformation is easier than it looks because many of the elements are zero. Even so, we shall not go through the details (you may do that if you wish), but we shall quote the result of a slightly more general Lorentz transformation in which frame \(S'\) has an arbitrary velocity V (not necessarily in the x -direction) in frame S. In this case the transformation rules are usually expressed for field components that are parallel在洛伦兹变换矩阵是通常的情况下，在标准配置中关联框架 S 和 \(S'\)，变换比看起来更容易，因为许多元素为零。即便如此，我们不会详细介绍(如果您愿意，您可以这样做)，但我们将引用稍微更一般的洛伦兹变换的结果，其中框架 \(S'\) 在框架 S 中具有任意速度 V(不一定在 x 方向)。在这种情况下，变换规则通常表示为平行的场分量

(indicated by 1) or perpendicular (indicated by ⊥) to the direction of V:(用 1 表示)或垂直于(用 ⊥ 表示)V 方向：

These equations beautifully illustrate the blending of electricity and magnetism that relativity demands. Looking back at the covariant Lorentz force law, you can see the electromagnetic four-tensor as the mathematical entity required to allow a velocity-dependent force to be consistent with Lorentz covariance. From this point of view, electromagnetism is as simple as it could be.这些方程完美地说明了相对论所要求的电和磁的混合。回顾协变洛伦兹力定律，您可以将电磁四张量视为允许与速度相关的力与洛伦兹协方差一致所需的数学实体。从这个角度来看，电磁学是非常简单的。

Exercise 2.13 Using Equation 2.85 and taking [\(\Lambda\) \(\mu\)练习2.13 使用公式2.85并取[\(\Lambda\) \(\mu\)

\(\nu\)] to represent the usual Lorentz transformation between frames in standard configuration, show that \(E'\) = E.\(\nu\)] 代表标准配置中框架之间通常的洛伦兹变换，表明 \(E'\) = E。

2.3.4 The Maxwell equations2.3.4 麦克斯韦方程组

The remaining laws of vacuum electromagnetism are the Maxwell equations. These are the laws that determine the electric and magnetic fields in a given region. They relate the electric and magnetic fields to the charge and current densities that are their sources, and also to each other since a changing magnetic field can produce an electric field, and a changing electric field can produce a magnetic field.真空电磁学的其余定律是麦克斯韦方程组。这些定律决定了给定区域的电场和磁场。它们将电场和磁场与其来源的电荷和电流密度相关联，并且也将它们彼此相关联，因为变化的磁场可以产生电场，而变化的电场可以产生磁场。

In elementary treatments, the Maxwell equations are usually presented as a set of four differential equations written in the compact language of vector calculus, or sometimes as the equivalent set of eight component equations. This book does not assume any detailed familiarity with the Maxwell equations. The vector calculus versions are shown below, but all that matters mathematically is that the left-hand sides of the equations represent various combinations of partial derivatives of the electric and magnetic field components with respect to the spatial coordinates x, y and z:在初等处理中，麦克斯韦方程通常表示为一组用向量微积分的紧凑语言编写的四个微分方程组，或者有时表示为一组八个分量方程组。本书不假设您对麦克斯韦方程有任何详细的了解。矢量微积分版本如下所示，但在数学上最重要的是方程的左侧表示电场和磁场分量相对于空间坐标 x、y 和 z 的偏导数的各种组合：

where ∇ represents the vector derivative其中 ∇ 表示向量导数

The invariant constants that appear in these equations are not independent. They are linked by the equation \(\mu\) 0 ε 0 \(c^2\) = 1.这些方程中出现的不变常数不是独立的。它们通过方程 \(\mu\) 0 ε 0 \(c^2\) = 1 联络。

The charge and current densities were introduced earlier as components of the current four-vector. The fields, of course, are components of the electromagnetic four-tensor. The way to covariantly construct eight component equations from these ingredients is as follows.电荷和电流密度之前作为电流四矢量量量的组成部分引入。当然，场是电磁四张量的组成部分。从这些成分协变构造八分量方程的方法如下。

The covariant Maxwell equations协变麦克斯韦方程组

The first of these covariant equations has one free index and represents four component equations. These include references to the charge density and the current density, and reproduce Equations 2.97 and 2.100. The interpretation of the second covariant equation is less clear. It has three free indices, which indicates 64 (= 4 × 4 × 4) component equations. However, if any two of the indices are the same, the equation concerned is identically zero. Furthermore, in those cases where all the indices are different, permutations such as \(\lambda\) = 1, \(\mu\) = 2, \(\nu\) = 3 and \(\lambda\) = 2, \(\mu\) = 3, \(\nu\) = 1 lead to the same equation. Taking these symmetries into account, the original 64 component equations are reduced to just four independently meaningful equations. This second set of four component equations reproduces Equations 2.98 and 2.99. Thus, taken together, the two covariant equations reproduce the complete set of Maxwell equations and conclude our rewriting of the laws of vacuum electromagnetism in a manifestly covariant form. All that remains is to draw some lessons that will be of value in future chapters.这些协变方程中的第一个有一个自由索引并表示四个分量方程。其中包括对电荷密度和电流密度的引用，并重现方程 2.97 和 2.100。第二个协变方程的解释不太清楚。它有 3 个自由索引，表示 64 (= 4 × 4 × 4) 个分量方程。然而，如果任意两个指数相同，则相关方程同样为零。此外，在所有索引都不同的情况下，诸如 \(\lambda\) = 1、\(\mu\) = 2、\(\nu\) = 3 和 \(\lambda\) = 2、\(\mu\) = 3、\(\nu\) = 1 等排列会导致相同的方程。考虑到这些对称性，原来的 64 个分量方程被简化为只有 4 个独立有意义的方程。第二组四分量方程再现了方程 2.98 和 2.99。因此，两个协变方程合在一起再现了完整的麦克斯韦方程组，并以明显的协变形式得出了我们对真空电磁定律的重写结论。剩下的就是吸取一些在以后的章节中有价值的教训。

2.3.5 Four-tensors2.3.5 四张量

Exposing the formal simplicity, almost the inevitability, of electromagnetism is one of the great triumphs of special relativity. However, from the point of view of relativity itself, the main development in this chapter has been the introduction of tensors. In this particular chapter the tensors have been called four-tensors. This indicates that they are specific to special relativity. You will meet a much more general class of tensors later, when we move on to general relativity, but a good understanding of four-tensors will be a valuable starting point for that more general experience.揭示电磁学的形式简单性，几乎是必然性，是狭义相对论的伟大胜利之一。然而，从相对论本身的角度来看，本章的主要发展是张量的引入。在这一章中，张量被称为四张量。这表明它们特定于狭义相对论。稍后，当我们转向广义相对论时，您将遇到更一般的张量类别，但是对四张量的良好理解将是更一般经验的宝贵起点。

The only four-tensor that we have formally introduced so far is the electromagnetic four-tensor [F \(\mu\)\(\nu\)] and its variants \([F^{\mu}]\) and [F], but you have already met some others. For instance, the vitally important Minkowski metric [\(\eta\) \(\mu\)\(\nu\)] is a fully covariant four-tensor, and the quantity [\(\eta\) \(\mu\)\(\nu\)] is a fully contravariant four-tensor. Moreover, the term four-tensor is used in such a general sense that these two-indexed examples represent only one particular class of four-tensors — technically referred to as four-tensors of rank 2. All four-vectors are also four-tensors, but they are of rank 1, and it is easy to define four-tensors of rank 3, rank 4, or any higher rank.到目前为止，我们正式介绍的唯一四张量是电磁四张量 [F \(\mu\)\(\nu\)] 及其变体 \([F^{\mu}]\) 和 [F]，但您已经遇到了其他一些。例如，极其重要的闵可夫斯基度规 [\(\eta\) \(\mu\)\(\nu\)] 是一个完全协变的四张量，而数量 [\(\eta\) \(\mu\)\(\nu\)] 是一个完全逆变的四张量。此外，术语“四张量”在一般意义上使用，即这些两个索引的示例仅代表一类特定的四张量 - 技术上称为 2 阶四张量。所有四向量也是四张量，但它们的阶为 1，并且很容易定义 3 阶、4 阶或任何更高阶的四张量。

The defining characteristic of any four-tensor, whatever its rank, is its behaviour under Lorentz transformations. If S and \(S'\) are two inertial frames linked by a general Lorentz transformation (i.e. not necessarily in standard configuration), then we, know that the coordinates in S will be related to those in \(S'\) by [\(\Lambda\) \(\mu\)]\([x^{\nu}]\). (Note that we are now using \(\Lambda\) \(\mu\) in a more general sense than before; we shall have to clarify this shortly.) Under such a general Lorentz rank m consists of 4 m transformation, a four-tensor [T \(\mu\) 1,\((\mu)^2\),...,\(\mu\) m] of contravariant components that transform according to任何四张量的定义特征，无论其等级如何，都是其在洛伦兹变换下的行为。如果 S 和 \(S'\) 是通过一般洛伦兹变换(即不一定在标准配置中)联络的两个惯性系，那么我们知道 S 中的坐标将通过 [\(\Lambda\) \(\mu\)]\([x^{\nu}]\) 与 \(S'\) 中的坐标相关。(请注意，我们现在使用的 \(\Lambda\) \(\mu\) 比以前更广泛；我们必须尽快澄清这一点。)在这样的一般洛伦兹秩 m 包含 4 m 变换下，逆变分量的四张量 [T \(\mu\) 1,\((\mu)^2\),...,\(\mu\) m] 根据以下公式进行变换

Under the same Lorentz transformation, a covariant four-tensor of rank n is a collection of 4 n components that transform according to在相同的洛伦兹变换下，秩为 n 的协变四张量是 4 n 个分量的集合，这些分量根据以下公式进行变换

where [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] is the matrix inverse of [\(\Lambda\) \(\mu\) \(\nu\)] in the usual sense that其中 [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] 是通常意义上的 [\(\Lambda\) \(\mu\) \(\nu\)] 的矩阵逆

A mixed four-tensor of contravariant rank m and covariant rank n consists of 4 m + n components that transform according to逆变秩 m 和协变秩 n 的混合四张量由 4 m + n 个分量组成，这些分量根据以下公式进行变换

All that remains is to specify the elements of the general Lorentz transformation matrix that is the basis of this general definition of a four-tensor. We already \(\Lambda\) 0 0 = \(\gamma(V)\), know that if S and \(S'\) are in standard configuration, then \(\Lambda\) 0 = − \(\gamma(V)\) V/c, \(\Lambda\) 1 = − \(\gamma(V)\) V/c and \(\Lambda\) 1 = \(\gamma(V)\), but what if the inertial frames S and \(S'\) are not in standard configuration? What if the axes are not aligned, for example, or the origin of \(S'\) never passes through the origin of S? What form do the matrix elements take under such general circumstances? We saw earlier, when deriving the Lorentz transformations in Chapter 1, that the primed coordinates have to be linear functions of the unprimed coordinates. In such circumstances, the constants that determine the transformation, the generalized analogues of \(\gamma(V)\) and \(\gamma(V)\) V/c, can be represented by partial derivatives of the coordinates, so the elements of the general Lorentz transformation matrix can be written as剩下的就是指定一般洛伦兹变换矩阵的元素，该矩阵是四张量一般定义的基础。我们已经 \(\Lambda\) 0 0 = \(\gamma(V)\)，知道如果 S 和 \(S'\) 处于标准配置，则 \(\Lambda\) 0 = − \(\gamma(V)\) V/c，\(\Lambda\) 1 = − \(\gamma(V)\) V/c 和 \(\Lambda\) 1 = \(\gamma(V)\)，但是如果惯性呢？车架S和\(S'\)不是标准配置吗？例如，如果轴未对齐，或者 \(S'\) 的原点从未经过 S 的原点，该怎么办？在这种一般情况下矩阵元素采取什么形式？我们之前在第一章推导洛伦兹变换时看到，带底数的坐标必须是未带底数的坐标的线性函数。在这种情况下，决定变换的常数，即 \(\gamma(V)\) 和 \(\gamma(V)\) V/c 的广义类似物，可以用坐标的偏导数来表示，因此一般洛伦兹变换矩阵的元素可以写为

and the elements of the corresponding inverse transformation will be相应逆变换的元素为

Substituting these expressions into Equation 2.107 gives将这些表达式代入公式 2.107 得出

This is the form of the general tensor transformation law that you will meet later. The main difference is that in the case of four-tensors and special relativity, the partial derivatives are all constants that are independent of spacetime position. This will not always be the case in general relativity, as will soon become clear.这就是稍后您将遇到的一般张量变换定律的形式。主要区别在于，在四张量和狭义相对论的情况下，偏导数都是与时空位置无关的常数。在广义相对论中，情况并不总是如此，这一点很快就会变得清楚。

Exercise 2.14 You are told that the 256 -component object [H \(\mu\)\(\nu\)\(\rho\)\(\eta\)] with练习 2.14 你被告知 256 分量对象 [H \(\mu\)\(\nu\)\(\rho\)\(\eta\)] 与

elements H \(\mu\)\(\nu\)\(\rho\)\(\eta\) is a fully covariant four-tensor of rank 4. Write down the general rule for transforming its components from frame S to frame \(S'\).elements H \(\mu\)\(\nu\)\(\rho\)\(\eta\) 是一个完全协变的 4 阶四张量。写下将其分量从框架 S 转换到框架 \(S'\) 的一般规则。

Summary of Chapter 2第 2 章总结

1. Invariants that take the same value in all inertial frames include the speed of1. 在所有惯性系中取相同值的不变量包括

light in a vacuum, the spacetime separation between events, the proper time between time-like separated events, the charge of a particle and the mass of a particle.真空中的光、事件之间的时空分离、类似时间的分离事件之间的固有时间、粒子的电荷和粒子的质量。

2. The principle of relativity demands that the laws of physics should be2. 相对性原理要求物理定律应该是

form-invariant under Lorentz transformations. Such laws are said to be Lorentz-covariant.洛伦兹变换下的形式不变。这些定律被称为洛伦兹协变。

3. The relativistic momentum of a particle of mass m and velocity v is3. 质量为 m、速度为 v 的粒子的相对论动量为

4. The relativistic kinetic energy of a particle of mass m and speed v is4. 质量为 m、速度为 v 的粒子的相对论动能为

5. The total relativistic energy of a particle of mass m and speed v is5. 质量为 m、速度为 v 的粒子的总相对论能量为

where \(E_0\) = \(mc^2\) is the mass energy of the particle.其中 \(E_0\) = mc 2 是粒子的质量能。

6. In the absence of external forces, relativistic total energy is conserved, but6.在没有外力的情况下，相对论总能量守恒，但

neither kinetic energy nor mass energy is necessarily conserved. This establishes an ‘equivalence’ of mass and energy, with many important consequences.neither kinetic energy nor mass energy is necessarily conserved.这建立了质量和能量的“等价”，并产生许多重要的后果。

7. The four-momentum \([P^{\mu}]\) = (E/c, p, p, p) brings together momentum7. 四动量 \([P^{\mu}]\) = (E/c, p, p, p) 汇聚动量

and energy. It transforms in the same way as a four-displacement:和能量。它的变形方式与四排量相同：

m is米是

8. The energy–momentum relation for a particle of mass8. 质量粒子的能量-动量关系

showing that for a massless particle p = E/c.表明对于无质量粒子 p = E/c。

9. Laws of conservation of total energy and momentum9.总能量和动量守恒定律

are combined in a manifestly covariant law of four-momentum conservation.结合在一个明显的四动量守恒协变定律中。

10. The four-force \([F^{\mu}]\) = ((γ/c) f · v, γ f) determines10. 四力 \([F^{\mu}]\) = ((γ/c) f · v, γ f) 确定

the rate of change of a particle’s four-momentum with respect to proper time. It transforms like the four-momentum, placing restrictions on the acceptable expressions for the three-force f. The electromagnetic Lorentz force meets these requirements; Newton’s gravitational force does not. 3 \(\nu\) =0 \(\Lambda\) \(\mu\) \(\nu\) x \(\nu\), a粒子四动量相对于原时间的变化率。它像四动量一样变换，对三力 f 的可接受表达式施加限制。电磁洛伦兹力满足这些要求；牛顿的万有引力则不然。 3 \(\nu\) =0 \(\Lambda\) \(\mu\) \(\nu\) x \(\nu\)，一个

11. Under a Lorentz transformation in which \(x'\) \(\mu\) =11. 在洛伦兹变换下，其中 \(x'\) \(\mu\) =

contravariant four-vector \([A^{\mu}]\) transforms in the same way as a four-displacement:逆变四向量 \([A^{\mu}]\) 的变换方式与四位移相同：

Under the same Lorentz transformation, a covariant transforms in the same way as a set of derivatives:在相同的洛伦兹变换下，协变的变换方式与一组导数相同：

where [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] is the matrix inverse of [\(\Lambda\) \(\mu\) \(\nu\)]. In the case of two frames in standard configuration,其中 [(\(\Lambda\) − 1) \(\mu\) \(\nu\)] 是 [\(\Lambda\) \(\mu\) \(\nu\)] 的矩阵逆。标准配置两框的情况下，

12. Indices on four-vectors may be lowered or raised12. 四向量指数可降低或提高

using the, Minkowski by \(\nu\) \(\eta\) \(\alpha\)\(\nu\) \(\eta\) \(\nu\)\(\beta\) = δ \(\alpha\) \(\beta\): metric \(\eta\) \(\mu\)\(\nu\) or the related inverse quantity \(\eta\) \(\mu\)\(\nu\) defined使用闵可夫斯基通过 \(\nu\) \(\eta\) \(\alpha\)\(\nu\) \(\eta\) \(\nu\)\(\beta\) = δ \(\alpha\) \(\beta\)：公制 \(\eta\) \(\mu\)\(\nu\) 或相关的反数量\(\eta\) \(\mu\)\(\nu\) 已定义

and和

13. Contraction involves summing over one raised and13. 收缩涉及将一个升高的和

one lowered index, and may be used to form invariants as in一个降低的索引，可用于形成不变量，如

14. The Lorentz-covariant laws of electromagnetism are:14. 电磁学的洛伦兹协变定律是：

the covariant equation of continuity连续性协变方程

the covariant Lorentz force law协变洛伦兹力定律

the covariant Maxwell equations协变麦克斯韦方程组

where \([J^{\mu}]\) = (cρ, J, J, J) is the contravariant current four-vector, and [F \(\mu\)\(\nu\)] is the fully contravariant electromagnetic four-tensor given by其中 \([J^{\mu}]\) = (cρ, J, J, J) 是逆变电流四矢量量量，[F \(\mu\)\(\nu\)] 是完全逆变电磁四张量，由下式给出

15. Under a Lorentz transformation, the electromagnetic four-tensor transforms15. 在洛伦兹变换下，电磁四张量变换

according to根据

This leads to the following transformation laws for the electric and magnetic fields:这导致了以下电场和磁场的变换定律：

16. Under a general Lorentz transformation, the components of a four-tensor16. 在一般洛伦兹变换下，四张量的分量

transform according to变换根据

Chapter 3 Geometry and curved spacetime第三章几何与弯曲时空

Introduction介绍

Einstein’s 1905 theory of special relativity concerns relationships between observations made by inertial observers in uniform relative motion. As you saw in the previous chapter, the theory is inconsistent with Newtonian gravitation. In 1907, in what he later described as ‘the happiest thought of my life’, Einstein realized that a theory of general relative motion — one that included relationships between observations made by accelerated observers — would also shed light on the problem of gravitation. It was not long after this that Minkowski introduced his four-dimensional spacetime approach to special relativity, which revealed the geometric basis of the theory. Under these influences, Einstein’s own thinking took on an increasingly geometric flavour, and by the middle of 1912 he realized that to make further progress in relativity and gravitation, he needed to find out what mathematicians knew about certain problems concerning invariants in geometry. At that point he asked his friend, the mathematician Marcel Grossman (1878–1936), to help him to find the required information. Grossman was soon able to tell Einstein that what he was looking for was contained in the subject known as Riemannian geometry — a branch of mathematics particularly concerned with the study of curved spaces.爱因斯坦 1905 年提出的狭义相对论关注的是惯性观察者在匀速相对运动中所做的观察之间的关系。正如您在上一章中看到的，该理论与牛顿引力不一致。1907 年，爱因斯坦意识到广义相对运动理论(包括加速观察者的观测结果之间的关系)也将揭示引力问题，后来他将其描述为“我一生中最幸福的想法”。不久之后，闵可夫斯基将他的四维时空方法引入了狭义相对论，揭示了该理论的几何基础。在这些影响下，爱因斯坦自己的思想越来越带有几何色彩，到1912年中期，他意识到，为了在相对论和万有引力方面取得进一步进展，他需要了解数学家对几何不变量的某些问题的了解。此时，他请他的朋友、数学家马塞尔·格罗斯曼(Marcel Grossman，1878-1936)帮助他找到所需的信息。格罗斯曼很快就告诉爱因斯坦，他正在寻找的东西包含在被称为黎曼几何的学科中——这是一个特别关注弯曲空间研究的数学分支。

Geometry is the study of shape and spatial relationships. The kind of geometry taught in high schools is known as Euclidean geometry, after Euclid of Alexandria who collected together the main results of the field in around 300 BC. Among the best known of those results (see Figure 3.1) are:几何学是对形状和空间关系的研究。高中教授的几何​​学被称为欧几里得几何，得名于亚历山大的欧几里得，他在公元前 300 年左右收集了该领域的主要成果。其中最著名的结果(见图 3.1)包括：

• • the internal angles of a triangle add up to 180 ◦• 三角形的内角之和为 180 °
• • a circle of radius R has a circumference of length C• 半径为 R 的圆的周长为 C
• • a sphere of radius R has a surface area A = 4 πR 2.• 半径为 R 的球体的表面积 A = 4 πR 2。

It was long thought that Euclidean geometry was the only kind of geometry, and that these results would therefore apply to all triangles, circles and spheres. mathematicians, Ja´nos However, in the first half of the nineteenth century, three Bolyai (1802–1860), Nikolai Lobachevsky (1792–1856), and Carl Friedrich Gauss (1777–1855; Figure 3.2), independently established that it was possible to formulate a kind of geometry that made mathematical sense but was quite different from traditional Euclidean geometry. In non-Euclidean geometry, none of the Euclidean results quoted above is necessarily true.长期以来，人们一直认为欧几里得几何是唯一的几何类型，因此这些结果适用于所有三角形、圆形和球体。然而，在十九世纪上半叶，三位博利亚伊(Bolyai，1802-1860)、尼古拉·洛巴切夫斯基(Nikolai Lobachevsky，1792-1856)和卡尔·弗里德里希·高斯(Carl Friedrich Gauss，1777-1855；图 3.2)独立地确立了建立一种具有数学意义但又与传统欧几里得几何截然不同的几何的可能性。在非欧几里德几何中，上面引用的欧几里德结果都不一定是正确的。

The realization that there was more than one kind of geometry meant that determining the geometric properties of the space around us was an experimental question, not just a mathematical one. Lobachevsky considered the possibility of using astronomical measurements to determine the true geometry of space, but concluded that they would not be sufficiently accurate. Gauss became involved in a land survey and examined the angles of the large triangle between three mountain tops. He failed to find any sign of non-Euclidean geometry, but he too realized that this might simply reflect the limited sensitivity of the technique that he was using.认识到存在不止一种几何形状意味着确定我们周围空间的几何特性是一个实验问题，而不仅仅是一个数学问题。罗巴切夫斯基考虑了使用天文测量来确定空间真实几何形状的可能性，但得出的结论是它们不够准确。高斯参与了一次土地测量，并检查了三个山顶之间的大三角形的角度。他未能找到任何非欧几里得几何的迹象，但他也意识到这可能只是反映了他所使用的技术的有限灵敏度。

Gauss was one of the greatest of all mathematicians. His many discoveries included several important contributions to the development of geometry. Not least was his part in helping to found differential geometry, the branch of mathematics that applies the techniques of calculus to the analysis of geometric problems. It was in furthering this subject that Gauss’s assistant Bernhard Riemann (1826–1866; Figure 3.3) introduced the geometry that now bears his name.高斯是最伟大的数学家之一。他的许多发现包括对几何学发展的几项重要贡献。尤其重要的是，他帮助建立了微分几何，这是将微积分技术应用于几何问题分析的数学分支。正是在进一步推进这一主题的过程中，高斯的助手伯恩哈德·黎曼(Bernhard 黎曼，1826-1866；图 3.3)引入了现在以他的名字命名的几何学。

Riemann (1826–1866), a protege´ of Gauss, was a great mathematician in his own right and the founder of Riemannian geometry.黎曼(1826-1866)是高斯的弟子，也是一位伟大的数学家，也是黎曼几何的创始人。

has been properly explained, it will be sufficient to think of a ‘flat’ space as one in which the conventional Euclidean geometrical results hold true, and a ‘curved’ space as one in which they fail. Note that the terms ‘flat’ and ‘curved’ are used to describe geometric properties and may be applied to spaces with any number of dimensions. They do not simply mean ‘curved like a bow’ or ‘flat like a pancake’.已经得到了正确的解释，将“平坦”空间视为传统欧几里得几何结果成立的空间，将“弯曲”空间视为它们失败的空间就足够了。请注意，术语“平坦”和“弯曲”用于描述几何属性，并且可以应用于具有任意维数的空间。它们不仅仅意味着“像弓一样弯曲”或“像煎饼一样平坦”。

3.1 Line elements and differential geometry3.1 线元和微分几何

3.1.1 Line elements in a plane3.1.1 平面内的线单元

In order to analyze the geometry of curved space, we need to clarify what we mean by the length of a curve. Figure 3.4 shows a smooth curve C linking two points P and Q in an ordinary (Euclidean) plane. The plane is equipped with Cartesian coordinates so that each point on the curve can be assigned coordinates (x, y). The length of the curve can be approximately determined by dividing it into n short segments, each of which can be regarded as a straight line of length \(\Delta l\) (i = 1, 2,..., n), and then adding together the lengths of all those short straight lines. The approximate length of the curve C from P to Q will then be given by为了分析弯曲空间的几何形状，我们需要澄清曲线长度的含义。图 3.4 显示了在普通(欧几里得)平面中联络两点 P 和 Q 的平滑曲线 C。该平面配备有笛卡尔坐标，因此曲线上的每个点都可以指定坐标(x，y)。曲线的长度可以通过将其分成 n 个短段来近似确定，每个短段可以视为长度为 \(\Delta l\) (i = 1, 2,..., n) 的直线，然后将所有这些短直线的长度相加。从 P 到 Q 的曲线 C 的近似长度将由下式给出

According to Pythagoras’s theorem, which is one of the fundamental results of Euclidean geometry, the length \(\Delta l\) of the straight line linking two points separated by the coordinate intervals \(\Delta x\) and \(\Delta y\) (see Figure 3.5) is given by根据毕达哥拉斯定理(欧几里德几何的基本结果之一)，联络由坐标间隔 \(\Delta x\) 和 \(\Delta y\) 分隔的两点的直线长度 \(\Delta l\)(见图 3.5)由下式给出：

Decreasing the length of those short segments will increase their number and improve the accuracy of the approximation to the total length of C. Taking the limit as \(\Delta x\) → 0 and \(\Delta y\) → 0, the sum will become an integral, and we can write the length of curve C from P to Q as减少这些短线段的长度会增加它们的数量，提高逼近C总长度的精度。以Δ x→0和Δ y→0为极限，其和将成为一个积分，我们可以将曲线C从P到Q的长度写为

where the line element, d l, is defined by其中线元素 d l 定义为

or或者

Unfortunately, this is not enough to let us actually work out the length of C; we need to know how to perform such an integral. In particular, in order to add up all the line elements along the curve, we need to take account of their differing directions, which will cause each element d l to correspond to differently-sized increments in the x - and y -directions.不幸的是，这还不足以让我们实际计算出 C 的长度；我们需要知道如何进行这样的积分。特别是，为了将沿曲线的所有线元素相加，我们需要考虑它们的不同方向，这将导致每个元素 d l 对应于 x 和 y 方向上不同大小的增量。

One powerful way of taking the shape of C into account involves representing it as a parameterized curve. This requires that every point on the curve should be identified with a unique value of some continuously varying parameter, u say, so that the x - and y -coordinates of any particular point on the curve represent specific values of two coordinate functions x (u) and y (u) that effectively define the curve. So, for example:考虑 C 形状的一种有效方法是将其表示为参数化曲线。这要求曲线上的每个点都应该用某个连续变化的参数 u 的唯一值来标识，以便曲线上任何特定点的 x 和 y 坐标代表有效定义曲线的两个坐标函数 x (u) 和 y (u) 的特定值。因此，例如：

• • the parabola y = \(x_{2}\) can be described in terms of a parameter u by the抛物线 y = \(x_{2}\) 可以用参数 u 来描述：

functions x (u) = u, y (u) = \(u^{2}\)函数 x (u) = u, y (u) = \(u^{2}\)

• • the circle \(x_{2}\) + \(y^{2}\) = 1 can be described in terms of a parameter u by the圆 \(x_{2}\) + \(y^{2}\) = 1 可以用参数 u 来描述：

functions x (u) = cos(u), y (u) = sin(u).函数 x (u) = cos(u), y (u) = sin(u)。

(Notice how in the first example, it is easy to parameterize a single-valued function y = f (x): we just write x (u) = u and y (u) = f (u).)(请注意，在第一个示例中，参数化单值函数 y = f (x) 很容易：我们只需写 x (u) = u 和 y (u) = f (u)。)

Adopting this parametric approach, it’s clear that any two points on the curve C that are separated by coordinate intervals \(\Delta x\) and \(\Delta y\), will also be separated by some corresponding parameter interval \(\Delta u\), and we can say that采用这种参数化方法，很明显，曲线 C 上被坐标区间 \(\Delta x\) 和 \(\Delta y\) 分开的任何两点，也将被某个相应的参数区间 \(\Delta u\) 分开，我们可以说

and和

As \(\Delta u\) → 0 (so that \(\Delta x\) → 0 and \(\Delta y\) → 0), the fractions \(\Delta x\)/\(\Delta u\) and \(\Delta y\)/\(\Delta u\) become the derivatives d x/d u and d y/d u of x (u) and y (u) with respect to u, and it follows that当 \(\Delta u\) → 0(使得 \(\Delta x\) → 0 和 \(\Delta y\) → 0)时，分数 \(\Delta x\)/\(\Delta u\) 和 \(\Delta y\)/\(\Delta u\) 变为 x (u) 和 y (u) 对 u 的导数 d x/d u 和 d y/d u，得出

and hence, from Equation 3.5,), y (u)) to So, finally, the length of the curve C from P = (x (u Q = (x (u), y (u)) is given by the following.因此，从方程 3.5,), y (u)) 到 So，最后，由 P = (x (u Q = (x (u), y (u))) 得出的曲线 C 的长度由下式给出。

Length of a curve in a Euclidean plane欧几里得平面中曲线的长度

Once we know the functions x (u) and y (u) that parameterize the curve C, and the values of u that correspond to the points P and Q, this expression for the length of a curve between two points in a Euclidean plane really can be evaluated. It is our first major result in this chapter.一旦我们知道参数化曲线 C 的函数 x (u) 和 y (u)，以及对应于点 P 和 Q 的 u 值，就可以真正评估欧几里得平面中两点之间的曲线长度表达式。这是我们本章的第一个主要成果。

Worked Example 3.1工作示例 3.1

(a) Parameterize the straight line y = 2(6 x + 1).(a) 参数化直线 y = 2(6 x + 1)。

(b) Using the line element method described above, calculate the length of the line from (0, 2) to (5, 14). Check your result using Pythagoras’s theorem.(b) 使用上述线元法，计算从 (0, 2) 到 (5, 14) 的线段长度。使用毕达哥拉斯定理检查您的结果。

Solution解决方案

is x = u, (a) This is a single-valued function, so a suitable parameterization y = 2(6 u + 1).是 x = u, (a) 这是一个单值函数，因此合适的参数化 y = 2(6 u + 1)。

(b) Differentiating with respect to u, we obtain(b) 对 u 求导，我们得到

Since x = u, we have u (0, 2) = 0 and u (5, 14) = 5, so Equation 3.6 gives由于 x = u，我们有 u (0, 2) = 0 和 u (5, 14) = 5，因此方程 3.6 给出

Pythagoras’s theorem gives the same answer:毕达哥拉斯定理给出了同样的答案：

Worked Example 3.2工作示例 3.2

Parameterize the circle \(x_{2}\) + \(y^{2}\) = \(R^2\), and find the length of the circumference in terms of the (constant) radius R.将圆 \(x_{2}\) + \(y^{2}\) = \(R^2\) 参数化，并根据(常数)半径 R 求出圆周的长度。

Solution解决方案

x (u) = R cos(u) and The simplest way to parameterize the circle is to set y (u) = R sin(u), as given earlier. Differentiating with respect to u, we obtainx (u) = R cos(u) 并且参数化圆的最简单方法是设置 y (u) = R sin(u)，如前所述。对 u 求导，我们得到

To get the circumference C, we need to let u vary from 0 to 2 \(\pi\), so using \(\sin^2 u\) + \(\cos^2 u\) = 1, we have为了得到周长 C，我们需要让 u 在 0 到 2 \(\pi\) 之间变化，因此使用 \(\sin^2 u\) + \(\cos^2 u\) = 1，我们有

Exercise 3.1 (a) Sketch the curve parameterized by x = 3 \(u^{2}\), y = 4 \(u^{2}\). (b) Calculate the length L of the curve from u = 0 to u = 3.练习 3.1 (a) 绘制由 x = 3 \(u^{2}\), y = 4 \(u^{2}\) 参数化的曲线。 (b) 计算从 u = 0 到 u = 3 的曲线长度 L。

There are always many ways to parameterize a curve, but it is usually best to choose the simplest. For example, in Exercise 3.1 we used the parameterization x = 3 \(u^{2}\), y = 4 \(u^{2}\), but this gives us no particular benefit and it would be simpler to use x = 3 u, y = 4 u. For the circle in Worked Example 3.2, another possibility is x = u, y = ± (\(R^2\) − \(u^{2}\)) 1/2, but this would make the calculations much more difficult.参数化曲线的方法总是有很多种，但通常最好选择最简单的。例如，在练习 3.1 中，我们使用了参数化 x = 3 \(u^{2}\), y = 4 \(u^{2}\)，但这并没有给我们带来特别的好处，而且使用 x = 3 u, y = 4 u 会更简单。对于工作示例 3.2 中的圆，另一种可能性是 x = u, y = ± (\(R^2\) – \(u^{2}\)) 1/2，但这会使计算变得更加困难。

When dealing with a general curve in the plane, instead of Cartesian coordinates, it is often more convenient to use plane polar coordinates (r, \(\phi\)), which can be defined in terms of (x, y) by当处理平面上的一般曲线时，使用平面极坐标 (r, \(\phi\)) 代替笛卡尔坐标通常更方便，它可以用 (x, y) 来定义：

as shown in Figure 3.6. Note that r is now a variable (not the constant radius R of Worked Example 3.2), so we can define any point in the plane by the coordinates (r, \(\phi\)), where r is the distance from the origin measured along a line that makes an angle \(\phi\) with the x -axis.如图3.6所示。请注意，r 现在是一个变量(不是工作示例 3.2 中的恒定半径 R)，因此我们可以通过坐标 (r, \(\phi\)) 定义平面中的任何点，其中 r 是沿着与 x 轴形成角度 \(\phi\) 的直线测量到的距原点的距离。

Using the rule for differentiating a product, it follows from the above definitions that使用区分产品的规则，从上述定义可以得出：

and so, from Equation 3.4, the line element in a Euclidean plane is also given by因此，根据方程 3.4，欧几里得平面中的线元素也由下式给出

This too is indicated in Figure 3.6.图 3.6 也表明了这一点。

Exercise 3.2 Use the parameterization r = R(a constant) and \(\phi\) = u练习 3.2 使用参数化 r = R(常数)和 \(\phi\) = u

(a variable parameter) together with Equation 3.7 to again find the circumference C of a circle of radius R.(可变参数)结合公式 3.7 再次求出半径为 R 的圆的周长 C。

3.1.2 Curved surfaces3.1.2 曲面

The differential approach to geometry that we have just been using can be generalized to higher dimensions. In three-dimensional Euclidean space with Cartesian coordinates, the definition of the line element in Equation 3.4 generalizes to我们刚刚使用的几何微分方法可以推广到更高的维度。在笛卡尔坐标的三维欧几里得空间中，方程 3.4 中线元素的定义可推广为

In spherical coordinates, as illustrated in Figure 3.7, x, y, z can be written as在球坐标系中，如图 3.7 所示，x、y、z 可以写为

which leads, after some algebra, to经过一些代数运算后，得出

Using these alternative expressions for the line element, we can give meaning to the length of a curve in three-dimensional Euclidean space, and from there we could start to build up the whole of three-dimensional Euclidean geometry, just as we started to do in the two-dimensional case. As Gauss realized, these line elements are really the key to unlocking an entire geometry.使用线元素的这些替代表达式，我们可以赋予三维欧几里德空间中曲线的长度含义，并且从那里我们可以开始构建整个三维欧几里德几何，就像我们在二维情况下开始做的那样。正如高斯意识到的那样，这些线元素确实是解锁整个几何图形的关键。

One topic that we can investigate is the geometry of two-dimensional surfaces in three-dimensional space. If, in Equation 3.9, we set r equal to a constant, R, then we are restricting ourselves to the surface of a sphere of radius R, and the equation for the line element reduces to我们可以研究的主题之一是三维空间中二维表面的几何形状。如果在方程 3.9 中，我们将 r 设置为常数 R，那么我们将自己限制在半径为 R 的球体表面，并且线元素的方程简化为

There are just two variables in Equation 3.10, \(\theta\) and \(\phi\), so it really does describe the geometry of a two-dimensional space. But the geometry of this two-dimensional space — the surface of the sphere — differs significantly from that of the plane, as the following example shows.方程 3.10 中只有两个变量：\(\theta\) 和 \(\phi\)，因此它确实描述了二维空间的几何形状。但这个二维空间的几何形状(球体的表面)与平面的几何形状显着不同，如以下示例所示。

Worked Example 3.3工作示例 3.3

Figure 3.8 shows a sphere of radius R and a spherical coordinate system. Suppose that we draw a circle on the sphere by sweeping round the ‘north pole’ at a fixed angle \(\theta\). Starting from Equation 3.10, find the length of the circumference C of the circle.图 3.8 显示了半径为 R 的球体和球坐标系。假设我们以固定角度 \(\theta\) 绕“北极”扫过，在球体上画了一个圆。从公式 3.10 开始，求出圆的周长 C。

Solution解决方案

Since \(\theta\) is constant, Equation 3.10 tells us that a line element along the circle’s circumference is given by \(dl^2\) = \(R^2\) \(\sin^2 \theta\) d \(\phi\) 2. Adding together (i.e. integrating) all the line elements around the circle is easy in this case, since each one points in the direction of increasing \(\theta\), so the circumference is由于 \(\theta\) 是常数，方程 3.10 告诉我们，沿圆圆周的线元由 \(dl^2\) = \(R^2\) \(\sin^2 \theta\) d \(\phi\) 2 给出。在这种情况下，将圆周围的所有线元加在一起(即积分)很容易，因为每个线元都指向 \(\theta\) 增加的方向，因此周长为

If the geometry of a spherical surface were the same as that of a plane, we would expect the circumference C to be 2 \(\pi\) times the radius of the circle, with both the circumference and the radius measured in the spherical surface. The radius measured in the spherical surface is Rθ, so the geometry of a plane would lead us to expect C = 2 πRθ. However, as the worked example showed, the circumference of the circle on the sphere is actually C = 2 πR sin \(\theta\), which is less than plane geometry implies. So the geometry of a spherical surface is different from that of a plane. This has been well known to mathematicians and navigators for a long time. (Euclid used spherical geometry in his writings on astronomy.) But its real significance was not properly appreciated until the discovery of non-Euclidean geometry (now sometimes called hyperbolic geometry) caused mathematicians to reconsider the nature of geometry in general.如果球面的几何形状与平面的几何形状相同，则我们预计周长 C 为 2 \(\pi\) 乘以圆的半径，并且周长和半径都是在球面中测量的。在球面中测量的半径为 Rθ，因此平面的几何形状将使我们预期 C = 2 πRθ。然而，正如工作示例所示，球体上圆的周长实际上是 C = 2 πR sin \(\theta\)，这小于平面几何所暗示的。因此球面的几何形状与平面的几何形状不同。长期以来，这一点已为数学家和航海家所熟知。(欧几里得在他的天文学著作中使用了球面几何。)但直到非欧几里得几何(现在有时称为双曲几何)的发现促使数学家重新考虑一般几何的本质，它的真正意义才得到适当的认识。

We shall not try to formulate spherical geometry here, but it is worth noting some key points that will be of significance later. A topic of great interest in spherical geometry is the behaviour of triangles. Obviously, there are no straight lines on a spherical surface, so before we can discuss spherical triangles, we need to know what are the spherical analogues of straight lines from which such triangles can be constructed. On a spherical surface this special role is played by the arcs of great circles. A great circle is a curve on the surface of a sphere created by the intersection of the sphere and a plane that passes through its centre. (On the Earth, the equator is an example of a great circle, and so are the meridian circles that pass through the North and South Poles.) In a Euclidean plane, the shortest path between any two points is the straight line that joins them. Similarly, on the surface of a sphere, the shortest path between any two points is the minor (i.e. shorter) arc of the great circle that passes through those points.我们不会在这里尝试制定球面几何，但值得注意的是一些稍后将具有重要意义的关键点。球面几何中一个非常有趣的话题是三角形的行为。显然，球面上不存在直线，因此在讨论球面三角形之前，我们需要知道可以构造此类三角形的直线的球面类似物是什么。在球面上，这种特殊的作用是由大圆弧发挥的。大圆是球体表面上的一条曲线，由球体与穿过其中心的平面相交而形成。(在地球上，赤道是大圆的一个例子，穿过北极和南极的子午线圆也是如此。)在欧几里得平面中，任意两点之间的最短路径是联络它们的直线。类似地，在球体表面上，任意两点之间的最短路径是穿过这些点的大圆的短(即较短)弧。

Figure 3.9 shows a spherical triangle constructed from the minor arcs of three great circles. In this case the spherical triangle is a rather special one since each of the interior angles is a right angle, but this illustrates another important difference between spherical geometry and plane geometry: the sum of the interior angles of a spherical triangle is greater than 180 ◦.图 3.9 显示了由三个大圆的短弧构成的球面三角形。在这种情况下，球面三角形是一个相当特殊的三角形，因为每个内角都是直角，但这说明了球面几何与平面几何之间的另一个重要区别：球面三角形的内角之和大于 180°。

What lies behind the differences between the geometries of a plane and a sphere is the simple fact that the plane is flat while the surface of a sphere is curved. At this stage it is easy to believe that the spherical surface is curved because we can ‘see’ it as a curved two-dimensional surface in a three-dimensional Euclidean space, but this is not generally a reliable guide nor is such visual information always obtainable. Later, a mathematical definition of curvature will be introduced that will confirm the curvature of the spherical surface. However, it’s important to note that we now have two tests for the presence of curvature that do not depend on being able to ‘see’, or even imagine, the curved surface in a space of higher dimension. Using the appropriate two-dimensional line element, we can compare the circumference of a circle with 2 \(\pi\) times the radius, or we can construct a triangle (using paths of shortest length as sides) and compare the sum of the interior angles with 180 ◦. Each of these tests for curvature could be carried out by two-dimensional beings — traditionally called bugs — who live on the two-dimensional surface and have no concept of any higher-dimensional space. From a mathematical point of view this is an indication that curvature is an intrinsic property of a surface that can be determined from measurements made in the surface, rather than an extrinsic property that depends on measurements made in some higher dimension.平面和球体的几何形状差异背后隐藏着一个简单的事实：平面是平坦的，而球体的表面是弯曲的。在这个阶段，很容易相信球面是弯曲的，因为我们可以将其“视为”三维欧几里得空间中的弯曲二维表面，但这通常不是可靠的指导，也并不总是可以获得这样的视觉信息。随后，将引入曲率的数学定义来确认球面的曲率。然而，值得注意的是，我们现在有两个关于曲率存在的测试，这些测试并不依赖于能够“看到”，甚至想象更高维度空间中的曲面。使用适当的二维线元素，我们可以将圆的周长与 2 \(\pi\) 乘以半径进行比较，或者我们可以构造一个三角形(使用最短长度的路径作为边)并将内角和与 180 ° 进行比较。这些曲率测试中的每一个都可以由二维生物(传统上称为虫子)进行，它们生活在二维表面上，没有任何高维空间的概念。从数学的角度来看，这表明曲率是表面的内在属性，可以通过在表面中进行的测量来确定，而不是取决于在某些更高维度中进行的测量的外在属性。

It is important to be aware of the intrinsic nature of curvature and our ability to detect it for at least three reasons. First, unlike spherical surfaces, not all surfaces that are of mathematical interest can be reproduced (the proper term is embedded) in three-dimensional Euclidean space. The ‘hyperbolic’ surface of the original non-Euclidean geometry is of this kind. The geometry exists, but the two-dimensional surface to which it applies cannot be embedded in three-dimensional Euclidean space. Second, when we come to deal with the curvature of the physical four-dimensional spacetime in which we live, it’s very hard to imagine that we might successfully visualize it as existing within some other space or spacetime of even higher dimension. Third, not everything that appears curved in three dimensions really is curved in the mathematical sense. This last point is illustrated by the example of the cylinder given below.出于至少三个原因，了解曲率的内在本质以及我们检测曲率的能力非常重要。首先，与球面不同，并非所有具有数学意义的表面都可以在三维欧几里德空间中再现(嵌入了专有术语)。原始非欧几里得几何的“双曲”曲面就是这种类型。几何体存在，但它所应用的二维表面不能嵌入三维欧几里得空间中。其次，当我们处理我们所生活的物理四维时空的曲率时，很难想象我们可以成功地将其想象为存在于其他更高维度的空间或时空中。第三，并非所有在三维空间中出现弯曲的东西实际上都是数学意义上的弯曲。下面给出的圆柱体示例说明了最后一点。

A cylinder is formed by taking a strip of a plane, say the \(xy\)-plane from x = a to x = b, and rolling it up so that the line x = a becomes identified with the line x = b, as shown in Figure 3.10. Before rolling up the strip, we can draw on it a circle with radius r and circumference 2 πr. We can also draw a triangle whose interior angles add up to 180 ◦. These two features don’t change when we roll up the strip of the plane, so our two-dimensional bugs carrying out local measurements of distances and angles would not be able to detect what we see as extrinsic curvature due to the rolling up in a third dimension. The process of ‘rolling up’ is what enables us to embed the cylindrical surface in three-dimensional space, but it does not produce any intrinsic curvature at all. In fact, the geometry of the cylinder is intrinsically flat.圆柱体是通过取一条平面(例如从 x = a 到 x = b 的 xy 平面)并将其卷起以使线 x = a 与线 x = b 一致而形成的，如图 3.10 所示。在卷起长条之前，我们可以在上面画一个半径为r、周长为2πr的圆。我们还可以画一个内角和为 180° 的三角形。当我们卷起平面条带时，这两个特征不会改变，因此我们进行距离和角度局部测量的二维错误将无法检测到由于在三维中卷起而导致的外在曲率。 “卷起”的过程使我们能够将圆柱表面嵌入三维空间中，但它根本不产生任何内在曲率。事实上，圆柱体的几何形状本质上是平坦的。

We can approach this idea more mathematically by using the appropriate two-dimensional line elements. The length L of the straight line from P to Q in the plane is given by \(L^{2}\) = (\(\Delta x\)) 2 + (\(\Delta y\)) 2, reminding us that the line element in a plane, expressed in Cartesian coordinates, is \(dl^2\) = \((dx)^2\) + \(dy^2\). Using the cylindrical coordinates (z, \(\phi\)) shown in Figure 3.10, where z is measured parallel to the axis of the cylinder and \(\phi\) is an angle measured in the plane perpendicular to the axis, we see that the distance from P to Q in the cylindrical surface is given by \(L^{2}\) = (\(\Delta z\)) 2 + \(R^2\) (Δ \(\phi\)) 2, where R = (a − b)/2 \(\pi\) is the radius of the cylinder. This shows that the line element in the cylindrical surface will be \(dl^2\) = \(dz^2\) + \(R^2\) d \(\phi\) 2. However, if we make the change of variables x = Rφ, y = z, we see that these two line elements are actually the same.我们可以通过使用适当的二维线元素以更数学的方式来实现这个想法。平面中从 P 到 Q 的直线长度 L 为 \(L^{2}\) = (\(\Delta x\)) 2 + (\(\Delta y\)) 2，这提醒我们平面中的线元以笛卡尔坐标表示为 \(dl^2\) = \((dx)^2\) + \(dy^2\)。使用图 3.10 中所示的圆柱坐标 (z, \(\phi\))，其中 z 是平行于圆柱体轴线测量的，而 \(\phi\) 是在垂直于圆柱体轴线的平面中测量的角度，我们看到圆柱表面中从 P 到 Q 的距离由 \(L^{2}\) = (\(\Delta z\)) 2 + \(R^2\) (Δ \(\phi\)) 2 给出，其中 R = (a − b)/2 \(\pi\) 是圆柱体的半径。这表明圆柱面上的线元将是 \(dl^2\) = \(dz^2\) + \(R^2\) d \(\phi\) 2。但是，如果我们改变变量 x = Rφ，y = z，我们看到这两个线元实际上是相同的。

As a final example of the importance of intrinsic curvature, consider a hotplate consisting of a circular region of the plane with a heat source at the centre point. The heat diffuses through the disc so that it gets cooler as the distance from the heat source increases. The two-dimensional bugs and their measuring sticks expand with the heat, so from our point of view they are bigger towards the centre of the disc (see Figure 3.11), although this is not noticeable to the bugs themselves. As a result of the temperature distribution, the shortest distance from P to Q as measured by the bugs will appear to us to curve in towards the centre, where fewer measuring sticks are needed to cover the distance (this too is shown in Figure 3.11). Hence the angles of the triangle PQR in Figure 3.11 add up to less than 180 ◦, and so, despite looking like a part of a flat plane to us, the hotplate has an intrinsically curved geometry according to the bugs that inhabit it.作为固有曲率重要性的最后一个例子，考虑一个由平面圆形区域组成的加热板，其中心点有一个热源。热量通过圆盘扩散，因此随着与热源距离的增加，圆盘会变得更冷。二维虫子和它们的量尺会随着热量而膨胀，因此从我们的角度来看，它们在靠近圆盘中心的地方会变大(见图 3.11)，尽管虫子本身并没有注意到这一点。由于温度分布的原因，虫子测量的从 P 到 Q 的最短距离在我们看来会向中心弯曲，需要更少的测量棒来覆盖该距离(这也如图 3.11 所示)。因此，图 3.11 中三角形 PQR 的角度加起来小于 180°，因此，尽管对我们来说看起来像是平面的一部分，但根据居住在其中的虫子，加热板具有本质上弯曲的几何形状。

It was Gauss who first recognized the intrinsic nature of the curvature of surfaces, but, as you will see in the next section, it was Riemann who enthusiastically embraced the idea and extended it to spaces of higher dimension.高斯首先认识到曲面曲率的内在本质，但是，正如您将在下一节中看到的那样，黎曼热情地接受了这个想法并将其扩展到更高维度的空间。

Exercise 3.3 Using the same sort of informal arguments练习 3.3 使用相同类型的非正式论证

as in the above examples, investigate the curvature of the following spaces.如上面的示例所示，研究以下空间的曲率。

(a) A cone, excluding the point at its apex. Note that this means that you shouldn’t consider circles and triangles drawn around the apex, as they are not completely contained in the space.(a) 圆锥体，不包括其顶点。请注意，这意味着您不应考虑围绕顶点绘制的圆形和三角形，因为它们并未完全包含在空间中。

(b) A circular ‘hotplate’ where the heat source is around the edge of the disc, so that it cools towards the centre (Figure 3.12).(b) 圆形“加热板”，其中热源位于圆盘边缘周围，以便向中心冷却(图 3.12)。

3.2 Metrics and connections3.2 指标和联络

Having informally introduced the idea of a curved space, we now focus on the branch of differential geometry known as Riemannian geometry that is mainly used to analyze such spaces. As we shall see in Chapter 4, it is Riemannian geometry that is particularly relevant to Einstein’s theory of general relativity.在非正式地介绍了弯曲空间的概念之后，我们现在关注微分几何的一个分支，即黎曼几何，它主要用于分析此类空间。正如我们将在第四章中看到的，黎曼几何与爱因斯坦的广义相对论特别相关。

3.2.1 Metrics and Riemannian geometry3.2.1 度规和黎曼几何

In the previous section we saw that in the differential approach to geometry, line elements hold the key to determining lengths of curves and paths of shortest distance, and through them to the properties of circles and triangles, and hence to the whole geometry of Euclidean space or the surface of a sphere. Several line elements were written down for two- and three-dimensional spaces, flat and curved, using a variety of coordinate systems (Equations在上一节中，我们看到，在几何的微分方法中，线元素是确定曲线长度和最短距离路径的关键，并通过它们确定圆和三角形的属性，从而确定欧几里得空间或球体表面的整个几何形状。使用各种坐标系，为二维和三维空间(平面和曲线)写下了几个线元素(方程

3.4, 3.7, 3.8, 3.9,3.4、3.7、3.8、3.9、

3.10). In each case, by analogy with Pythagoras’s theorem,3.10)。在每种情况下，通过与毕达哥拉斯定理类比，

the line element was expressed as a sum of squares of coordinate differentials, such as d x, d y, d r and d \(\theta\). In all those cases the line element was deduced from the known geometrical properties of the space concerned. Riemann’s great insight was to recognize that line elements could be used not merely to summarize a geometry but rather as the starting point for the consideration of a geometry. He realized that by constructing line elements in accordance with certain simple general principles, it would be possible to develop a whole family of geometries that could describe flat and curved spaces with any desired number of dimensions. This is the basis of Riemannian geometry.线元素表示为坐标微分的平方和，例如d x、d y、d r 和d \(\theta\)。在所有这些情况下，线元素都是根据相关空间的已知几何特性推导出来的。黎曼的伟大洞察力是认识到线元素不仅可以用来概括几何，而且可以作为考虑几何的起点。他意识到，通过根据某些简单的一般原理构建线元素，可以开发出一整套几何图形，可以描述具有任何所需维数的平坦和弯曲空间。这是黎曼几何的基础。

An n-dimensional Riemann space is a space in which the line element takes the general formn 维黎曼空间是其中线元素采用一般形式的空间

where d \(x^1\), \((dx)^2\),..., d x n are the differentials of the n coordinates that describe the space, and the various \(g_{ij}\) are functions of the coordinates known as metric coefficients that are required to be symmetric in the sense that \(g_{ij}\) = \(g_{ji}\).其中 d x 1, d x 2,..., d x n 是描述空间的 n 个坐标的微分，各个 \(g_{ij}\) 是被称为度规系数的坐标的函数，这些系数要求在 \(g_{ij}\) = \(g_{ji}\) 的意义上是对称的。

Each of the line elements that we examined in the previous section was a special case of this general Riemannian line element. In the case of the Euclidean plane described by plane polar coordinates, for example, we saw in Equation 3.7 that我们在上一节中研究的每个线单元都是这个一般黎曼线单元的一个特例。例如，在由平面极坐标描述的欧几里得平面的情况下，我们在方程 3.7 中看到：

which corresponds to the choices n = 2, \(x^1\) = r, \(x_{2}\) = \(\phi\) and the metric coefficients g = 1, g = \(r^2\) and g = g = 0.对应于选择 n = 2、\(x^1\) = r、\(x_{2}\) = \(\phi\) 以及度规系数 g = 1、g = \(r^2\) 和 g = g = 0。

In an n-dimensional Euclidean space described by n Cartesian coordinates (\(x^1\), \(x_{2}\), \(x^{3}\),..., x n), the line element is在由 n 个笛卡尔坐标 (x 1, x 2, x 3,..., x n) 描述的 n 维欧几里得空间中，线元素为

and the metric coefficients can be written as \(g_{ij}\) = \(\delta_{ij}\), where \(\delta_{ij}\) is the Kronecker delta defined by度规系数可以写为 \(g_{ij}\) = \(\delta_{ij}\)，其中 \(\delta_{ij}\) 是克罗内克增量，定义为

1 if i = j, 0 if \(i\ne j\).如果 i = j，则为 1；如果 \(i\ne j\)，则为 0。

In general, the metric coefficients can be regarded as forming an n × n array with \(n^2\) elements, though due to the symmetry requirement \(g_{ij}\) = \(g_{ji}\), the number of independent elements is only n (n + 1)/2, i.e. half the number of off-diagonal elements, plus the n diagonal ones. The complete set of metric coefficients \(g_{\mu\nu}\) is called the metric or sometimes the metric tensor. (We shall not be much concerned with coordinate transformations in this chapter, but you will see later that the metric does transform in the way required of a rank 2 covariant tensor.) Consequently, the metric tensor for the three-dimensional Euclidean space defined by the line element of Equation 3.8 can be written as一般来说，度规系数可以看作是由 \(n^2\) 个元素组成的 n × n 数组，但由于对称性要求 \(g_{ij}\) = \(g_{ji}\)，独立元素的数量仅为 n (n + 1)/2，即非对角元素数量加上 n 个对角元素的一半。度规系数 \(g_{\mu\nu}\) 的完整集合称为度规或有时称为度规张量。(本章中我们不会太关心坐标变换，但是稍后您将看到度规确实按照 2 阶协变张量所需的方式进行变换。)因此，由方程 3.8 的线元素定义的三维欧几里得空间的度规张量可以写为

where the i, j simply indicate the positions of the indices and have no other significance. (In much of the literature on general relativity, no explicit distinction is made between a tensor and its components. Rather than follow this potentially confusing practice, we use brackets [] to indicate the full tensor.)其中i、j仅表示索引的位置，没有其他意义。 (在许多关于广义相对论的文献中，张量及其分量之间没有明确的区别。我们不遵循这种可能令人困惑的做法，而是使用括号 [] 来指示完整的张量。)

Note that, in general, the metric coefficients are not constants, but are functions of the coordinates x i. Once the coordinates being used to describe a space have been specified, it is the metric coefficients that perform the crucially important task of relating the coordinate differentials to lengths and thereby determine the geometry of the space. This point is so important to all that follows that it deserves special emphasis. Once you know the metric, the geometry of the space is entirely determined. However, the converse is not true. The geometry does not uniquely determine the metric; this is simply because there are many possible coordinate systems and hence many different ways of writing the metric.注意，一般来说，度规系数不是常数，而是坐标 x i 的函数。一旦指定了用于描述空间的坐标，度规系数就执行将坐标差与长度相关联的至关重要的任务，从而确定空间的几何形状。这一点对于接下来的一切都非常重要，值得特别强调。一旦知道了度规，空间的几何形状就完全确定了。然而，反之则不然。几何形状并不唯一地确定度规；这仅仅是因为有许多可能的坐标系，因此有许多不同的书写度规的方式。

Exercise 3.4 Writing \(x^1\) = \(\theta\), \(x_{2}\) = \(\phi\), find the metric that defines the curved练习 3.4 写出 \(x^1\) = \(\theta\), \(x_{2}\) = \(\phi\)，求定义曲线的度规

geometry of the surface of a sphere of radius R with the line element given by Equation 3.10.半径为 R 的球体表面的几何形状，其线元素由公式 3.10 给出。

We have now seen that both flat and curved spaces can be represented by metrics that are diagonal arrays. In fact, diagonal metrics occur whenever we have orthogonal coordinate systems, in which the different sets of grid lines corresponding to the directions of the x i are at right angles to each other. All the coordinate systems that we have used so far, Cartesian, plane polar and spherical, have been of this kind. It turns out that the metrics of interest in general relativity and cosmology are usually orthogonal, so most of the examples of metrics that we use in this book will be diagonal, but non-diagonal arrays are possible.我们现在已经看到，平坦和弯曲的空间都可以用对角数组的度规来表示。事实上，只要我们有正交坐标系，就会出现对角度规，其中与 x i 方向相对应的不同网格线组彼此成直角。迄今为止我们使用过的所有坐标系，笛卡尔坐标系、平面极坐标系和球坐标系，都是这种类型的。事实证明，广义相对论和宇宙学中感兴趣的度规通常是正交的，因此我们在本书中使用的大多数度规示例都是对角的，但非对角数组也是可能的。

Exercise 3.5 Here we consider the metric of three-dimensional练习3.5 这里我们考虑三维的度规

Euclidean space in spherical coordinates. With \(x^1\) = r, \(x_{2}\) = \(\theta\), \(x^{3}\) = \(\phi\), write down the metric coefficients \(g_{ij}\) that correspond to Equation 3.9, i.e.球坐标中的欧几里得空间。令 \(x^1\) = r、\(x_{2}\) = \(\theta\)、\(x^{3}\) = \(\phi\)，记下对应于公式 3.9 的度规系数 \(g_{ij}\)，即

Exercise 3.6 The metric coefficients for a plane in练习 3.6 中平面的度规系数

polar coordinates have already been given. Rewrite them as an array using appropriate notation.极坐标已经给出。使用适当的符号将它们重写为数组。

Notice that in both of these exercises, the metric is a function of one or more of the coordinates, even though the spaces are certainly flat. This demonstrates that simply observing that the metric is a function of the coordinates is not sufficient to conclude that the space is curved; we may merely have a flat space in a non-Cartesian coordinate system.请注意，在这两个练习中，度规都是一个或多个坐标的函数，即使空间肯定是平坦的。这表明，仅仅观察度规是坐标的函数并不足以得出空间是弯曲的结论；我们可能只是在非笛卡尔坐标系中拥有一个平坦的空间。

We can summarize the main results of this subsection as follows.我们可以将本小节的主要结果总结如下。

Metrics指标

In an n-dimensional Riemann space, the line element is given by在 n 维黎曼空间中，线元素由下式给出

where the \(n^2\) metric coefficients \(g_{ij}\) that define the geometry of the space are symmetric in the sense that \(g_{ij}\) = \(g_{ji}\), and transform as the components of a rank 2 covariant tensor [\(g_{ij}\)] called the metric tensor.其中定义空间几何形状的 \(n^2\) 个度规系数 \(g_{ij}\) 在 \(g_{ij}\) = \(g_{ji}\) 的意义上是对称的，并且变换为称为度规张量的 2 阶协变张量 [\(g_{ij}\)] 的分量。

3.2.2 Connections and parallel transport3.2.2 联络和并行传输

The main purpose of this subsection is to introduce an important set of quantities known as connection coefficients. In an n-dimensional Riemannian space there 1, 2,..., n), though due to are \(n^3\) such coefficients, usually denoted Γ i (i, j, k = symmetry they are not all independent. Despite the indices, the connection coefficients are not the components of a tensor; under a coordinate transformation they do not transform in the way that tensor components must. The connection coefficients are directly related to the metric coefficients and are important in several contexts, including differentiation in curved space and a related process known as parallel transport. We shall start with a physical discussion of parallel transport and then go on to a more mathematical discussion that includes the connection coefficients.本小节的主要目的是介绍一组重要的量，称为联络系数。在 n 维黎曼空间中，有 1, 2,..., n)，尽管由于有 \(n^3\) 个这样的系数，通常表示为 Γ i (i, j, k = 对称性，但它们并不都是独立的。尽管有索引，但联络系数并不是张量的组成部分；在坐标变换下，它们不会按照张量分量必须的方式进行变换。联络系数与度规系数直接相关，并且在多种情况下都很重要，包括弯曲空间中的微分和称为并行传输的相关过程。我们将从并行传输的物理讨论开始，然后继续进行包括联络系数在内的更数学的讨论。

Imagine a scientist studying the distribution of wind velocity in the Earth’s atmosphere. The scientist might well want to compare the wind velocity v P at some point P with the wind velocity v Q at some other point Q. To do this, the scientist really needs to convey a copy of v P along some chosen path C to the point Q, preserving the direction of the original v P throughout each infinitesimal step. This is the process of parallel transport. It is illustrated in Figure 3.13, where the copy of v P that has been parallel transported to Q is denoted v # Q.想象一下一位科学家正在研究地球大气层中风速的分布。科学家可能很想将某个点 P 处的风速 v P 与其他点 Q 处的风速 v Q 进行比较。为此，科学家确实需要沿着某个选定的路径 C 将 v P 的副本传送到点 Q，并在每个无穷小步骤中保留原始 v P 的方向。这就是并行传输的过程。如图 3.13 所示，其中已并行传输到 Q 的 v P 的副本表示为 v # Q。

The mathematical difficulty of performing such a parallel transport of a vector along a curve depends very much on the nature of the space and coordinates involved. If the space is Euclidean and the coordinates Cartesian, the process is very simple. The wind velocity at any point can be written as v = v 1 i + \(v^{2}\) j + \(v^{3}\) k, where the unit vectors i, j and k in the \(x^1\) = x, \(x_{2}\) = y and \(x^{3}\) = z directions are said to be coordinate basis vectors, since they point in the direction of increasing coordinate values, and v 1, \(v^{2}\) and \(v^{3}\) are the components of v in the coordinate basis. Since we are using Cartesian coordinates in Euclidean space, a vector may be parallel transported by simply keeping its components constant, so the components of v will be v 1 = v 1, \(v^{2}\) = \(v^{2}\) and \(v^{3}\) = \(v^{3}\).沿着曲线执行矢量的这种平行传输的数学难度很大程度上取决于所涉及的空间和坐标的性质。如果空间是欧几里得坐标，坐标是笛卡尔坐标，则过程非常简单。任意点的风速可以写为 v = v 1 i + \(v^{2}\) j + \(v^{3}\) k，其中 \(x^1\) = x、\(x_{2}\) = y 和 \(x^{3}\) = z 方向上的单位向量 i、j 和 k 被称为坐标基向量，因为它们指向坐标值增加的方向，并且 v 1、\(v^{2}\) 和\(v^{3}\) 是坐标系中 v 的分量。由于我们在欧几里得空间中使用笛卡尔坐标，因此只需保持其分量不变即可并行传输向量，因此 v 的分量将为 v 1 = v 1、\(v^{2}\) = \(v^{2}\) 和 \(v^{3}\) = \(v^{3}\)。

The situation is not so simple if the Cartesian coordinates are replaced by spherical coordinates with \(x^1\) = r, \(x_{2}\) = \(\theta\) and \(x^{3}\) = \(\phi\). The reason for the extra complexity is easy to see and is illustrated in Figure 3.14. Spherical coordinates belong to the family vector along of curvilinear coordinates. That means that the coordinate basis vectors r E, \(\theta\) E and \(\phi\) E change their direction from place to place. As a consequence, in these coordinates, the components of the parallel transported vector at Q, v # Q, will be different from those of the original vector v P at P. So, in order to parallel transport a vector in this case, we need to know exactly how the components must change during each infinitesimal displacement along the curve.如果将笛卡尔坐标替换为球坐标(\(x^1\) = r、\(x_{2}\) = \(\theta\) 和 \(x^{3}\) = \(\phi\))，情况就不那么简单了。额外复杂性的原因很容易看出，如图 3.14 所示。球坐标属于曲线坐标系向量。这意味着坐标基向量 r E、\(\theta\) E 和 \(\phi\) E 的方向会随着位置的不同而改变。因此，在这些坐标中，Q 处的平行传输矢量 v # Q 的分量将不同于 P 处的原始矢量 v P 的分量。因此，为了在这种情况下并行传输矢量，我们需要准确地知道在沿曲线的每个无穷小的位移期间分量必须如何变化。

Now let’s generalize this problem to a three-dimensional Riemann space (which may be intrinsically curved) with coordinates \(x^1\), \(x_{2}\), \(x^{3}\) and metric [g] (i, j = 1, 2, 3) in which we want to parallel transport a vector specified at point P along a curve C to point Q. We shall suppose that positions along the curve are described by a parameter u and that the curve is therefore described by three coordinate functions \(x^1\) (u), \(x_{2}\) (u) and \(x^{3}\) (u). If we denote the coordinate basis vectors (the analogues of i, j, k or r E, \(\theta\) E, \(\phi\) E) by e, e, e, then at any point on C that corresponds to the parameter value u, we can write the local value of an arbitrary vector field v(u) in terms of its components in the coordinate basis and the coordinate basis vectors at that point. Thus现在让我们将这个问题推广到一个三维黎曼空间(可能本质上是弯曲的)，坐标为 \(x^1\), \(x_{2}\), \(x^{3}\) 和度规 [g] (i, j = 1, 2, 3)，其中我们希望将点 P 指定的向量沿着曲线 C 平行传输到点 Q。我们假设沿曲线的位置由参数 u 描述，因此曲线由三个坐标函数 \(x^1\) (u)、\(x_{2}\) (u) 描述和x 3 (u)。如果我们用 e, e, e 表示坐标基向量(i、j、k 或 r E、\(\theta\) E、\(\phi\) E 的类似物)，那么在 C 上对应于参数值 u 的任意点，我们可以根据其在坐标基中的分量和该点的坐标基向量来写出任意向量场 v(u) 的局部值。因此

Applying the rule for differentiating a product, we see that the rate of change of the vector field with respect to u as we move along the curve is given by应用对乘积进行微分的规则，我们可以看到，当我们沿着曲线移动时，向量场相对于 u 的变化率由下式给出

where the first term on the right represents the effect of changing the components, while the second term represents the effect of the changing basis vectors. Using the chain rule we can express the last term as a sum of terms, giving其中右侧第一项表示更改分量的影响，而第二项表示更改基向量的影响。使用链式法则，我们可以将最后一项表示为各项之和，给出

Consider the term ∂ e j/∂x k — note that this is a vector quantity. It represents the rate of change of e j with respect to x k and will have components in the direction of each of the basis vectors. This means that we can write it as a sum:考虑术语 ∂ e j/∂x k — 请注意，这是一个向量。它表示 e j 相对于 x k 的变化率，并且具有每个基向量方向上的分量。这意味着我们可以将其写成总和：

where, at any point, \(\Gamma^i{}_{jk}\) represents the component in the direction of basis vector e i of the rate of change of e j with respect to x k. It is the \(n^3\) quantities \(\Gamma^i{}_{jk}\) defined by this equation that are the connection coefficients for the space and coordinates concerned. Since each connection coefficient involves only unit vectors and coordinates, it is clear that it must be determined by the metric; we shall see how a little later.式中，在任意点， \(\Gamma^i{}_{jk}\) 表示 e j 相对于 x k 的变化率在基向量 e i 方向上的分量。该方程定义的 \(n^3\) 个量 \(\Gamma^i{}_{jk}\) 是相关空间和坐标的联络系数。由于每个联络系数只涉及单位向量和坐标，显然必须由度规来确定；我们稍后会看到如何进行。

Substituting Equation 3.14 into Equation 3.13, we see that将公式 3.14 代入公式 3.13，我们可以看到

All of the indices on the right-hand side are summed over, so they are all dummy indices. This means that we can change any of them, provided that we do so consistently. Using this freedom we can rewrite the equation as右侧所有指数都经过求和，因此它们都是虚拟指数。这意味着我们可以改变其中任何一个，只要我们始终如一地这样做。利用这种自由度，我们可以将方程重写为

If we now require that the vector field that we have been discussing represents the same parallel transported vector at every point, then we can say that its rate of change must be zero. So the condition that must be satisfied if the vector v is actually being parallel transported along the curve is that如果我们现在要求我们一直在讨论的向量场在每个点都表示相同的并行传输向量，那么我们可以说它的变化率必须为零。因此，如果向量 v 实际上沿着曲线平行传输，则必须满足的条件是

Thus, even in the case of a curved space, where the geometric interpretation is not simple, we can ensure the parallel transport of a vector by requiring that for each component,因此，即使在几何解释并不简单的弯曲空间的情况下，我们也可以通过要求每个分量确保向量的并行传输，

So, given the components v i (u) of a vector at some point on the curve, the components of the parallel transported vector at a neighbouring point are因此，给定曲线上某个点处向量的分量 v i (u)，相邻点处并行传输向量的分量为

All that remains is to determine the expression for the connection coefficient \(\Gamma^i{}_{jk}\) in terms of the metric.剩下的就是确定联络系数 \(\Gamma^i{}_{jk}\) 的度规表达式。

If we consider two nearby points, we can write their infinitesimal vector separation as如果我们考虑两个附近的点，我们可以将它们的无穷小向量间隔写为

and consequently因此

Comparing this with the original line element (Equation 3.11)将其与原始线元素进行比较(公式 3.11)

we see that我们看到

So the basis vectors are directly related to the metric coefficients.因此基向量与度规系数直接相关。

Now, if we partially differentiate Equation 3.20 with respect to x k, we see that现在，如果我们对方程 3.20 对 x k 进行部分微分，我们会看到

Using Equation 3.14 again, this can be rewritten as再次使用公式 3.14，可以将其重写为

After several lines of additional algebra, this leads to the final result经过几行附加代数后，得出最终结果

metric tensor [\(g_{ij}\)]. This where g il is a component of the contravariant form of the that is, [\(g_{ij}\)][\(g_{ij}\)] is equal latter quantity is the inverse of [\(g_{ij}\)] regarded as a matrix; to the identity matrix or, more explicitly,度规张量 [\(g_{ij}\)]。其中 g il 是的逆变形式的一个分量，即 [\(g_{ij}\)][\(g_{ij}\)] 等于后一个量，是被视为矩阵的 [\(g_{ij}\)] 的逆；到单位矩阵，或者更明确地说，

Since [\(g_{ij}\)] is the inverse of the metric [\(g_{ij}\)], it too must contain all the information about the geometry of the space. It is sometimes referred to as the dual metric.由于 [\(g_{ij}\)] 是度规 [\(g_{ij}\)] 的倒数，因此它也必须包含有关空间几何形状的所有信息。它有时被称为双度规。

Our findings regarding parallel transport can now be summarized as follows.我们关于并行传输的发现现在可以总结如下。

Parallel transport and connection coefficients并行传输和联络系数

Given the components v i of a vector at some point on a curve specified by x i (u) in a Riemann space with coordinates x i,..., x n and metric [g], the components of the parallel transported vector at some neighbouring point on the curve are given by给定在坐标 x i,..., x n 和度规 [g] 的黎曼空间中由 x i (u) 指定的曲线上某个点处向量的分量 v i，曲线上某个相邻点处的平行传输向量的分量由下式给出

where the connection coefficient \(\Gamma^i{}_{jk}\) is given by其中联络系数 \(\Gamma^i{}_{jk}\) 由下式给出

and the dual metric [\(g_{ij}\)], the matrix inverse of [\(g_{ij}\)], is defined by the requirement that对偶度规 [\(g_{ij}\)]([\(g_{ij}\)] 的矩阵逆)由以下要求定义：

i jk for:我想：

Exercise 3.7 Calculate the connection coefficients Γ练习 3.7 计算联络系数 Γ

(a) a two-dimensional Euclidean space using Cartesian coordinates; (b) the surface of a sphere of radius R = 1, using polar coordinates.(a) 使用笛卡尔坐标的二维欧几里得空间； (b) 半径 R = 1 的球体表面，使用极坐标。

As mentioned earlier, connection coefficients and parallel transport are important in several contexts, particularly in connection with differentiation in curved spaces. However, as Exercise 3.7 shows, they also provide an important indicator of the curvature of a space. Two-dimensional surfaces provide some easily visualized examples of this. In the case of the cylinder shown in Figure 3.15, parallel transport does exactly what it says: if we transport a vector v around a closed curve, it stays parallel to itself all the way around and gets back to the initial point exactly as it started out. That’s because the surface of a cylinder is actually a flat space in terms of its intrinsic geometry.如前所述，联络系数和平行传输在多种情况下都很重要，特别是与弯曲空间的微分相关。然而，正如练习 3.7 所示，它们还提供了空间曲率的重要指标。二维表面提供了一些易于可视化的示例。在图 3.15 所示的圆柱体中，平行传输的作用正如其所言：如果我们绕一条闭合曲线传输一个向量 v，它自始至终都保持与自身平行，并准确地回到起始点。这是因为，就其内在几何形状而言，圆柱体的表面实际上是一个平坦的空间。

However, as shown in Figure 3.16, there are no parallel lines in the curved geometry of a spherical surface, so we can’t really expect that even a vector that is parallel transported over infinitesimal steps will manage to stay ‘parallel’ to itself when transported around a loop of finite size. And indeed, after being moved around a closed spherical triangle by ‘parallel’ transport, the vector in Figure 3.16 arrives back at its starting position pointing in a different direction.然而，如图 3.16 所示，球面的弯曲几何形状中不存在平行线，因此我们不能真正期望即使是在无穷小的步长上平行传输的向量，在绕有限大小的环传输时也能设法保持与其自身“平行”。事实上，在通过“平行”传输围绕闭合球形三角形移动后，图 3.16 中的矢量返回到指向不同方向的起始位置。

vector v around the triangle ABC drawn on the surface of a cylinder.围绕在圆柱体表面绘制的三角形 ABC 的矢量 v。

Thus parallel transport of a vector around a closed curve or path gives us another test for whether a particular geometry is intrinsically flat or curved. Indeed, as you will see later, the difference between the initial and final directions of the vector gives us a measure of just how curved the geometry is in the vicinity of the closed path.因此，矢量围绕闭合曲线或路径的平行传输为我们提供了另一种测试，以确定特定几何形状本质上是平坦的还是弯曲的。事实上，正如您稍后将看到的，矢量的初始方向和最终方向之间的差异使我们能够衡量闭合路径附近几何形状的弯曲程度。

3.3 Geodesics3.3 测地线

In a flat space, straight lines are of particular importance. A straight line represents the most direct route between two points and also the path of shortest distance between those points. Great circles play a similar role in the curved surface of a sphere. The analogues of straight lines and great circles in a general Riemannian space are referred to as geodesics. In this section we generalize the notions of ‘most direct path’ and ‘shortest distance’ in order to present two different derivations of the equations that are used to determine geodesics.在平坦的空间中，直线尤为重要。直线代表两点之间最直接的路线，也是两点之间距离最短的路径。大圆在球体的曲面中起着类似的作用。一般黎曼空间中直线和大圆的类似物被称为测地线。在本节中，我们概括了“最直接路径”和“最短距离”的概念，以呈现用于确定测地线的方程的两种不同推导。

3.3.1 Most direct route between two points3.3.1 两点之间最直接的路线

One way of defining a straight line in Euclidean space is as a curve that always goes in the same direction. In order to extend this definition to the more general spaces of Riemannian geometry, we need to analyze the concept of ‘direction along a curve’ and what it means to ‘always go in the same direction’. At any coordinate functions x i (u) point on a curve parameterized by u and defined by the (i = 1,..., n), we can define the tangent vector t to be the vector that points along the curve, as shown in Figure 3.17. The components of such a vector are t i = d x i/d u. If the curve is always going to go in the same direction, then the tangent vector should not change its direction as the parameter u varies and the tangent vector travels along the curve. In other words, if we parallel transport the by u + d u, the resulting tangent vector at u along the curve C to the point specified vector should be proportional to the tangent vector at u在欧几里得空间中定义直线的一种方法是定义为始终沿同一方向行进的曲线。为了将这个定义扩展到黎曼几何的更一般空间，我们需要分析“沿曲线的方向”的概念以及“始终沿同一方向行进”的含义。在由 u 参数化并由 (i = 1,..., n) 定义的曲线上的任意坐标函数 x i (u) 点，我们可以将切向量 t 定义为沿曲线指向的向量，如图 3.17 所示。该向量的分量为 t i = d x i/d u。如果曲线始终沿同一方向行进，则切向量不应随着参数 u 的变化而改变其方向，并且切向量沿曲线行进。换句话说，如果我们通过 u + d u 平行传输，则沿曲线 C 到点指定向量在 u 处得到的切向量应该与 u 处的切向量成正比

where f (u) is some function of u. It then follows from the condition for parallel transport that for each component of t,其中 f (u) 是 u 的某个函数。然后根据并行传输的条件得出对于 t 的每个分量，

Recalling that t i = d x i/d u, this gives回想一下 t i = d x i/d u，这给出

Now this can be simplified by choosing the parameter u in such a way that the function f (u) is equal to zero. When the parameter is chosen in this particular way, it is said to be an affine parameter and will be denoted by the symbol \(\lambda\). (This choice ensures that the tangent vector will preserve its magnitude as well as its direction as we move along the curve.) So, provided that we choose to use an affine parameter \(\lambda\), the condition for a parameterized curve defined by a set of coordinate functions x i (\(\lambda\)) to always point in the same direction is that现在可以通过选择参数 u 使函数 f (u) 等于 0 来简化。当以这种特定方式选择参数时，它被称为仿射参数，并用符号 \(\lambda\) 表示。(此选择确保当我们沿着曲线移动时，切向量将保留其大小和方向。)因此，假设我们选择使用仿射参数 \(\lambda\)，则由一组坐标函数 x i (\(\lambda\)) 定义的参数化曲线始终指向同一方向的条件是：

These are called the geodesic equations. Any parameterized pathway defined by a set of n functions x i (\(\lambda\)), i = 1,..., n, that satisfies these differential equations is said to be a geodesic in the n-dimensional Riemannian space with metric [\(g_{ij}\)] and connection coefficients \(\Gamma^i{}_{jk}\). This is the analogue of a straight line in the curved space.这些称为测地线方程。任何由一组 n 个函数 x i (\(\lambda\)), i = 1,..., n 定义且满足这些微分方程的参数化路径被称为 n 维黎曼空间中的测地线，其度规为 [\(g_{ij}\)] 和联络系数 \(\Gamma^i{}_{jk}\)。这类似于弯曲空间中的直线。

3.3.2 Shortest distance between two points3.3.2 两点之间的最短距离

We saw earlier that in a two-dimensional Euclidean space, the length of a curve, and y (u), between the parameterized by u and defined by the functions x (u) d l from P to Q points P and Q is given by integrating the line element (Equation 3.6) so that我们之前看到，在二维欧几里得空间中，曲线的长度和 y (u)(由 u 参数化并由函数 x (u) d l 定义，从 P 到 Q 点 P 和 Q 是通过对线元素进行积分而给出(方程 3.6)，因此

In an n-dimensional Riemannian space, Equation 3.11 extends the definition of the line element to include the metric via在 n 维黎曼空间中，方程 3.11 扩展了线元素的定义以包括度规通过

so we must correspondingly extend the formula for the length of the curve to所以我们必须相应地将曲线长度的公式推广为

What we want to find is the parameterized curve (\(x^1\) (u), \(x_{2}\) (u),..., x n (u)) between P and Q that gives the smallest value for L (P, Q), i.e. the shortest distance between the two points. Such a curve would be the analogue of a straight line, and therefore a geodesic. We use a method that is analogous to finding the minimum of a function f (x) by differentiating it and looking for points at which d f/d x = 0. The full mathematical treatment uses the calculus of variations, which is beyond the scope of this book, although a flavour of it is sketched below. You are not expected to follow the details, unless you have prior knowledge of the calculus of variations.我们想要找到的是 P 和 Q 之间的参数化曲线 (\(x^1\) (u), \(x_{2}\) (u),..., x n (u))，它给出 L (P, Q) 的最小值，即两点之间的最短距离。这样的曲线类似于直线，因此是测地线。我们使用的方法类似于通过对函数 f (x) 求导并寻找 d f/d x = 0 处的点来查找函数 f (x) 的最小值。完整的数学处理使用了变分法，这超出了本书的范围，尽管下面概述了它的一些风格。除非您事先了解变分法，否则您不需要遵循细节。

We can see from Figure 3.18 that since the geodesic between P and Q is the path of shortest length L, the curves that are close to it are of almost the same length. Now, if we consider all possible curves linking P and Q, and in each case we imagine changing the curve very slightly by making an infinitesimal variation of the form x i (u) → x i (u) + δx i (u), then in each case the length of the curve will change by a small amount δL. However, in the case of the true geodesic, where the length is a minimum, we will find that δL is zero. So, writing从图3.18中我们可以看出，由于P和Q之间的测地线是最短长度L的路径，因此靠近它的曲线的长度几乎相同。现在，如果我们考虑联络 P 和 Q 的所有可能的曲线，并且在每种情况下，我们想象通过进行 x i (u) → x i (u) + δx i (u) 形式的无穷小变化来非常轻微地改变曲线，那么在每种情况下，曲线的长度都会改变少量 δL。然而，在真正的测地线的情况下，长度最小，我们会发现 δL 为零。所以，写

it can be shown that可以证明

Integrating the second part of the sum by parts, and noting that将总和的第二部分按部分积分，并注意到

leads to导致

However, δx m = 0 at P and Q for all m, so the first bracket is zero. Consequently, for δL = 0, we have然而，对于所有 m，在 P 和 Q 处 δx m = 0，因此第一个括号为零。因此，对于 δL = 0，我们有

Since this is true for arbitrary variation δx m, we obtain由于对于任意变化 δx m 都是如此，我们得到

These are known as the Euler–Lagrange equations and are of fundamental importance to the study of the calculus of variations. If we substitute the expression for F(Equation 3.29) into the Euler–Lagrange equations, and choose u so that it is an affine parameter \(\lambda\), it can be shown that这些方程被称为欧拉-拉格朗日方程，对于变分法的研究至关重要。如果我们将 F(方程 3.29)的表达式代入欧拉-拉格朗日方程，并选择 u 使其成为仿射参数 \(\lambda\)，则可以证明：

These are just the geodesic equations again (Equation 3.27), which shows that both methods of generalizing the definition of a straight line lead to the same concept of the geodesic.这些又只是测地线方程(方程 3.27)，这表明两种推广直线定义的方法都得出相同的测地线概念。

So, to summarize, we have the following.因此，总而言之，我们有以下内容。

Geodesics and the geodesic equations测地线和测地线方程

In an n-dimensional Riemannian space, the analogues of straight lines are known as geodesics. A geodesic can be represented by a curve parameterized by an affine parameter \(\lambda\) and defined by a set of n coordinate functions x i (\(\lambda\)) that satisfy the geodesic equations在 n 维黎曼空间中，直线的类似物称为测地线。测地线可以用一条曲线表示，该曲线由仿射参数 \(\lambda\) 参数化，并由满足测地线方程的一组 n 个坐标函数 x i (\(\lambda\)) 定义

Exercise 3.8 Solve the geodesic equations for two-dimensional练习 3.8 求解二维测地线方程

Euclidean space and verify that the geodesics are indeed straight lines.欧几里得空间并验证测地线确实是直线。

Exercise 3.9 Figure 3.19 shows three curves on the练习 3.9 图 3.19 显示了 3 条曲线

surface of a sphere:球体表面：

• • a portion of a meridian A, with end-points (\(\pi\), 0) and• 子午线 A 的一部分，其端点为 (\(\pi\), 0) 和
• • the equator B, defined by \(\theta\) = \(\pi\) and 0 ≤ \(\phi\) < 2 \(\pi\)赤道 B，定义为 \(\theta\) = \(\pi\) 且 0 ≤ \(\phi\) < 2 \(\pi\)
• • a line of latitude C, defined by \(\theta\) = \(\pi\) and 0 ≤ \(\phi\) < 2• 一条纬度线 C，定义为 \(\theta\) = \(\pi\) 且 0 ≤ \(\phi\) < 2

\(\pi\).\(\pi\)。

Starting from the geodesic equations (Equation 3.27), show that for the curves A, B and C in Figure 3.19:从测地线方程(方程 3.27)开始，显示对于图 3.19 中的曲线 A、B 和 C：

(a) curve A is a geodesic;(a) 曲线A是测地线；

(b) curve B is also a geodesic;(b) 曲线B也是测地线；

(c) the line of latitude C is not a geodesic.(c) 纬线 C 不是测地线。

3.4 Curvature3.4 曲率

In this section we formalize and quantify the notion of the curvature of space. In particular we learn how to measure the curvature in an intrinsic way that does not depend on being able to embed the space being studied in some other space of higher dimension.在本节中，我们形式化并量化了空间曲率的概念。特别是，我们学习如何以内在的方式测量曲率，而不依赖于能够将正在研究的空间嵌入到其他更高维度的空间中。

3.4.1 Curvature of a curve in a plane3.4.1 平面内曲线的曲率

We start with the comparatively simple idea of a curved line in a plane. Looking at the curve AH in Figure 3.20, it makes sense to say that the section around BCD is ‘more curved’ than the section around EFG. Our first objective is to associate a quantity k with this curvature at each point, such that k C > k F.我们从平面中的曲线这一相对简单的想法开始。观察图 3.20 中的曲线 AH，可以说 BCD 周围的部分比 EFG 周围的部分“更弯曲”。我们的第一个目标是将量 k 与每个点的曲率相关联，使得 k C > k F。

and as l gets smaller, the approximations get better. We can do the same thing with the section EFG of the curve, also of length l, although this time the angle \(\theta\) F between the tangents is smaller than \(\theta\) C, and the radius R F of the approximating circle is larger than R C. This gives the relation随着 l 变小，近似值变得更好。我们可以对曲线的截面 EFG 做同样的事情，长度也是 l，尽管这次切线之间的角度 \(\theta\) F 小于 \(\theta\) C，并且近似圆的半径 R F 大于 R C。这给出了关系

Using the angles between the tangents as the measure of curvature, we can say that使用切线之间的角度作为曲率的度规，我们可以说

curvature at C > curvature at FC 处的曲率 > F 处的曲率

because因为

But但

and \(\theta\) ≈和 \(\theta\) ≈

so所以

and therefore因此

Consequently, the quantity因此，数量

is a measure of the curvature at any point X of a curve C in the plane, where R X is the radius of the circle that best approximates C in the region close to X.是平面中曲线 C 的任意点 X 处曲率的度规，其中 R X 是靠近 X 的区域中最接近 C 的圆的半径。

\(k = 0\).2 c\(\mathrm{m^{-1}}\)?\(k = 0\).2 厘米 - 1？

Exercise 3.10 (a) What is the curve of constant curvature练习 3.10 (a) 什么是常曲率曲线

(b) What is the curvature k = 1/R of a straight line?(b) 直线的曲率 k = 1/R 是多少？

For more complicated curves, a better way of finding the radius of the approximating circle is needed. It can be shown that if a curve is parameterized by the coordinate functions (x (\(\lambda\)), y (\(\lambda\))), then its curvature对于更复杂的曲线，需要更好的方法来找到近似圆的半径。可以证明，如果一条曲线由坐标函数 (x (\(\lambda\)), y (\(\lambda\))) 参数化，那么它的曲率

where a single dot over a function indicates the first derivative of that function with respect to \(\lambda\), and a double dot indicates the second derivative.其中函数上的单点表示该函数相对于 \(\lambda\) 的一阶导数，双点表示二阶导数。

\(x_{2}\) at x = 0. Where is\(x_{2}\) 在 x = 0 处。其中

Exercise 3.11 Find the curvature of the parabola y =练习 3.11 求抛物线的曲率 y =

the centre of the circle that best approximates the parabola in the region close to x = 0?在接近 x = 0 的区域中最接近抛物线的圆心？

Exercise 3.12 Find the curvature at any point on an练习 3.12 求曲线上任意点的曲率

ellipse parameterized by x = a cos \(\lambda\), y = b sin \(\lambda\). Use your answer to show that it leads to the expected result for the curvature of a circle of radius R.椭圆由 x = a cos \(\lambda\)、y = b sin \(\lambda\) 参数化。使用你的答案来证明它可以得出半径为 R 的圆的曲率的预期结果。

3.4.2 Gaussian curvature of a two-dimensional3.4.2 二维的高斯曲率

surface表面

We now consider the curvature of a two-dimensional surface embedded in three-dimensional Euclidean space. Suppose that we want to measure the curvature at a point A on the two-dimensional surface. From the point of view of the three-dimensional space, we can construct a vector N, normal to the surface at A, and this partly defines a plane PL containing N. As shown in Figure 3.21, the plane intersects the two-dimensional surface to give a curve C. (If we fix the two end-points of C in the plane PL, then the curve C is in fact a geodesic.) The curvature of C can be measured, as in the previous subsection, by finding the circle that best approximates the curve at A and then taking the reciprocal of the radius of that circle to obtain the curvature k. However, the plane PL is not completely defined since it can have any orientation with respect to N: different orientations will give different curves C and hence different curvatures k. We can get a measure of the curvature of the two-dimensional surface (rather than just a single curve C) at A by letting the plane PL rotate about N and picking the largest and smallest values of k, which we can denote by k max and k min. The curvature of the two-dimensional surface at A is then characterized by what is known as the Gaussian curvature, which is defined by我们现在考虑嵌入三维欧几里得空间中的二维表面的曲率。假设我们要测量二维表面上 A 点的曲率。从三维空间的角度来看，我们可以构造一个向量N，垂直于A处的曲面，这部分定义了包含N的平面PL。如图3.21所示，该平面与二维曲面相交得到曲线C。(如果我们将 C 的两个端点固定在平面 PL 上，则曲线 C 实际上是测地线。)如上一小节所述，可以通过找到最接近 A 处曲线的圆来测量 C 的曲率，然后取该圆的半径的倒数以获得曲率 k。然而，平面 PL 并未完全定义，因为它可以具有相对于 N 的任何方向：不同的方向将给出不同的曲线 C，从而给出不同的曲率 k。通过让平面 PL 绕 N 旋转并选取 k 的最大和最小值(我们可以用 k max 和 k min 表示)，我们可以测量 A 处二维表面(而不仅仅是一条曲线 C)的曲率。然后，A 处二维表面的曲率用所谓的高斯曲率来表征，其定义为

is the intersection between One important subtlety is that for different curves at the same point A on a the surface and the plane PL surface, the approximating circles may lie on opposite sides of the surface: for through N. instance, this occurs in Figure 3.22(a) but not in Figure 3.22(b). In order to distinguish between these situations, we define the curvature k to be positive if the centre of the approximating circle is on the opposite side to the arrowhead of the normal vector N, and negative if it is on the same side. To ensure a unique result, negative curvatures are always taken as being smaller than positive ones in the search for k max and k min. (Of course, the orientation of N is arbitrary, but this doesn’t matter.)一个重要的微妙之处是，对于曲面和平面 PL 曲面上同一点 A 处的不同曲线，近似圆可能位于曲面的相对两侧：对于通过 N 的实例，这种情况出现在图 3.22(a) 中，但不在图 3.22(b) 中。为了区分这些情况，如果近似圆的中心与法向量N的箭头相反一侧，我们定义曲率k为正，如果在同一侧，则曲率k为负。为了确保结果唯一，在搜索 k max 和 k min 时始终将负曲率视为小于正曲率。 (当然，N的方向是任意的，但这并不重要。)

• ● What is the Gaussian curvature for the surface of a two-dimensional sphere of二维球体表面的高斯曲率是多少

radius R?半径R？

❍ For the sphere,❍ 对于球体，

so the Gaussian curvature is所以高斯曲率是

So far, our arguments have depended on being able to embed the surface being studied in a three-dimensional space so that it has an obvious extrinsic curvature. In 1828 Gauss discovered a result regarding the curvature of surfaces that surprised him so much that he called it the ‘remarkable theorem’ (theorema egregium). The theorem provided a formula for working out the Gaussian curvature K of a two-dimensional surface, but the remarkable aspect of the result was that it showed K to be an invariant, independent of the coordinate system used. This was one of the inspirations for Riemann’s work, and is now seen as an indication that curvature is an intrinsic property; it can be calculated directly from the metric [\(g_{ij}\)] and does not require any embedding in a space of higher dimension. We shall not prove the theorema egregium here — even an outline proof requires four pages of dense mathematics — but we shall return to its main outcome once we have considered the curvature of spaces with three or more dimensions, in the next subsection.到目前为止，我们的论点依赖于能够将所研究的表面嵌入到三维空间中，使其具有明显的外在曲率。 1828 年，高斯发现了一个关于曲面曲率的结果，这让他非常惊讶，以至于他将其称为“非凡定理”(theorema egregium)。该定理提供了计算二维表面高斯曲率 K 的公式，但结果的显着之处在于它表明 K 是一个不变量，与所使用的坐标系无关。这是黎曼工作的灵感之一，现在被视为曲率是一种内在属性的迹象；它可以直接从度规 [\(g_{ij}\)] 计算出来，不需要在更高维度的空间中进行任何嵌入。我们不会在这里证明这个定理——即使是一个大纲证明也需要四页的密集数学——但是一旦我们在下一小节中考虑了具有三个或更多维度的空间的曲率，我们将回到它的主要结果。

3.4.3 Curvature in spaces of higher dimensions3.4.3 高维空间中的曲率

Now consider an n-dimensional Riemann space with metric [\(g_{ij}\)] that can be used = 1,..., n). Suppose that to determine the space’s connection coefficients Γ i (i we have a vector v specified at some point P and that we parallel transport that specified by d x j vector around an infinitesimal rectangle PQRS with sides and d x k. This process is illustrated in Figure 3.23, where the parallel transported vector that arrives back at P is denoted v # P and is shown as being different from v because of the curvature of the space. We should expect the difference现在考虑一个 n 维黎曼空间，其度规 [\(g_{ij}\)] 可以使用 = 1,..., n)。假设为了确定空间的联络系数 Γ i (i，我们在某个点 P 处指定了一个向量 v，并且我们将由 d x j 向量指定的向量围绕一个边长为 d x k 的无穷小矩形 PQRS 进行并行传输。这个过程如图 3.23 所示，其中返回 P 的并行传输向量表示为 v # P，并且由于空间的曲率而显示为与 v 不同。我们应该期待差异

v # P at point P.v # P 在 P 点。

between v # P and v to have components that are proportional to the infinitesimal displacements and to the components of the original vector, so we can write any given component of the difference asv # P 和 v 之间的分量与无穷小位移和原始向量的分量成比例，因此我们可以将差值的任何给定分量写为

where \(R^l{}_{ijk}\) will be some measure of the curvature. (In a flat space we know that v # P = v, so in that case we know that \(R^l{}_{ijk}\) = 0 for all choices of i, j, k and l.) When the parallel transport is actually carried out, it can be shown that其中 \(R^l{}_{ijk}\) 是曲率的某种度规。 (在平坦空间中，我们知道 v # P = v，因此在这种情况下，我们知道对于 i、j、k 和 l 的所有选择，\(R^l{}_{ijk}\) = 0。)当并行传输实际进行时，可以证明：

It turns out that under a general coordinate transformation, the quantity \(R^l{}_{ijk}\) transforms in the manner required of a rank 4 tensor with one contravariant index and three covariant indices. Consequently, \(R^l{}_{ijk}\) is known as the Riemann curvature tensor or the Riemann tensor. It is possible to show that the vanishing of the Riemann tensor [\(R^l{}_{ijk}\)] at all points in a space is a necessary and sufficient condition for a space to be flat, i.e. not curved. So we finally have a quantitative measure of curvature, and — since it is related directly to the metric, albeit in a complicated way — it is clearly an intrinsic quantity that does not require any embedding in higher dimensions.事实证明，在一般坐标变换下，量 \(R^l{}_{ijk}\) 按照具有一个逆变索引和三个协变索引的 4 阶张量所需的方式进行变换。因此，\(R^l{}_{ijk}\) 被称为黎曼曲率张量或黎曼张量。可以证明，空间中所有点的黎曼张量[\(R^l{}_{ijk}\)]消失是空间平坦的充要条件，即不弯曲。因此，我们最终有了曲率的定量测量，而且——因为它与度规直接相关，尽管方式很复杂——它显然是一个内在的量，不需要任何更高维度的嵌入。

In n dimensions the Riemann tensor has \(n^4\) components, giving \(2^4=16\) components in two dimensions and \(3^4=81\) in three dimensions. However, because of the definition of the connection (Equation 3.23) and Equation 3.35 itself, the Riemann tensor has many symmetries involving its indices. For example,在 n 维中，黎曼张量有 \(n^4\) 个分量，在二维中给出 \(2^4=16\) 个分量，在三维中给出 \(3^4=81\) 个分量。然而，由于联络的定义(方程 3.23)和方程 3.35 本身，黎曼张量具有许多涉及其索引的对称性。例如，

These symmetries reduce the number of in dependent components to 6 in three-dimensional spaces and only one in two-dimensional spaces. In two dimensions, the single independent component can be related to the Gaussian curvature K. From the point of view of Riemannian geometry, this is the explanation of Gauss’s theorema egregium, with its implication that Gaussian curvature is intrinsic.这些对称性将三维空间中独立分量的数量减少到 6 个，而在二维空间中仅减少一个。在二维中，单个独立分量可以与高斯曲率K联系起来。从黎曼几何的角度来看，这就是高斯定理egregium的解释，意味着高斯曲率是内在的。

Exercise 3.13 Use Equation 3.35 to show that R l = − R l.练习 3.13 使用方程 3.35 证明 R l = − R l。

Exercise 3.14 Find the Riemann tensor for two-dimensional Euclidean练习 3.14 求二维欧几里得的黎曼张量

space with the line element given by Equation 3.4. Extend your result to an n-dimensional Euclidean space. (Hint: Use Equation 3.35 and the results of Exercise 3.7(a).)空间与方程 3.4 给出的线元素。将结果扩展到 n 维欧几里得空间。 (提示：使用公式 3.35 和练习 3.7(a) 的结果。)

Exercise 3.15 Find the component \(R^{1}\) 212 of the Riemann tensor for a练习 3.15 求黎曼张量的分量 \(R^1{}_{212}\)

two-dimensional sphere of radius R with the line element given in Equation 3.10. (Hint: Use Equation 3.35 and the results of Exercise 3.7(b).)半径为 R 的二维球体，其线元素如公式 3.10 所示。 (提示：使用公式 3.35 和练习 3.7(b) 的结果。)

Exercise 3.16 The Gaussian curvature K for a two-dimensional surface is练习3.16 二维表面的高斯曲率K为

related to the Riemann tensor by where与黎曼张量相关，其中

and the first index of R 1212 is ‘lowered’ by means of the metric tensor (see Chapter 2). Use the result of the earlier in-text question concerning the Gaussian curvature for the surface of a two-dimensional sphere and the results of Exercises 3.7 and 3.15 to verify this relationship for a two-dimensional sphere of radius a. (We use a for the radius of the sphere in order to avoid possible confusion with the Riemann tensor.)R 1212 的第一个索引通过度规张量“降低”(参见第 2 章)。使用先前有关二维球体表面高斯曲率的文内问题的结果以及练习 3.7 和 3.15 的结果来验证半径为 a 的二维球体的这种关系。 (我们使用 a 表示球体的半径，以避免与黎曼张量混淆。)

So, to summarize, we have the following.因此，总而言之，我们有以下内容。

The Riemann tensor黎曼张量

In an n-dimensional Riemannian space, the curvature is described by the在 n 维黎曼空间中，曲率由下式描述

The necessary and sufficient condition for a space to be flat (i.e. not curved) is that all the components of this tensor should vanish at every point.空间平坦(即不弯曲)的充分必要条件是该张量的所有分量在每个点都应该消失。

3.4.4 Curvature of spacetime3.4.4 时空曲率

So far, we have considered curved spaces that are Riemannian. In a strict mathematical sense, such spaces are defined by a line element taking the form到目前为止，我们已经考虑了黎曼曲线空间。从严格的数学意义上讲，此类空间由采用以下形式的线元素定义

where \(dl^2\) > 0. As you will see in the next chapter, Einstein’s general theory of relativity is a geometric theory of gravity that makes essential use of the Riemann tensor. However, in searching for a geometric theory of gravity, Einstein needed to generalize the Minkowski spacetime of special relativity, which is defined by a line element of the form其中 \(dl^2\) > 0。正如您将在下一章中看到的，爱因斯坦的广义相对论是一种几何引力理论，它本质上利用了黎曼张量。然而，在寻找引力的几何理论时，爱因斯坦需要推广狭义相对论的闵可夫斯基时空，该时空由以下形式的线元素定义

where在哪里

More explicitly, the line element in Minkowski spacetime is更明确地说，闵可夫斯基时空中的线元是

separation (\(\Delta s\)) 2 that was This is the infinitesimal generalization of the spacetime introduced in Chapter 1. It is clearly possible to choose the differentials so that \(ds^2\) is negative, breaking the \(dl^2\) > 0 requirement of a Riemannian geometry.分离 (\(\Delta s\)) 2 这是第 1 章中介绍的时空的无穷小推广。显然可以选择微分以使 \(ds^2\) 为负，从而打破了黎曼几何的 \(dl^2\) > 0 要求。

Spaces for which the squared line element can be positive, negative or zero (null) are called pseudo-Riemannian spaces by mathematicians. However, physicists often don’t bother to make this distinction and use the term Riemannian space to cover any space (or spacetime) defined by a metric as in Equation 3.11.平方线元素可以为正、负或零(空)的空间被数学家称为伪黎曼空间。然而，物理学家通常懒得做这种区分，而是使用术语黎曼空间来涵盖由度规定义的任何空间(或时空)，如公式 3.11 所示。

The generalization from the flat Minkowski spacetime of special relativity to the curved spacetime of general relativity is made by replacing the Minkowski spacetime metric coefficients \(\eta\) \(\mu\)\(\nu\), which are constants, with metric coefficients \(g_{\mu\nu}\) that are function of the coordinates, so that从狭义相对论的平坦闵可夫斯基时空到广义相对论的弯曲时空的推广是通过将作为常数的闵可夫斯基时空度规系数 \(\eta\) \(\mu\)\(\nu\) 替换为坐标函数的度规系数 \(g_{\mu\nu}\) 来实现的，使得

Notice that it is traditional to use Greek letters for the indices of four-dimensional Minkowski spacetime and for its extension to the curved spacetime of general relativity, with 0 representing the time coordinate. Latin letters are reserved for indices relating to space coordinates, usually taking the values 1, 2, 3.请注意，传统上使用希腊字母作为四维闵可夫斯基时空的索引并将其扩展到广义相对论的弯曲时空，其中 0 代表时间坐标。拉丁字母保留用于与空间坐标相关的索引，通常取值1、2、3。

Many of the properties of Riemannian spaces carry over to pseudo-Riemannian ones. Most importantly, the vanishing of the Riemann tensor R δ αβγ is a necessary and sufficient condition for a spacetime to be flat. For such spacetimes, it is possible to choose a coordinate system so that the metric reduces to that of Minkowski spacetime at every point. For a curved spacetime, it is possible to choose a coordinate system so that the metric reduces to the Minkowski metric in the vicinity of any specific point P, but it is not generally possible to find a coordinate system in which this happens everywhere. Thus in general relativity we shall find that the results of special relativity will continue to hold true in the neighbourhood of any point but cannot be relied on generally. Special relativity will apply locally but not globally. This is similar to the finding that any small part of the Earth’s surface can be treated as flat, but any extensive investigation will soon show that the Earth is actually curved.黎曼空间的许多性质都延续到了伪黎曼空间。最重要的是，黎曼张量 R δ αβγ 的消失是时空平坦的充要条件。对于这样的时空，可以选择一个坐标系，使得每个点的度规都简化为闵可夫斯基时空的度规。对于弯曲时空，可以选择一个坐标系，使得度规在任何特定点 P 附近简化为闵可夫斯基度规，但通常不可能找到一个到处都发生这种情况的坐标系。因此，在广义相对论中，我们会发现狭义相对论的结果在任何点附近仍然成立，但不能普遍依赖。狭义相对论适用于局部，但不适用于全球。这类似于地球表面的任何一小部分都可以被视为平坦的发现，但任何广泛的调查很快就会表明地球实际上是弯曲的。

One important property of a pseudo-Riemannian space is that it is possible to have curves for which \(ds^2\) is zero at all points along the curve. Such curves are known as null curves since they have zero ‘length’ in the generalized sense of length residing in Equation 3.40. An important example of a null curve is a null geodesic. A null geodesic cannot therefore be defined as the shortest distance between the end-points of the curve (as in Subsection 3.3.2), but the definition as a curve along which the tangent always points in the same direction (as in Subsection 3.3.1) is still valid. Null geodesics are important in general relativity since, as you will see in the next chapter, they represent the possible paths of light rays in curved spacetime.伪黎曼空间的一个重要性质是，可能存在 \(ds^2\) 在曲线上所有点都为零的曲线。此类曲线被称为零曲线，因为在公式 3.40 中的广义长度意义上，它们的“长度”为零。零曲线的一个重要示例是零测地线。因此，零测地线不能定义为曲线端点之间的最短距离(如第 3.3.2 节中所示)，但切线始终指向同一方向的曲线的定义(如第 3.3.1 节中所示)仍然有效。零测地线在广义相对论中很重要，因为正如您将在下一章中看到的那样，它们代表了弯曲时空中光线的可能路径。

Exercise 3.17 (a) Find the connection coefficients for the Minkowski metric练习 3.17 (a) 求闵可夫斯基度规的联络系数

of Equation 3.37.公式 3.37 的结果。

(b) Find the component \(R^{1}\) 212 of the Riemann tensor for the Minkowski metric of Equation 3.37.(b) 求公式 3.37 的闵可夫斯基度规的黎曼张量的分量 \(R^1{}_{212}\)。

Exercise 3.18 A two-dimensional Minkowski spacetime has the metric练习3.18 二维闵可夫斯基时空有度规

(a) Setting \(x^0\) = t and \(x^1\) = x, find the connection coefficients.(a) 设x 0 = t，\(x^1\) = x，求联络系数。

(b) Hence find the component \(R_0\) 101 of the Riemann tensor.(b) 因此找到黎曼张量的分量 \(R_0\) 101。

Summary of Chapter 3第 3 章总结

1. The line element for a Riemannian space is given1. 给出黎曼空间的线元

by经过

[\(g_{ij}\)] represents the where the \(g_{ij}\) are the metric coefficients and the array metric tensor.[\(g_{ij}\)] 表示其中 \(g_{ij}\) 是度规系数和数组度规张量。

2. The metric tensor completely defines the geometry2. 度规张量完全定义了几何形状

of the space. The converse is not true due to the freedom to choose different coordinates.的空间。由于可以自由选择不同的坐标，反之亦然。

3. The line element in Cartesian coordinates for a plane3. 平面直角坐标系中的线元素

is given by由下式给出

4. The line element in spherical coordinates for the surface4. 曲面球坐标系中的线元

of a sphere with radius R is given by半径为 R 的球体由下式给出

5. On a parameterized curve, each point corresponds5. 在参数化曲线上，每个点对应

to a unique value of a single parameter u. The curve can be described in an n-dimensional space by specifying a set of coordinate functions x i (u) that assign to each point coordinates \(x^1\), \(x_{2}\),..., x n that depend on the value of u.为单个参数 u 的唯一值。通过指定一组坐标函数 x i (u)，可以在 n 维空间中描述曲线，这些函数分配给取决于 u 值的每个点坐标 \(x^1\)、\(x_{2}\)、...、x n。

6. A vector v that is moved along a curve while remaining6. 向量 v 沿曲线移动，同时保持不变

parallel to its original direction is said to undergo parallel transport.与其原始方向平行的称为平行传输。

7. When a vector v is parallel transported along a curve7. 当向量 v 沿曲线平行传输时

specified by the coordinate functions x i (u), its components in the coordinate basis must change (to compensate for any changes in the coordinate basis vectors) at the rate由坐标函数 x i (u) 指定，其在坐标基础中的分量必须以以下速率变化(以补偿坐标基础向量中的任何变化)

in the direction of朝着

8. The connection coefficient \(\Gamma^i{}_{jk}\) describes the component8. 联络系数 \(\Gamma^i{}_{jk}\) 描述了组件

basis vector e i of the rate of change of the basis vector e j with respect to changes in the coordinate x k. It is directly related to the metric by the expression基向量e i 表示基向量e j 相对于坐标x k 的变化的变化率。它通过表达式与度规直接相关

9. [\(g_{ij}\)], is called the dual metric, and is the inverse of9. [\(g_{ij}\)]，称为对偶度规，是

[\(g_{ij}\)] regarded as a matrix, i.e. i,j [\(g_{ij}\)][\(g_{ij}\)] is equal to the identity matrix. Or, more explicitly,[\(g_{ij}\)] 视为矩阵，即 i,j [\(g_{ij}\)][\(g_{ij}\)] 等于单位矩阵。或者，更明确地说，

where δ i j is known as the Kronecker delta, which is defined by其中 δ i j 称为克罗内克增量，其定义为

1 if i = j, 0 if \(i\ne j\).如果 i = j，则为 1；如果 \(i\ne j\)，则为 0。

10. In a curved space, the geodesic between two points is the most direct path10.在弯曲空间中，两点之间的测地线是最直接的路径

between those points (its tangent vector always points in the same direction) and also the path of shortest distance between them. Geodesics are analogous to straight lines in Euclidean space and minor arcs of great circles on the surface of a sphere. Geodesics are affinely parameterized curves described by coordinate functions x i (\(\lambda\)) that satisfy the geodesic equations这些点之间(其切向量始终指向同一方向)以及它们之间的最短距离的路径。测地线类似于欧几里得空间中的直线和球体表面上大圆的小弧。测地线是由满足测地线方程的坐标函数 x i (\(\lambda\)) 描述的仿射参数化曲线

11. The curvature k at a point P of a curve in the plane is defined by11. 平面中曲线的点 P 处的曲率 k 定义为

where R is the radius of the circle that best approximates the curve in the region of P.其中 R 是最接近 P 区域中曲线的圆的半径。

12. The Gaussian curvature K of a two-dimensional surface at a point P is12. 二维曲面在点 P 处的高斯曲率 K 为

defined by定义为

where k max and k min are the maximum and minimum curvatures obtained by considering all possible geodesics through P.其中 k max 和 k min 是通过 P 考虑所有可能的测地线而获得的最大和最小曲率。

13. The (intrinsic) curvature of an n-dimensional Riemannian space is13. n 维黎曼空间的(本征)曲率是

characterized by the \(n^4\) components of the Riemann tensor, which are directly related to the metric by the expression由黎曼张量的 \(n^4\) 个分量表征，它们通过表达式与度规直接相关

14. The Riemann tensor has many symmetries with respect to interchanging its14. 黎曼张量在互换方面具有许多对称性

indices, and this considerably restricts the number of independent components. In four dimensions there are 20 independent components, in three dimensions 6, and in two dimensions only 1.指数，这大大限制了独立成分的数量。在四维中有 20 个独立分量，在三维中有 6 个，在二维中只有 1 个。

15. The vanishing of the Riemann tensor is a necessary and sufficient condition15.黎曼张量消失是充要条件

for a space to be flat.使空间变得平坦。

16. Strictly speaking, one requirement for a Riemannian space is that the line16. 严格来说，黎曼空间的一个要求是直线

element satisfies \(dl^2\) > 0. Spaces for which the line element can be positive, negative or zero (null) are technically known as pseudo-Riemannian. The four-dimensional Minkowski spacetime of special relativity in which \(ds^2\) = \(c^2\) \((dt)^2\) − \((dx)^2\) − \(dy^2\) − \(dz^2\) is a pseudo-Riemannian space, as is its generalization to the curved spacetime of general relativity.元素满足 \(dl^2\) > 0。线元素可以为正、负或零(空)的空间在技术上称为伪黎曼。狭义相对论的四维闵可夫斯基时空，其中 \(ds^2\) = \(c^2\) \((dt)^2\) − \((dx)^2\) − \(dy^2\) − \(dz^2\) 是一个伪黎曼空间，它推广到广义相对论的弯曲时空也是如此。

17. In pseudo-Riemannian spaces, a geodesic for which \(ds^2\) vanishes at all17. 在伪黎曼空间中，\(ds^2\) 完全消失的测地线

points along the curve is known as a null geodesic. It remains true that the tangent vector at any point along a null geodesic always points in the same direction.沿曲线的点称为零测地线。沿零测地线任意点的切向量始终指向同一方向，这一点仍然成立。

Chapter 4 General relativity and gravitation第四章广义相对论和引力

Introduction介绍

Gravitation is an observable phenomenon; unsupported objects have a general tendency to fall downwards. In the Aristotelian physics of ancient Greece this was explained in terms of the composition of a body and the idea that objects had a ‘natural place’ in an Earth-centred universe. An apple released from a tree would fall downwards because its earthy composition gave it a natural place below the ground and its ‘gravity’ was the result of its tendency to move towards that place when free to do so. Likewise, smoke from a fire rose upwards because its airy composition gave it a natural place above the Earth that its innate ‘levity’ (the opposite of gravity) caused it to seek. Newton wrote scathingly of these ancient ideas. He offered a more mechanistic explanation of the phenomenon. Gravitation, according to Newton, was the result of a gravitational force that acted between massive bodies. In the case of two massive particles separated by a distance r, the gravitational force acting on each particle varied in proportion to 1/\(r^2\), so the Newtonian law that described this force became known as the inverse square law.万有引力是一种可观察到的现象；不受支撑的物体通常有向下坠落的趋势。在古希腊的亚里士多德物理学中，这是根据物体的构成以及物体在以地球为中心的宇宙中拥有“自然位置”的观念来解释的。从树上释放的苹果会向下掉落，因为它的泥土成分给了它在地面以下的自然位置，而它的“引力”是它在自由时倾向于向那个地方移动的结果。同样，火产生的烟雾会向上上升，因为它的空气成分使其在地球上方拥有一个自然的位置，而其固有的“轻浮”(与引力相反)促使它寻找这个位置。牛顿严厉地批评了这些古老的思想。他对这一现象提供了更机械的解释。根据牛顿的说法，万有引力是大质量物体之间引力作用的结果。在两个相隔距离 r 的大质量粒子的情况下，作用在每个粒子上的引力按 1/\(r^2\) 的比例变化，因此描述这种力的牛顿定律被称为平方反比定律。

Neither Newton nor any of his followers was ever able to give a convincing explanation of the origin of this force. Newton tried to do so using ideas that were in vogue at the time, but he found that they did not work, so he said instead that he would ‘feign no hypothesis’ as to the origin of the gravitational force. The inverse square law of Newtonian gravitation simply described the way things were — it was a phenomenological law, based on experience, with no deeper justification than the fact that it worked. But it worked supremely well.牛顿和他的任何追随者都无法对这种力的起源给出令人信服的解释。牛顿试图利用当时流行的想法来做到这一点，但他发现这些想法不起作用，因此他说他不会对引力的起源“假装没有假设”。牛顿万有引力的平方反比定律简单地描述了事物的本来面目——它是一个基于经验的现象学定律，除了它起作用的事实之外，没有任何更深层次的理由。但效果非常好。

Over the generations that followed, innumerable scientists and engineers used the Newtonian concept of a gravitational force to explain a vast array of phenomena. Nowhere was this more true than in the field of celestial mechanics — the application of mechanical principles to the study of the motion of celestial bodies. Newton himself had shown that his notion of a gravitational force could explain the gross features of the Moon’s motion but it fell to others, particularly French investigators such as Pierre-Simon Laplace (Figure 4.1), his pupil Sime´on-Denis Poisson (Figure 4.2), and later still Charles Delaunay (1816–1872) to develop powerful ways of exploiting Newton’s insights and working out their detailed consequences. That line of work continues to this day, particularly among the astrodynamicists who devise the trajectories of interplanetary spacecraft. These often include several ‘gravity assist’ manoeuvres in which a probe is helped on its way to a distant target by energy that it gathers from the planets that it encounters en route (Figure 4.3).在随后的几代人中，无数的科学家和工程师使用牛顿引力的概念来解释大量的现象。天体力学领域最能体现这一点——将机械原理应用于天体运动的研究。牛顿本人已经证明，他的引力概念可以解释月球运动的总体特征，但这落在了其他人的肩上，特别是法国研究人员，如皮埃尔-西蒙·拉普拉斯(图 4.1)、他的学生西蒙-丹尼斯·泊松(图 4.2)，以及后来的查尔斯·德劳内(1816-1872)，他们开发了强有力的方法来利用牛顿的见解并计算出其详细结果。这项工作至今仍在继续，特别是在设计行星际航天器轨道的天体动力学家中。这些通常包括几种“引力辅助”机动，其中探测器通过从途中遇到的行星收集的能量来帮助探测器到达遥远的目标(图 4.3)。

The Newtonian approach to gravitation has been so successful that many confuse Newton’s proposed explanation of gravitation with the phenomenon itself. The question ‘What is gravitation?’ deserves an answer that speaks of the general tendency of massive bodies to draw together, yet even today a common answer is that it is an attractive force described by an inverse square law. Newton’s brilliant and highly successful concept of a gravitational force has taken over gravitation in much the same way that the term ‘Hoover’ has replaced ‘vacuum cleaner’.牛顿的引力方法非常成功，以至于许多人将牛顿提出的引力解释与现象本身混为一谈。 “什么是引力？”这个问题需要一个关于大质量物体相互吸引的一般趋势的答案，但即使在今天，一个常见的答案仍然是它是一种由平方反比定律描述的吸引力。牛顿出色且非常成功的引力概念已经取代了万有引力，就像“胡佛”一词取代了“真空吸尘器”一样。

However, as Chapter 2 started to show, the development of Einsteinian relativity exposed problems deep in the heart of the Newtonian approach to gravitation. Under a change of inertial reference frame, a force described by an inverse square law does not transform in the way that a (three-) force should according to special relativity. Perhaps even more seriously, the Newtonian requirement that for every action there is an equal and opposite reaction implies that the gravitational forces linking two widely separated bodies should act instantly, irrespective of the distance between the two bodies. This is clearly at odds with the special relativistic requirement that such effects should not travel faster than the speed of light. Such arguments showed that Newtonian gravitation was not consistent with special relativity, and it soon became clear that no minor modification would make the two consistent.然而，正如第二章开始表明的那样，爱因斯坦相对论的发展暴露了牛顿引力理论的核心问题。在惯性参考系变化的情况下，由平方反比定律描述的力不会按照狭义相对论中的(三)力应有的方式变化。也许更严重的是，牛顿要求每一个作用都有一个相等且相反的反应，这意味着联络两个相距较远的物体的引力应该立即起作用，无论两个物体之间的距离如何。这显然与特殊相对论的要求相矛盾，即这种效应的传播速度不应超过光速。这些论证表明牛顿引力与狭义相对论不一致，而且很快人们就清楚，任何微小的修改都不能使两者一致。

The aim of this chapter is to introduce the core ideas of general relativity — Einstein’s relativistic theory of gravity. We start with the principles that guided Einstein in his search for the theory, then go on to examine the basic mathematical ingredients of the theory, and finally present the Einstein field equations that relate those ingredients and use them to provide a new explanation of gravitation that does not require the existence of any gravitational force.本章的目的是介绍广义相对论的核心思想——爱因斯坦的相对论引力理论。我们从引导爱因斯坦探索理论的原理开始，然后继续研究该理论的基本数学成分，最后提出将这些成分联系起来的爱因斯坦场方程，并用它们提供一种不需要任何引力存在的新的引力解释。

4.1 The founding principles of general relativity4.1 广义相对论的基本原理

Formulating a new theory in fundamental physics is not an entirely logical process. The search usually involves some general fundamental principles, consistency with known experimental facts, elegance and economy of ideas, and, inevitably, some guesswork. Of course, the ultimate test of any theory is provided by confronting its predictions with new experiments, and we shall come to this in Chapter 7; first we have to formulate the theory. Einstein was motivated in his search by three basic principles:在基础物理学中制定新理论并不是一个完全合乎逻辑的过程。搜索通常涉及一些一般的基本原则、与已知实验事实的一致性、思想的优雅和经济性，以及不可避免的一些猜测。当然，任何理论的最终检验都是通过新的实验来检验其预测，我们将在第七章中讨论这一点；首先我们必须阐述理论。爱因斯坦的研究受到三个基本原则的推动：

1. The principle of equivalence1. 等价原则

2. The principle of covariance2.协方差原理

3. The principle of consistency.3、一致性原则。

We shall discuss each of these in turn.我们将依次讨论这些内容。

4.1.1 The principle of equivalence4.1.1 等效原则

It was in 1907, just two years after the formulation of special relativity, that Einstein had the sudden insight that he later described as ‘the happiest thought of my life’. That thought was the realization that for an individual who was falling freely, accelerating downwards from a roof, say, or some other high place, it was almost as if gravity had been turned off. This idea, linking gravitation and acceleration, gave Einstein his start on extending relativity theory to include gravitation and showed him that a theory of general relative motion — one that included accelerations as well as uniform relative motions — could also be a theory of gravitation. This idea, that Einstein would later formalize as the principle of equivalence, also shed light on a troubling aspect of Newtonian mechanics.1907 年，即狭义相对论提出两年后，爱因斯坦突然有了顿悟，他后来将其描述为“我一生中最幸福的想法”。这个想法是这样的认识：对于一个自由落体的人来说，从屋顶或其他高处加速下降，几乎就像引力已经被关闭一样。这个将引力和加速度联系起来的想法使爱因斯坦开始将相对论扩展到包括引力，并向他表明广义相对运动理论(包括加速度和匀速相对运动)也可以是引力理论。爱因斯坦后来将这个想法形式化为等效原理，也揭示了牛顿力学的一个令人不安的方面。

The equality of gravitational and inertial mass引力质量和惯性质量相等

Newtonian mechanics involves two different concepts of mass:牛顿力学涉及两种不同的质量概念：

1. Inertial mass, m, which describes a particle’s resistance1. 惯性质量 m，描述粒子的阻力

to being accelerated by a force. The inertial mass of a particle is defined, according to Newton’s second law, by the ratio of the magnitude of the force on the m = | F |/| a |. particle to the magnitude of the acceleration it produces,被力加速。根据牛顿第二定律，粒子的惯性质量由作用在 m = | 上的力的大小之比来定义。 F|/|一个|。粒子产生的加速度的大小，

2. Gravitational mass, \(\mu\), which determines the force2. 引力质量，\(\mu\)，它决定了力

that a given particle experiences due to, or exerts on, another particle as a result of gravity. The gravitational mass is defined through Newton’s law of gravitation for the force \(F^{12}\) on particle 1 of gravitational mass \(\mu\) 1 due to particle 2 of gravitational mass \((\mu)^2\). The magnitude of this force can be written as一个给定的粒子由于引力而受到另一个粒子的影响或施加在另一个粒子上。引力质量通过牛顿万有引力定律定义，即引力质量为 \(\mu\) 1 的粒子 1 由于引力质量为 \((\mu)^2\) 的粒子 2 所受的力 \(F^{12}\)。该力的大小可以写为

Now, as will be discussed later, it is a well-established experimental fact that the ratio \(\mu\)/m is the same for all bodies, to an accuracy of at least one part in \(10^{11}\). In Newtonian mechanics, this is simply an extraordinary coincidence with no explanation. However, for Einstein it was something that cried out for a fundamental explanation. Of course, once we accept that the ratio of gravitational to inertial mass is a constant, then we can (and do) choose to use units of measurement that make the two masses for any body equal, so that \(\mu\)/m = 1. This is why we can ignore the distinction between gravitational and inertial masses for almost all practical purposes.现在，正如稍后将讨论的，这是一个公认的实验事实，即比率 \(\mu\)/m 对于所有物体都是相同的，精度至少为 \(10^{11}\) 的一部分。在牛顿力学中，这只是一个非凡的巧合，无需解释。然而，对于爱因斯坦来说，这是迫切需要一个基本解释的事情。当然，一旦我们接受引力质量与惯性质量之比是一个常数，那么我们就可以(并且确实)选择使用使任何物体的两个质量相等的测量单位，使得 \(\mu\)/m = 1。这就是为什么我们可以在几乎所有实际目的中忽略引力质量和惯性质量之间的区别。

Freely falling frames are locally inertial frames自由落体框架是局部惯性系

In Newtonian physics, the equality of inertial and gravitational mass implies that the acceleration of any body due to a gravitational force is independent of the mass of the body.在牛顿物理学中，惯性质量和引力质量的相等意味着任何物体由于引力而产生的加速度与物体的质量无关。

• ● Prove the above statement.● 证明上述说法。

❍ The equality of inertial and gravitational mass implies that \(\mu\) i in Equation 4.1 may be replaced by m i, so❍ 惯性质量和引力质量相等意味着方程 4.1 中的 \(\mu\) i 可以用 m i 代替，因此

The acceleration a 1 of particle 1 due to this force is given by由于该力，粒子 1 的加速度 a 1 由下式给出

and hence因此

Clearly, the mass m 1 cancels, and consequently the acceleration of any body subject only to gravitational forces will be independent of the mass of the body.显然，质量 m 1 抵消，因此任何仅受引力作用的物体的加速度将与物体的质量无关。

This result leads us to consider a famous ‘thought experiment’ in which it is supposed that a frictionless (non-rotating) lift is falling freely down an airless lift shaft (see Figure 4.4). The acceleration of the lift or any object in the vicinity of the lift is independent of its mass. Consequently, for an observer inside the lift, an object released from rest (relative to the observer) would remain stationary; that is, according to the freely falling observer, the object would be free of any force and would continue in its state of rest. Moreover, if the observer were to exert a force on the object, it would move according to Newton’s laws of motion. In other words, from the point of view of the observer in the freely falling lift, a frame of reference fixed in the lift is an inertial frame of reference.这个结果让我们考虑一个著名的“思想实验”，其中假设一个无摩擦(非旋转)的电梯沿着无空气的电梯井自由下落(见图 4.4)。电梯或电梯附近任何物体的加速度与其质量无关。因此，对于电梯内的观察者来说，从静止状态释放的物体(相对于观察者)将保持静止；也就是说，根据自由落体观察者的说法，该物体将不受任何力的影响，并将继续处于静止状态。此外，如果观察者对物体施加力，物体就会根据牛顿运动定律移动。换句话说，从自由落体升降机中的观察者的角度来看，固定在升降机中的参考系是惯性参考系。

Such a frame is properly described as a locally inertial frame (as opposed to a globally inertial frame) because we need to restrict our measurements to sufficiently small regions of space and sufficiently small intervals of time if we are not to observe departures from inertial behaviour. This is because the gravitational field in which the lift and its contents are located is not uniform. Two objects released simultaneously from the same height on opposite sides of the lift will each fall towards the centre of the Earth, so instead of falling along parallel paths and maintaining a constant separation, as they would in a uniform gravitational field, they will in fact fall along converging paths and gradually approach each other. The horizontal forces responsible for this non-inertial behaviour are examples of the tidal forces that cause neighbouring particles in any non-uniform gravitational field to have different accelerations. Such effects are usually small but they can have observable consequences (such as the tides in the Earth’s oceans!), and even within a freely falling lift they would be observable if experiments were performed with sufficient precision or over a sufficiently long period of time. Nonetheless, the point remains that a freely falling frame in a gravitational field is a locally inertial frame where the laws of special relativity will hold true.这样的框架被正确地描述为局部惯性系(而不是全局惯性系)，因为如果我们不观察惯性行为的偏离，我们需要将我们的测量限制在足够小的空间区域和足够小的时间间隔内。这是因为电梯及其内容物所在的引力场并不均匀。从升力两侧相同高度同时释放的两个物体将各自落向地球中心，因此它们不会像在均匀引力场中那样沿着平行路径下落并保持恒定的距离，而是实际上会沿着会聚路径下落并逐渐相互接近。导致这种非惯性行为的水平力是潮汐力的例子，潮汐力会导致任何非均匀引力场中的相邻粒子具有不同的加速度。这种效应通常很小，但可以产生可观察到的后果(例如地球海洋中的潮汐！)，即使在自由落体的升力内，如果实验以足够的精度或在足够长的时间内进行，它们也是可以观察到的。尽管如此，问题仍然是，引力场中的自由落体框架是局部惯性系，狭义相对论定律在其中成立。

Exercise 4.1 Two objects are 2.00 m apart in a freely练习 4.1 两个物体在自由空间中相距 2.00 m

falling lift near to the surface of the Earth (which has a radius of \(6.38\times10^{6}\) m).接近地球表面的下降升力(半径为 \(6.38\times10^{6}\) 米)。

(a) Calculate the magnitude of their acceleration towards each other when their separation is horizontal.(a) 计算当它们水平分离时它们相对于彼此的加速度大小。

(b) Calculate the magnitude of their acceleration towards each other when their separation is vertical.(b) 计算当它们垂直分离时它们相对于彼此的加速度大小。

Of course, you might well ask what is meant by ‘sufficiently small’ for a frame to be locally inertial. The answer is that we assume that having decided on limits to the accuracy of a particular experiment, we can always choose a small enough region and a short enough time interval so that a freely falling frame will appear to be inertial to within this accuracy.当然，您可能会问框架具有局部惯性的“足够小”是什么意思。答案是，我们假设在决定了特定实验精度的限制后，我们总是可以选择足够小的区域和足够短的时间间隔，以便自由落体框架在这个精度内看起来是惯性的。

Another thought experiment involves a rocket in a region in which there is no gravitational field. If the rocket is accelerated with a uniform acceleration of magnitude g, no sufficiently localized experiment within the rocket can distinguish between the consequences of the acceleration and the gravitational field on the surface of the Earth. An object released from rest within the rocket would accelerate downwards, just as an object on Earth would do (see Figure 4.5).另一个思想实验涉及在没有引力场的区域中发射火箭。如果火箭以 g 级的均匀加速度加速，则火箭内没有足够的局部实验可以区分加速度和地球表面引力场的后果。火箭内从静止状态释放的物体会向下加速，就像地球上的物体一样(见图 4.5)。

Principle of equivalence等效原理

In 1907, Einstein elevated to a formal principle the idea that locally one cannot distinguish between gravity and acceleration. What is now known as the weak equivalence principle can be stated as follows.1907 年，爱因斯坦将局部无法区分引力和加速度的想法提升为正式原则。现在所谓的弱等效原理可以表述如下。

Weak equivalence principle弱等价原理

Within a sufficiently localized region of spacetime adjacent to a concentration of mass, the motion of bodies subject to gravitational effects alone cannot be distinguished by any experiment from the motion of bodies within a region of appropriate uniform acceleration.在与质量集中相邻的足够局域化的时空区域内，任何实验都无法将仅受到引力作用的物体的运动与适当均匀加速度区域内的物体的运动区分开来。

The weak equivalence principle is a direct consequence of the fact that the acceleration of freely falling objects does not depend on their composition, and it is therefore sometimes referred to as the principle of universality of free fall. Note that this does not apply to very massive objects that would substantially change the gravitational field in their vicinity. Moreover, it only relates to gravitational forces, so experiments involving electromagnetic forces or nuclear interactions are specifically excluded.弱等效原理是自由落体物体的加速度不取决于其组成这一事实的直接结果，因此有时被称为自由落体普遍性原理。请注意，这不适用于会显着改变其附近引力场的非常大的物体。此外，它只与引力有关，因此涉及电磁力或核相互作用的实验被明确排除。

The restriction to gravitational forces does not apply to the strong equivalence principle.对引力的限制不适用于强等效原理。

Strong equivalence principle强等价原则

Within a sufficiently localized region of spacetime adjacent to a concentration of mass, the physical behaviour of bodies cannot be distinguished by any experiment from the physical behaviour of bodies within a region of appropriate uniform acceleration.在与质量集中相邻的足够局部的时空区域内，任何实验都无法将物体的物理行为与适当均匀加速度区域内的物体的物理行为区分开来。

This statement (which is often simply referred to as the equivalence principle) clearly goes beyond the universality of free fall, although that is included as a special case.这种说法(通常简称为等效原理)显然超出了自由落体的普遍性，尽管它被列为特例。

Both versions of the equivalence principle have been subject to many direct experimental tests. Galileo is often said to have demonstrated the universality of free fall by dropping different objects from the leaning tower of Pisa. It is unlikely that he actually performed such an experiment, but the experiments that he did perform — rolling bodies down inclined planes — should have made him aware of the outcome to expect. The first high-precision tests were carried out over many years with steadily improving sensitivity, eventually reaching better than one part in 10 8, by the Hungarian scientist Lora`nd Eo¨tvo¨s (pronounced ‘ert-vos’) in the late nineteenth and early twentieth centuries. These results were quoted by Einstein in his first complete formulation of general relativity. Currently, the most rigorous test of the weak equivalence principle is provided by the Eo¨t-Wash experiments, which provide agreement to better than one part in \(10^{12}\) (see Figure 4.6). Projected satellite experiments could provide even more stringent tests. For instance, the proposed Satellite Test of the Equivalence Principle (STEP), a space mission that is still in the design stage, could provide an accuracy of one part in \(10^{18}\).等效原理的两个版本都经过了许多直接的实验测试。人们常说伽利略通过从比萨斜塔上扔下不同的物体来证明自由落体的普遍性。他不太可能真的进行过这样的实验，但他所做的实验——让物体沿着斜面滚动——应该让他意识到预期的结果。十九世纪末二十世纪初，匈牙利科学家洛兰·埃奥特沃斯(Lora`nd Eo´tvo´s，发音为“ert-vos”)进行了多年的首次高精度测试，灵敏度稳步提高，最终达到了 10 8 分之一以上。爱因斯坦在他第一个完整的广义相对论中引用了这些结果。目前，Eo et-Wash 实验提供了对弱等效原理最严格的测试，该实验提供了比 \(10^{12}\) 中的一个部分更好的一致性(见图 4.6)。预计的卫星实验可以提供更严格的测试。例如，拟议的等效原理卫星测试(STEP)是一项仍处于设计阶段的太空任务，可以提供 \(10^{18}\) 中一个零件的精度。

Experimental tests of the strong equivalence principle are much less clear-cut, but most theories that violate it predict that the locally measured value of the gravitational constant, G, may vary with time. Current constraints on the rate of change of G are approaching one part in \(10^{13}\) \(\mathrm{year^{-1}}\). Einstein’s theory of general relativity is thought to be the only theory of gravity that is consistent with the strong equivalence principle.强等效原理的实验测试不太明确，但大多数违反该原理的理论都预测，当地测量的引力常数 G 的值可能会随时间而变化。目前对G变化率的限制正在接近\(10^{13}\)年−1的一部分。爱因斯坦的广义相对论被认为是唯一符合强等效原理的引力理论。

Although the strong equivalence principle is certainly in need of additional tests, the weak equivalence principle alone was sufficient to lead Einstein to predict two new effects that eventually became part of general relativity. First, consider a horizontally travelling beam of light that enters and crosses the interior of a rocket that is accelerating vertically upwards — at right angles to the beam of light. From the point of view of the accelerated observer travelling with the rocket, the light ray follows a downward-curving path. The local equivalence of gravitation and acceleration therefore led Einstein to predict that one effect of gravitation would be the deflection of light rays towards concentrations of mass. The second effect was based on the fact that an observer in an upward-accelerating rocket would find that the frequency of light waves emitted from the floor of the rocket would be redshifted (i.e. their frequency would be decreased) as successive wave peaks took longer and longer to reach the ceiling. (These effects are illustrated in Figure 4.7.) This led Einstein to predict that light escaping from a concentration of mass should exhibit a redshift due to gravity. As you will see later, these two predicted effects, the gravitational deflection of light and the gravitational redshift of light, both became the subject of refined calculations in the full theory of general relativity and both eventually became important tests of the theory.尽管强等效原理肯定需要额外的测试，但仅弱等效原理就足以让爱因斯坦预测出两个新效应，最终成为广义相对论的一部分。首先，考虑一束水平行进的光束，该光束进入并穿过火箭内部，火箭垂直向上加速——与光束成直角。从随火箭行进的加速观察者的角度来看，光线遵循向下弯曲的路径。因此，引力和加速度的局部等效性使爱因斯坦预测引力的一种效应是使光线偏向质量集中。第二个效应基于这样一个事实：向上加速的火箭中的观察者会发现，随着连续的波峰到达天花板所需的时间越来越长，从火箭底部发射的光波的频率会发生红移(即它们的频率会降低)。(这些效应如图 4.7 所示。)这使得爱因斯坦预测，从质量集中逸出的光应该会因引力而表现出红移。正如你稍后将看到的，光的引力偏转和光的引力红移这两种预测效应都成为了整个广义相对论中精细计算的主题，并最终成为了该理论的重要检验。

The final form of general relativity was not clear to Einstein in 1907, but his realization that gravitation was in some sense locally equivalent to acceleration made the notion of a gravitational force suspect and the equivalence of gravitational and inertial mass almost a matter of course. The idea that a freely falling (accelerated) observer was equivalent to an inertial observer, at least locally, raised again the issue of coordinate transformations but made it clear that in general relativity the class of relevant coordinate transformations would have to be much broader than the Lorentz transformations of special relativity.1907 年，爱因斯坦还不清楚广义相对论的最终形式，但他意识到引力在某种意义上局部等同于加速度，这使得引力的概念受到怀疑，而引力和惯性质量的等同性几乎是理所当然的。自由落体(加速)观察者至少在局部相当于惯性观察者的想法再次提出了坐标变换的问题，但清楚地表明，在广义相对论中，相关坐标变换的类别必须比狭义相对论的洛伦兹变换广泛得多。

4.1.2 The principle of general covariance4.1.2 广义协方差原理

General covariance一般协方差

The principle of general covariance is an extension of the principle of relativity that was introduced in Chapter 1. According to the principle of relativity, the laws of physics should take the same form in all inertial frames. As you saw in Chapter 2, that implied that physical laws should be form-invariant under Lorentz transformations, and a way of ensuring that was to write the laws as properly balanced four-tensor relations. We saw how to do that for the laws of electromagnetism using scalar invariants (four-tensors of rank zero), contravariant and covariant four-vectors (four-tensors of rank 1), and some four-tensors of [F \(\mu\)\(\nu\)], the mixed field rank 2 — specifically, the contravariant field four-tensor four-tensor [F \(\mu\) \(\nu\)], and the covariant field four-tensor [F \(\mu\)\(\nu\)]. (Remember that when we enclose a tensor component in square brackets, it indicates that we are discussing the entire tensor, not just the individual component.) You will also recall that it was the principle of relativity that excluded Newtonian gravitation from being a viable relativistic theory of gravity; the Newtonian gravitational force cannot be described as part of a four-vector, because it does not transform in the right way.广义协变原理是第一章中介绍的相对论原理的延伸。根据相对论原理，物理定律在所有惯性系中应采取相同的形式。正如您在第二章中看到的，这意味着物理定律在洛伦兹变换下应该是形式不变的，确保这一点的一种方法是将定律写成适当平衡的四张量关系。我们看到了如何使用标量不变量(秩为 0 的四张量)、逆变和协变四向量(秩为 1 的四个张量)以及 [F \(\mu\)\(\nu\)] 的一些四张量、混合场秩 2 — 具体而言，逆变场四张量四张量 \([F^{\mu}]\) 来实现电磁定律。 \(\nu\)]，以及协变场四张量 [F \(\mu\)\(\nu\)]。(请记住，当我们将张量分量括在方括号中时，这表明我们正在讨论整个张量，而不仅仅是单个分量。)您还会记得，正是相对论原理将牛顿引力排除在可行的相对论引力理论之外。牛顿引力不能被描述为四矢量量量的一部分，因为它没有以正确的方式变换。

The principle of general covariance extends the principle of relativity by requiring the physical equivalence of all frames, including non-inertial ones.广义协变原理通过要求所有框架(包括非惯性系)的物理等效性来扩展相对论原理。

There is still debate about the significance of this principle and the extent to which Einstein was successful in implementing it in general relativity. However, what he did in practice was to require that physical laws should retain their form under a broad class of coordinate transformations, and he did this by requiring that the laws should be expressed in terms of mathematical objects called general tensors, or more often just tensors. Most of this section will be devoted to making clear what tensors are, how they differ from the more restricted four-tensors that you met in Chapter 2, and how they may be combined to form tensor equations that might describe generally covariant laws of physics, including gravitation.关于这一原理的重要性以及爱因斯坦在广义相对论中成功应用该原理的程度，仍然存在争议。然而，他在实践中所做的是要求物理定律应该在广泛的坐标变换下保持其形式，并且他通过要求物理定律应该用称为广义张量或更常见的张量的数学对象来表达来做到这一点。本节的大部分内容将致力于阐明张量是什么，它们与您在第 2 章中遇到的更受限制的四张量有何不同，以及如何将它们组合起来形成张量方程，这些方程可以描述一般的协变物理定律，包括万有引力。

Defining general tensors定义一般张量

The study of tensors can be approached in several ways, but for our purposes tensors are multi-component mathematical objects that can be recognized and classified according to the way their components behave under general coordinate transformations — that is, under coordinate transformations in which the new coordinates \(x'\) \(\mu\) are functions of the old coordinates x \(\nu\), as in \(x'\) \(\mu\) = \(x'\) \(\mu\) (x \(\nu\)) for \(\mu\), \(\nu\) = 0, 1, 2, 3. These functions are required to be sufficiently well-behaved that they can be differentiated, but they are still more general than the Lorentz transformations of special relativity, which were restricted to linear functions. In the case of the Lorentz transformations, the linearity ensured that derivatives such as ∂\(x'\) \(\mu\)/∂x \(\nu\) would be constants (such as c, V, γ or combinations of those parameters). In the case of a general coordinate transformation \(x'\) \(\mu\) = \(x'\) \(\mu\) (x \(\nu\)), the sixteen functions ∂\(x'\) \(\mu\)/∂x \(\nu\) (\(\mu\), \(\nu\) = 0, 1, 2, 3) and the sixteen functions ∂x \(\beta\)/∂\(x'\) \(\alpha\) (\(\alpha\), \(\beta\) = 0, 1, 2, 3) are free of such restrictions. Having explained what is meant by a general coordinate transformation, we can say that a tensor of contravariant rank m and covariant rank n has components \(T'\) \(\mu\) \(\alpha\) 1 \(\mu\) \(\alpha\) 2...\(\mu\)...\(\alpha\) m that transform according to张量的研究可以通过多种方式进行，但就我们的目的而言，张量是多分量数学对象，可以根据其分量在一般坐标变换下的行为方式进行识别和分类 - 也就是说，在坐标变换下，新坐标 \(x'\) \(\mu\) 是旧坐标 x \(\nu\) 的函数，如 \(x'\) \(\mu\) = \(x'\) \(\mu\) (x \(\nu\)) 对于 \(\mu\)，\(\nu\) = 0, 1, 2, 3。这些函数需要表现得足够好才能被微分，但它们仍然比狭义相对论的洛伦兹变换更普遍，后者仅限于线性函数。在洛伦兹变换的情况下，线性确保诸如 ∂\(x'\) \(\mu\)/∂x \(\nu\) 之类的导数将是常数(例如 c、V、γ 或这些参数的组合)。在一般坐标变换 \(x'\) \(\mu\) = \(x'\) \(\mu\) (x \(\nu\)) 的情况下，十六个函数 ∂\(x'\) \(\mu\)/∂x \(\nu\) (\(\mu\), \(\nu\) = 0, 1, 2, 3) 和十六个函数∂x \(\beta\)/∂\(x'\) \(\alpha\) (\(\alpha\), \(\beta\) = 0, 1, 2, 3) 不受此类限制。解释了一般坐标变换的含义后，我们可以说逆变秩 m 和协变秩 n 的张量的分量 \(T'\) \(\mu\) \(\alpha\) 1 \(\mu\) \(\alpha\) 2...\(\mu\)...\(\alpha\) m 根据以下变换

Expressed in such general terms this looks very complicated, but the simple fact is that you have already met many of the most important tensor quantities that will be needed in this book. In particular, you are already familiar with the notion of a scalar invariant, S say, that remains unchanged under a general coordinate transformation. And you are also familiar with the infinitesimal displacement [d x \(\mu\)] = (d \(x^0\), d \(x^1\), \((dx)^2\), d \(x^{3}\)). This is actually a contravariant tensor of rank 1 with components that transform according to用这样的通用术语来表达，这看起来非常复杂，但简单的事实是，您已经满足了本书中需要的许多最重要的张量数量。特别是，您已经熟悉标量不变量的概念，例如 S，它在一般坐标变换下保持不变。您也熟悉无穷小位移 [d x \(\mu\)] = (d \(x^0\), d \(x^1\), \((dx)^2\), d \(x^{3}\))。这实际上是一个 1 阶逆变张量，其分量根据以下公式进行变换

You have also met the vastly important rank 2 metric tensor \(g_{\mu\nu}\). In its contravariant (dual) form its components transform according to您还遇到了极其重要的 2 阶度规张量 \(g_{\mu\nu}\)。在其逆变(对偶)形式中，其组件根据以下公式进行变换：

and in the covariant form they transform according to在协变形式中，它们根据

The metric tensor components satisfy the useful relationship度规张量分量满足有用关系

where δ \(\alpha\) \(\beta\) is a four-dimensional version of the Kronecker delta and is itself defined by其中 δ \(\alpha\) \(\beta\) 是克罗内克 delta 的四维版本，其本身定义为

1 if \(\alpha\) = \(\beta\), 0 if \(\alpha\) 3 = \(\beta\).如果 \(\alpha\) = \(\beta\)，则为 1；如果 \(\alpha\)，则为 0 3 = \(\beta\)。

You have even met the Riemann curvature tensor [R \(\alpha\) βγδ], a mixed tensor of contravariant rank 1 and covariant rank 3. In four-dimensional spacetime this tensor has 256 components, though due to symmetries, only 20 are independent. Each component transforms according to您甚至还遇到过黎曼曲率张量 [R \(\alpha\) βγδ]，它是逆变 1 级和协变 3 级的混合张量。在四维时空中，该张量有 256 个分量，但由于对称性，只有 20 个分量是独立的。每个组件根据以下方式进行变换

A final point to note — or rather to recall, since it was mentioned in Chapter 3 — is that not all multi-component objects are tensors. It was pointed out earlier that the 64 connection coefficients Γ \(\alpha\) βγ of a four-dimensional spacetime do not satisfy the appropriate transformation law for a mixed rank 3 tensor, so they simply do not form a tensor.最后要注意的一点——或者更确切地说是回忆一下，因为它在第 3 章中提到过——是并非所有多分量对象都是张量。前面已经指出，四维时空的 64 个联络系数 Γ \(\alpha\) βγ 不满足混合阶 3 张量的适当变换定律，因此它们根本不构成张量。

Exercise 4.2 Suppose that in a two-dimensional Euclidean space with练习 4.2 假设在二维欧几里得空间中

coordinates x \(\mu\) (\(\mu\) = 1, 2) the coordinates \(x^1\) and \(x_{2}\) correspond to the polar coordinates r and \(\theta\). Also suppose that the coordinates \(x'\) \(\mu\) correspond to the usual Cartesian coordinates x, y.坐标x \(\mu\) (\(\mu\) = 1, 2) 坐标x 1 和x 2 对应于极坐标r 和\(\theta\)。还假设坐标 \(x'\) \(\mu\) 对应于通常的笛卡尔坐标 x, y。

(a) If A \(\mu\) is a general tensor component in r, \(\theta\) coordinates, and \(A'\) \(\mu\) is the corresponding tensor component in x, y coordinates, find the transformation that expresses \(A'\) \(\mu\) in terms of A \(\mu\) for each value of \(\mu\).(a) 如果 A \(\mu\) 是 r、\(\theta\) 坐标中的一般张量分量，并且 \(A'\) \(\mu\) 是 x、y 坐标中的相应张量分量，则对于 \(\mu\) 的每个值，找到用 A \(\mu\) 表示 \(A'\) \(\mu\) 的变换。

(b) Confirm that this transformation law is satisfied by the two-dimensional infinitesimal displacement vector that has components (d \(x^1\), \((dx)^2\)) = (d r, d \(\theta\)) and (d \(x'_{1}\), d \(x'_{2}\)) = (d x, d y).(b) 确认具有分量 (d \(x^1\), \((dx)^2\)) = (d r, d \(\theta\)) 和 (d \(x'_{1}\), d \(x'_{2}\)) = (d x, d y) 的二维无穷小位移矢量满足该变换定律。

Raising and lowering general tensor indices提高和降低一般张量指数

It is the metric tensor that relates contravariant and covariant tensor components via它是通过以下方式将逆变和协变张量分量关联起来的度规张量

and和

In other words, the contravariant metric tensor ‘raises’ indices and the covariant metric tensor ‘lowers’ them.换句话说，逆变度规张量“提高”指数，协变度规张量“降低”指数。

Exercise 4.3 Show that if we use the covariant metric tensor to ‘lower’ the练习 4.3 证明如果我们使用协变度规张量来“降低”

index on A \(\mu\) and then we use the contravariant metric tensor to ‘raise’ the index again, we get back to A \(\mu\).A \(\mu\) 上的索引，然后我们使用逆变度规张量再次“提高”索引，我们回到 A \(\mu\)。

• ● If we have a mixed tensor with some indices up and some down, it is usually如果我们有一个混合张量，其中一些索引向上，一些索引向下，通常是

important to leave spaces so that, for example, we write R \(\alpha\) βγδ rather than R βγδ \(\alpha\). Explain why.留出空格很重要，例如，我们写 R \(\alpha\) βγδ 而不是 R βγδ \(\alpha\)。解释一下为什么。

❍ Suppose that we start with R αβγδ, then use the contravariant metric tensor to raise the \(\alpha\) index without paying attention to the order of the indices. We will obtain the result R βγδ \(\alpha\). The problem is that the individual indices are just placeholders and have no special significance. This means that if we subsequently use the covariant metric tensor to lower the \(\alpha\) index, it is impossible to tell if the lowered index should be put in the first or second ‘slot’, i.e. whether the result should be R αβγδ or R βαγδ. Unless the tensor happens to be symmetric with respect to interchange of the first two indices, the two possible results will be different. It is therefore usually important to preserve the order of the indices despite any raising or lowering that may be \(\alpha\) βγδ rather than R βγδ \(\alpha\). performed. That’s why we should generally write R❍ 假设我们从 R αβγδ 开始，然后使用逆变度规张量来提高 \(\alpha\) 索引，而不注意索引的顺序。我们将得到结果 R βγδ \(\alpha\)。问题是各个指数只是占位符，没有特殊意义。这意味着，如果我们随后使用协变度规张量来降低 \(\alpha\) 指数，则无法判断降低后的指数应该放在第一个还是第二个“槽”中，即结果应该是 R αβγδ 还是 R βαγδ。除非张量恰好关于前两个索引的互换对称，否则两个可能的结果将会不同。因此，尽管可能是 \(\alpha\) βγδ 而不是 R βγδ \(\alpha\)，但保持索引的顺序通常很重要。执行。这就是为什么我们一般应该写 R

The rules of tensor algebra张量代数规则

Einstein’s aim was to use tensors to write down a theory of gravity in a generally covariant form — in other words, following the rules of general tensor algebra for multiplying tensors by scalars, adding and subtracting tensors, multiplying tensors together and reducing the rank of a tensor through contraction. These rules are similar to those that we have already used to manipulate four-tensors in special relativity, but to make them completely clear, we now list them in their general forms.爱因斯坦的目标是使用张量以一般协变的形式写下引力理论——换句话说，遵循一般张量代数的规则，将张量乘以标量、加减张量、将张量相乘以及通过收缩降低张量的阶。这些规则与我们已经用来操纵狭义相对论中的四张量的规则类似，但为了使它们完全清晰，我们现在以一般形式列出它们。

1. Scaling A tensor [T \(\mu\) 1 \(\alpha\) \((\mu)^2\) \(\alpha\)...\(\mu\)...\(\alpha\) m] of contravariant1. 逆变张量 [T \(\mu\) 1 \(\alpha\) \((\mu)^2\) \(\alpha\)...\(\mu\)...\(\alpha\) m] 的缩放

rank m and covariant rank n may be multiplied by a scalar S to produce a new tensor [U \(\mu\) 1 \(\alpha\) \((\mu)^2\) \(\alpha\)...\(\mu\)...\(\alpha\) m] of the same rank. Each component of the new tensor is obtained by multiplying the corresponding component of the original tensor by the same scalar S. So, for example, for all values of \(\mu\) and \(\alpha\),阶 m 和协变阶 n 可以乘以标量 S 以产生相同阶的新张量 [U \(\mu\) 1 \(\alpha\) \((\mu)^2\) \(\alpha\)...\(\mu\)...\(\alpha\) m]。新张量的每个分量都是通过将原始张量的相应分量乘以相同的标量 S 获得的。因此，例如，对于 \(\mu\) 和 \(\alpha\) 的所有值，

2. Addition and subtraction Tensors may be added2. 加法和减法张量可以相加

or subtracted to form new tensors, but those being added or subtracted must be of the same type, i.e. with the same contravariant rank and the same covariant rank. Again the addition or subtraction is carried out component by component. So, for example, for all values of \(\mu\) and \(\alpha\),或相减以形成新的张量，但相加或相减的张量必须是同一类型，即具有相同的逆变秩和相同的协变秩。同样，加法或减法是逐个分量地进行的。因此，例如，对于 \(\mu\) 和 \(\alpha\) 的所有值，

3. Multiplication Tensors may be multiplied together3. 乘法张量可以相乘

by forming products of \([X^{\mu}]\), [Y \(\alpha\)] and [Z \(\beta\)], their components. So, for example, given three tensors we can form a new tensor [A \(\mu\) αβ] with components通过形成产品\([X^{\mu}]\)、\([Y^{\alpha}]\)和\([Z^{\beta}]\)及其组件。因此，例如，给定三个张量，我们可以形成一个新的张量 [A \(\mu\) αβ]，其分量

The rank of the new tensor is then the sum of the ranks of the tensors being multiplied together (e.g. A \(\mu\) αβ has rank 3). The tensors being multiplied together may even be the same, as in新张量的秩就是张量的秩相乘的总和(例如 \(\mu\) αβ 的秩为 3)。相乘的张量甚至可能是相同的，如

4. Contraction In the case of a single tensor with4. 收缩在单个张量的情况下

contravariant rank m and covariant rank n, or in the case of a product of tensors with combined contravariant rank m and covariant rank n, it is possible to form another tensor, of contravariant rank \(\mathrm{m^{-1}}\) and covariant rank n − 1, by summing over one raised index and one lowered index. So, for example,逆变等级 m 和协变等级 n，或者在张量与逆变等级 m 和协变等级 n ​​组合的乘积的情况下，可以通过对一个升高的索引和一个降低的索引求和来形成逆变等级 \(\mathrm{m^{-1}}\) 和协变等级 n ​​− 1 的另一个张量。所以，举例来说，

These rules imply that tensors can appear in expressions only in certain well-defined ways. In order to illustrate this, consider the following (fairly arbitrary) equation involving tensors:这些规则意味着张量只能以某些明确定义的方式出现在表达式中。为了说明这一点，请考虑以下涉及张量的(相当任意的)方程：

The right-hand side of Equation 4.8 consists of the sum of three ‘terms’, which we can use to emphasize some important general properties of tensor equations.方程 4.8 的右侧由三个“项”之和组成，我们可以用它来强调张量方程的一些重要的一般性质。

• • The only indices that are not ‘summed over’ are \(\mu\) and \(\nu\). These are the free唯一未“汇总”的指数是 \(\mu\) 和 \(\nu\)。这些都是免费的

indices. They exhibit the following properties:指数。它们具有以下特性：

(a) The \(\mu\) and \(\nu\) indices are consistently ‘up’ (contravariant) or ‘down’ (covariant).(a) \(\mu\) 和 \(\nu\) 指数始终“上升”(逆变)或“下降”(协变)。

(b) The \(\mu\) and \(\nu\) indices appear once and only once in every term on each side of the equation.(b) \(\mu\) 和 \(\nu\) 指数在方程每边的每一项中出现且仅出现一次。

(c) The letters \(\mu\) and \(\nu\) have no special significance. We can replace either (or both) of them with a different (Greek) letter provided that we carry out the replacement in every term (on both sides of the equation) and the new letter does not clash with one that is already in use. For example, we could replace \(\mu\) with \(\lambda\), but replacing \(\mu\) with \(\alpha\) would cause confusion.(c) 字母 \(\mu\) 和 \(\nu\) 没有特殊意义。我们可以用不同的(希腊)字母替换其中一个(或两个)，前提是我们在每一项(等式两边)中进行替换，并且新字母不会与已使用的字母发生冲突。例如，我们可以将 \(\mu\) 替换为 \(\lambda\)，但将 \(\mu\) 替换为 \(\alpha\) 会导致混乱。

• • Some indices (\(\alpha\) and \(\beta\) in this example) appear precisely twice in a term. These某些索引(本例中为 \(\alpha\) 和 \(\beta\))在一个术语中恰好出现两次。这些

are the dummy indices.是虚拟指数。

(a) Such indices are always summed over.(a) 这些指数总是相加的。

(b) One appearance is always ‘up’ and the other is ‘down’.(b) 一种外观总是“向上”，另一种外观总是“向下”。

(c) The letter used has no special significance and can always be replaced with another (Greek) letter provided that we replace both occurrences within any one term and the new letter doesn’t clash with one that is already in use. For example, \(\alpha\) in the third term on the right-hand side could be replaced with γ, but not with \(\beta\).(c) 所使用的字母没有特殊意义，并且始终可以用另一个(希腊)字母替换，前提是我们替换任何一个术语中出现的两个字母，并且新字母不会与已使用的字母发生冲突。例如，右侧第三项中的α可以用γ代替，但不能用β代替。

As you can see, the indices within a covariant equation form very distinct patterns that you will soon become adept at spotting. Expressions such as Equation 4.8 are said to be generally covariant or, more simply, in covariant form. This means that the equation will take the same form in any coordinate system; it does not, of course, mean that the numerical values of the components are necessarily the same. It is worth noticing how the word ‘covariant’ is a bit over-used. A rank 1 ‘covariant tensor’ is one with components that transform according to正如您所看到的，协变方程中的索引形成了非常独特的模式，您很快就会熟练地发现这些模式。诸如方程 4.8 之类的表达式通常被认为是协变的，或者更简单地说，是协变形式的。这意味着该方程在任何坐标系中都将采用相同的形式；当然，这并不意味着各分量的数值必须相同。值得注意的是，“协变”这个词有点被过度使用了。 1 阶“协变张量”是一个其分量根据以下公式进行变换的张量：

and is denoted by having the indices ‘down’. A ‘covariant equation’ is an equation that takes the same form in different coordinate systems, and may or may not involve covariant tensors. Indeed, a covariant equation may involve contravariant tensors.并通过指数“下降”来表示。 “协变方程”是在不同坐标系中采用相同形式的方程，并且可能涉及也可能不涉及协变张量。事实上，协变方程可能涉及逆变张量。

• ● What is the analogous equation to Equation 4.9 that describes how the与方程 4.9 类似的方程描述了如何

components of a rank 1 contravariant tensor transform?1 阶逆变张量变换的组成部分？

❍ From Equation 2.110 or from the rank 1 example that follows it in Equation 4.2, the required transformation rule is❍ 根据公式 2.110 或公式 4.2 中的秩 1 示例，所需的变换规则为

Exercise 4.4 Explain why each of the following is not练习 4.4 解释为什么下列各项不成立

a generally covariant tensor equation. Y \(\mu\)\(\nu\) Z \(\nu\) (c) A = (a) A \(\mu\) = B + K (b) X \(\mu\) =一般协变张量方程。 Y \(\mu\)\(\nu\) Z \(\nu\) (c) A = (a) A \(\mu\) = B + K (b) X \(\mu\) =

The rules of covariant differentiation协变微分的规则

When we wrote down the laws of Lorentz-covariant electromagnetism in Chapter 2, in addition to scaling, adding, multiplying and contracting four-tensors, we also formed four-tensors by taking partial derivatives of existing tensors. Being able to represent derivatives of four-tensors was important because the basic laws of electromagnetism (the Maxwell equations and the equation of continuity) were differential equations. We should expect the generally covariant theory of gravitation to involve differential equations, so we need to know how to differentiate a general tensor in a covariant way. This turns out to be more complicated in general relativity than it was in special relativity because simple partial derivatives of tensors are not generally covariant.当我们在第二章写下洛伦兹协变电磁学定律时，除了对四张量进行缩放、加法、乘法和收缩之外，我们还通过对现有张量求偏导数来形成四张量。能够表示四张量的导数非常重要，因为电磁学的基本定律(麦克斯韦方程和连续性方程)是微分方程。我们应该期望广义协变引力理论涉及微分方程，因此我们需要知道如何以协变的方式微分广义张量。事实证明，广义相对论中的情况比狭义相对论中的情况更为复杂，因为张量的简单偏导数通常不是协变的。

Defining the derivative of a function involves evaluating the function at some point, x say, and at a nearby point, x + δx say, and then taking the difference. In a flat space this does not present any particular problem. Nor is it particularly complicated in a curved space as long as we are only considering functions. vector [v \(\alpha\)] (i.e. a rank 1 However, we know from Chapter 3 that transporting a tensor) requires some care since the parallel transport of a vector generally involves the connection coefficients定义函数的导数涉及在某个点(例如 x)和附近的点(例如 x + δx)评估函数，然后求差。在平坦的空间中，这不会出现任何特殊问题。只要我们只考虑函数，在弯曲空间中也不是特别复杂。向量 \([v^{\alpha}]\) (即a 等级 1 然而，我们从第 3 章知道，传输张量需要小心，因为向量的并行传输通常涉及联络系数

For a vector with components v \(\alpha\), the expression对于具有分量 v \(\alpha\) 的向量，表达式

simply does not transform in the right way under general coordinate transformations to be a component of a rank 2 tensor. Nor does the expression在一般坐标变换下，根本无法以正确的方式变换为 2 阶张量的组成部分。表达式也不

However, sums of the form然而，形式的总和

arise when considering the limit of a difference in a vector and its parallel transported version, and this quantity does transform as a component of a rank 2 tensor. Expressions of this kind occur so frequently in general relativity that it is useful to give them a name and a symbol. Consequently, we write当考虑向量及其并行传输版本的差异极限时，就会出现这种情况，并且该量确实会变换为 2 阶张量的组成部分。这种表达式在广义相对论中出现得非常频繁，因此给它们命名和符号是很有用的。因此，我们写

and say that ∇ \(\beta\) v \(\alpha\) represents the covariant derivative of v \(\alpha\). In effect, the behaviour of ∂v \(\alpha\)/∂x \(\beta\) is cancelled by the non-tensorial behaviour non-tensorial, of \(\lambda\) Γ \(\alpha\) \(\lambda\)\(\beta\) v \(\lambda\). At this stage, you should regard ∇ \(\beta\) v \(\alpha\) as no more than a shorthand for the right-hand side of Equation 4.11. Of course, we don’t just want to differentiate rank 1 contravariant tensors. We also need to know how to covariantly differentiate rank 1 covariant tensors and tensors of higher rank, so that the result is a tensor in each case. It can be shown that Equation 4.11 implies that the covariant derivative of a covariant tensor v \(\alpha\) can be expressed as并称 ∇ \(\beta\) v \(\alpha\) 表示 v \(\alpha\) 的协变导数。实际上，∂v \(\alpha\)/∂x \(\beta\) 的行为被 \(\lambda\) Γ \(\alpha\) \(\lambda\)\(\beta\) v \(\lambda\) 的非张量行为抵消。在此阶段，您应该将 ∇ \(\beta\) v \(\alpha\) 视为公式 4.11 右侧的简写。当然，我们不仅仅想区分 1 阶逆变张量。我们还需要知道如何协变区分 1 阶协变张量和更高阶的张量，以便在每种情况下结果都是一个张量。可以看出，方程 4.11 意味着协变张量 v \(\alpha\) 的协变导数可以表示为

Note that in this case the final term is subtracted from the partial derivative, whereas in the case of a contravariant vector it was added. The covariant derivatives of higher-rank tensors are direct generalizations of Equations 4.11 and 4.12, as appropriate. For instance,请注意，在这种情况下，从偏导数中减去最后一项，而在逆变向量的情况下，最后一项被添加。高阶张量的协变导数是方程 4.11 和 4.12 的直接推广(视情况而定)。例如，

• ● Write down the expression for ∇ \(\lambda\) T \(\mu\) \(\nu\) in terms of the connection coefficients.用联络系数写出 ∇ \(\lambda\) T \(\mu\) \(\nu\) 的表达式。

❍ From Equations 4.11 and 4.12, we have❍ 根据方程 4.11 和 4.12，我们有

This is a good point at which to restate the principle of general covariance and summarize its significance in the formulation of general relativity.这是重申广义协变原理并总结其在广义相对论表述中的重要性的好时机。

General covariance, tensors and covariant differentiation一般协方差、张量和协变微分

According to the principle of general covariance, the laws of physics should take the same form in all frames of reference. In practice this means that they should be expressed as balanced tensor relationships that are covariant under general coordinate transformations.根据广义协变原理，物理定律在所有参考系中应采取相同的形式。实际上，这意味着它们应该表示为在一般坐标变换下协变的平衡张量关系。

Legitimate algebraic operations involving tensors include scaling, addition and subtraction (provided that the types are identical), multiplication and contraction. The partial differentiation of a tensor does not generally produce another tensor, but the process of covariant differentiation does. This may be applied to a tensor of any rank and is exemplified by涉及张量的合法代数运算包括缩放、加法和减法(前提是类型相同)、乘法和收缩。张量的偏微分通常不会产生另一个张量，但协变微分的过程会产生。这可以应用于任何阶的张量，例如

Exercise 4.5 What is the covariant derivative of the invariant scalar function练习 4.5 什么是不变标量函数的协变导数

S (ct, x, y, z)? (Hint: This is a tensor of rank 0.)S(ct、x、y、z)？ (提示：这是一个 0 阶张量。)

4.1.3 The principle of consistency4.1.3 一致性原则

The principle of consistency asserts that a new theory that aims to replace or supersede earlier theories should account for the successful predictions of those earlier theories. In the particular case of general relativity, we should expect consistency with the successes of Einstein’s own special relativity and Newtonian gravitation. The former requirement is guaranteed by using a spacetime that is locally equivalent to Minkowski spacetime; the latter provides a useful constraint on the kinds of tensor equations that can be used in the formulation of general relativity.一致性原则主张，旨在取代或取代早期理论的新理论应该解释那些早期理论的成功预测。在广义相对论的特殊情况下，我们应该期望与爱因斯坦自己的狭义相对论和牛顿引力的成功保持一致。前一个要求是通过使用局部等价于闵可夫斯基时空的时空来保证的；后者对可用于广义相对论公式的张量方程类型提供了有用的约束。

For the purposes of establishing consistency with Newtonian predictions, it is helpful to first see how Newton’s theory of gravity, as expressed by the inverse square law, can be reformulated as a field theory, based on the idea of a gravitational field that obeys differential equations similar to those satisfied by the electric and magnetic fields of electromagnetism.为了与牛顿预测保持一致，首先了解牛顿的引力理论(以平方反比定律表示)如何可以重新表述为场论，基于引力场的思想，该引力场服从与电磁场的电场和磁场所满足的微分方程类似的微分方程，这是有帮助的。

field g (r) to be a To this end, we first define the Newtonian gravitational function of position r = (x, y, z) that specifies the Newtonian gravitational force per unit mass that would act on a test particle at the point r. This means that the be m g (r). It follows gravitational force on a particle of mass m at r would from Newton’s law of gravitation (Equation 4.1) that in the case of a uniform coordinates (r = 0), the spherical body of total mass M centred on the origin of gravitational field is given by场 g (r) 为 a 为此，我们首先定义位置 r = (x, y, z) 的牛顿引力函数，该函数指定在 r 点作用于测试粒子的每单位质量的牛顿引力。这意味着是 m g (r)。根据牛顿万有引力定律(方程 4.1)，在统一坐标(r = 0)的情况下，以引力场原点为中心的总质量为 M 的球体由下式给出：

where e r is a unit vector in the radial direction, pointing away from the origin. The minus sign in Equation 4.14 means that g (r) is directed towards the origin at every point, as shown in Figure 4.8.其中 e r 是径向方向的单位向量，指向远离原点的方向。公式 4.14 中的负号表示 g(r) 在每一点都指向原点，如图 4.8 所示。

If we suppose that the sphere of mass M is enclosed by a larger sphere of radius R also centred on the origin, we can define the flux of the gravitational field leaving the larger sphere by a surface integral:如果我们假设质量为 M 的球体被一个同样以原点为中心、半径为 R 的较大球体包围，我们可以通过表面积分定义离开较大球体的引力场通量：

outward gravitational flux = g · n E d S,向外引力通量 = g·n E d S，

where n E is an outward-pointing unit vector normal to the spherical surface at every point. From the spherical symmetry of the situation, it is easy to see that in this case the surface integral will be given by the surface area of the sphere (4 πR 2) multiplied by the constant field strength on the surface of the sphere (GM/\(R^2\)), multiplied by − 1 because in this case the field points inwards, so e · n E = − 1. Thus其中 n E 是每个点垂直于球面的指向外的单位向量。从球对称的情况，很容易看出，在这种情况下，表面积分将由球体表面积 (4 πR 2) 乘以球体表面上的恒定场强 (GM/\(R^2\)) 得出，再乘以 − 1，因为在这种情况下场指向内，因此 e · n E = − 1。

Now, according to the divergence theorem of vector calculus, this kind of surface integral of the field can be rewritten as a volume integral of a quantity known as the divergence of the field, ∇ · g, throughout the volume现在，根据矢量微积分的散度定理，这种场的表面积分可以重写为整个体积中称为场散度 ∇·g 的量的体积积分

! where, in terms B of Cartesian components, the vector operator ∇ represents！其中，用笛卡尔分量 B 表示，向量算子 ∇ 表示

∂, ∂, ∂, so the divergence is defined by∂, ∂, ∂，因此散度定义为

If we now write the mass of the sphere as an integral over its density \(\rho\), we have如果我们现在将球体的质量写成其密度 \(\rho\) 的积分，我们有

Though not a proof, this last relation at least makes plausible a general relationship that can be proved by more rigorous methods, namely the differential relationship虽然不是证明，但最后一个关系至少使可以通过更严格的方法证明的一般关系变得合理，即微分关系

This is actually one of the fundamental equations of Newtonian gravitation, relating derivatives of the gravitational field to the mass density that is the source of the field. It is not restricted to spherical bodies, nor even to cases where the density is uniform. Nor is it quite the end of our argument.这实际上是牛顿引力的基本方程之一，将引力场的导数与作为场源的质量密度联系起来。它不限于球体，甚至也不限于密度均匀的情况。我们的争论还没有结束。

The gravitational force is conservative. That means that the work done against the gravitational force in moving a body from one point to another is independent of the path followed. That’s why it is possible to associate the gravitational force with a gravitational potential energy. The gravitational field g (r) can be similarly related to a gravitational potential field Φ(r) that describes the gravitational potential energy per unit mass located at r. The precise relationship is usually written in terms of a gradient as万有引力是保守的。这意味着将物体从一点移动到另一点时抵抗引力所做的功与所遵循的路径无关。这就是为什么可以将引力与引力势能联系起来。引力场 g (r) 可以类似地与描述位于 r 处每单位质量的引力势能的引力势场 Φ(r) 相关。精确的关系通常用梯度表示为

Substituting Equation 4.19 into Equation 4.18 leads to将公式 4.19 代入公式 4.18 得出

The combination ∇ · ∇ occurs so frequently in some areas of mathematics and physics that it is given a name, the Laplacian operator, and denoted by the symbol ∇ 2. Following this convention we can say that组合 ∇ · ∇ 在数学和物理的某些领域中出现得如此频繁，以至于它被命名为拉普拉斯算子，并用符号 ∇ 2 表示。按照这个约定，我们可以说：

Written out in full, in terms of Cartesian coordinates, this equation says that用笛卡尔坐标完整写出，该方程表示

Equation 4.21 is called Poisson’s equation. It provides the essential summary of Newtonian gravitation in terms of a differential equation that we have been seeking. It is entirely equivalent to Newton’s inverse square law but has the advantage that it is a differential equation for a scalar quantity that may be straightforward to solve. The gravitational field (which is a vector) can then be obtained via Equation 4.19, which involves differentiating the scalar field Φ(r). Notice that both the gravitational potential Φ and the mass density \(\rho\) are functions of the same position variable r.方程 4.21 称为泊松方程。它以我们一直在寻找的微分方程形式提供了牛顿引力的基本总结。它完全等价于牛顿平方反比定律，但优点是它是一个标量的微分方程，可以直接求解。然后可以通过方程 4.19 获得引力场(矢量)，其中涉及标量场 Φ(r) 的微分。请注意，引力势 Φ 和质量密度 \(\rho\) 都是同一位置变量 r 的函数。

Poisson’s equation and gravitation泊松方程和引力

The essence of Newtonian gravitation as a field theory is expressed in the Poisson equation牛顿引力场论的本质用泊松方程来表达

which relates a combination of second derivatives of the Newtonian gravitational potential Φ to the mass density \(\rho\) that is the source of the Newtonian gravitational field. The Newtonian gravitational field g is related to Φ by它将牛顿引力势 Φ 的二阶导数的组合与质量密度 \(\rho\) 联系起来，质量密度是牛顿引力场的来源。牛顿引力场 g 与 Φ 的关系为

It will be shown later that general relativity predicts that an equation of this type provides an approximate description of gravitation under appropriate circumstances (usually referred to as the Newtonian limit). It is in this sense that general relativity is consistent with the successful predictions of Newtonian gravitation, even though it makes no use of gravitational forces. General relativity is also consistent with special relativity in the sense that the results of special relativity hold true locally in general relativity.稍后将表明，广义相对论预测此类方程在适当的情况下(通常称为牛顿极限)提供了对引力的近似描述。正是在这个意义上，广义相对论与牛顿引力的成功预言是一致的，尽管它没有使用引力。广义相对论也与狭义相对论一致，因为狭义相对论的结果在广义相对论中局部成立。

4.2 The basic ingredients of general relativity4.2 广义相对论的基本要素

The principles outlined in the previous section led Einstein to formulate general relativity using covariant tensor equations. But what tensor quantities should be involved in those equations? It was obvious that a theory of gravity should involve the distribution of matter, and it was part of Einstein’s genius to realize that if gravity was somehow built into the geometric structure of spacetime, then it would act equally on all forms of matter and the universality of free fall would cease to be an unexplained accident. All forms of matter are subject to the same spacetime geometry, even though they may not be subject to identical forces. Such thoughts eventually led Einstein to consider two particular tensors as basic ingredients of general relativity — one describing the properties of matter, the other concerned with aspects of spacetime geometry. This section introduces those two tensor quantities and relates them to other tensors with which you are already familiar.上一节概述的原理引导爱因斯坦使用协变张量方程来制定广义相对论。但是这些方程中应该涉及哪些张量呢？显然，引力理论应该涉及物质的分布，爱因斯坦的天才之一是意识到，如果引力以某种方式构建到时空的几何结构中，那么它就会平等地作用于所有形式的物质，自由落体的普遍性将不再是一个无法解释的偶然事件。所有形式的物质都受到相同的时空几何形状的影响，尽管它们可能不会受到相同的力的影响。这些想法最终导致爱因斯坦将两个特定的张量视为广义相对论的基本成分——一个描述物质的属性，另一个涉及时空几何的各个方面。本节介绍这两个张量，并将它们与您已经熟悉的其他张量联系起来。

4.2.1 The energy–momentum tensor4.2.1 能量-动量张量

In Newton’s theory of gravity, mass, or more generally mass density, is a conserved quantity that is the ‘source’ of gravitation. (See, for instance Equation 4.21.) In special relativity, the mass m of a particle is no longer conserved, but it is related to the energy and momentum magnitude of the particle by在牛顿的引力理论中，质量，或更普遍的质量密度，是一个守恒量，是引力的“来源”。 (例如，参见方程 4.21。)在狭义相对论中，粒子的质量 m 不再守恒，但它与粒子的能量和动量大小相关：

and there are conservation laws that relate to energy (including mass–energy) and to momentum. Hence we should expect that in a relativistic theory, the source of gravitation cannot be mass alone but must also involve energy and momentum. Since these sources of gravitation must somehow appear in a tensor, you will not be surprised to learn that one of the basic ingredients of general relativity is known as the energy–momentum tensor. The only issues are: what is it, what is its rank, what are its symmetries, and how is it defined?并且存在与能量(包括质能)和动量相关的守恒定律。因此，我们应该预料到，在相对论理论中，引力的来源不能只是质量，还必须涉及能量和动量。由于这些引力源必须以某种方式出现在张量中，因此当您得知广义相对论的基本成分之一被称为能量-动量张量时，您不会感到惊讶。唯一的问题是：它是什么，它的等级是什么，它的对称性是什么，以及它是如何定义的？

The energy–momentum tensor describes the distribution and flow of energy and momentum in a region of spacetime. It is a rank 2 tensor, so at an event (i.e. any ‘point’ in spacetime) it is specified by sixteen components, usually denoted \(T_{\mu\nu}\) (\(\mu\), \(\nu\) = 0, 1, 2, 3). It is a symmetric tensor, so \(T_{\mu\nu}\) = T \(\nu\)\(\mu\), and that means that only ten of its components are independent (the four components T \(\mu\)\(\mu\) and six of the twelve components \(T_{\mu\nu}\) where \(\mu\) 3 = \(\nu\)). Each component can be measured in units of energy density (\(\mathrm{J\,m^{-3}}\)), though it is sometimes appropriate to use other equivalent units. Each component is a function of the spacetime coordinates, with the following general significance in the neighbourhood of each event in spacetime:能量-动量张量描述了时空区域中能量和动量的分布和流动。它是一个 2 阶张量，因此在一个事件(即时空中的任何“点”)上，它由 16 个分量指定，通常表示为 \(T_{\mu\nu}\)(\(\mu\)、\(\nu\) = 0, 1, 2, 3)。它是一个对称张量，因此 \(T_{\mu\nu}\) = T \(\nu\)\(\mu\)，这意味着它只有 10 个分量是独立的(四个分量 T \(\mu\)\(\mu\) 和 12 个分量中的 6 个 \(T_{\mu\nu}\)，其中 \(\mu\) 3 = \(\nu\))。每个成分都可以用能量密度单位 (\(\mathrm{J\,m^{-3}}\)) 来测量，尽管有时使用其他等效单位也比较合适。每个分量都是时空坐标的函数，在时空中每个事件的邻域中具有以下一般意义：

• • \(T^{00}\) is the local energy density, including any mass–energy contribution.\(T^{00}\) 是局部能量密度，包括任何质能贡献。
• • \(T^{0i}\) = \(T^{i0}\) is the rate of flow of energy per unit area at right angles to the\(T^{0i}\) = \(T^{i0}\) 是与方向成直角的每单位面积的能量流率

i -direction, divided by c, or, equivalently, the density of the i -component of momentum, multiplied by c.i 方向除以 c，或者等效地，动量的 i 分量的密度乘以 c。

• \(T^{ij}=T^{ji}\) is the rate of flow of the \(i\)-component of momentum per unit area at right angles to the \(j\)-direction.\(T^{ij}=T^{ji}\) 是与 \(j\) 方向成直角的每单位面积动量的 \(i\) 分量的流速。

right angles to the j -direction.与 j 方向成直角。

Figure 4.10 tries to give some feeling for the meaning of these components by considering the special case of a group of identical, non-interacting particles, each of mass m and velocity v = (\(v_{x}\), \(v_{y}\), 0), where we identify x, y and z with the 1 -, 2 - and 3 -directions, respectively. Each of these particles will have a relativistic momentum mγ (v) v and a total relativistic energy mγ (v) \(c^2\), where v = | v | represents the common speed of the particles and \(\gamma(v)\) = 1/1 − \(v^{2}\)/\(c^2\).图 4.10 试图通过考虑一组相同的、不相互作用的粒子的特殊情况来对这些分量的含义给出一些感觉，每个粒子的质量为 m，速度 v = (\(v_x\), \(v_y\), 0)，其中我们分别将 x、y 和 z 标识为 1、2 和 3 方向。这些粒子中的每一个都将具有相对论动量 mγ (v) v 和总相对论能量 mγ (v) \(c^2\)，其中 v = | v |代表粒子的共同速度，γ (v) = 1/1 − v 2/\(c^2\)。

number crossing an area A perpendicular to the \(x\)-direction in time t will be nv x At; and since each carries energy mγ (v) \(c^2\), the rate of flow of energy per unit area through a surface at right angles to the \(x\)-direction, divided by c, will be \(T^{01}\) = nv Atmγ (v) \(c^2\)/Atc = nmv \(\gamma(v)\) c. Since each of the particles has an x -component of momentum given by mγ (v) \(v_{x}\), you can see that the density of the x -component of momentum, multiplied by c, is given by the same expression, so \(T^{10}\) = nmv \(\gamma(v)\) c. A similar argument shows that \(T^{02}\) = \(T^{20}\) = nmv \(\gamma(v)\) c, while \(T^{03}\) = \(T^{30}\) = 0 because we have chosen to consider particles with \(v_{z}\) = 0. Finally, we note that in a time t, particles with y -component of momentum mγ (v) \(v_{y}\) are crossing an area A perpendicular to the \(x\)-direction at a rate given by nv x At/At = nv x, so the rate of flow of the y -component of momentum per unit area through a surface at right angles to the \(x\)-direction is \(T^{21}\) = nv mγ (v) v = nmv v \(\gamma(v)\), which is also the value of \(T^{12}\). By similar arguments, \(T^{11}\) = nmv 2 \(\gamma(v)\) and \(T^{22}\) = nmv 2 \(\gamma(v)\), but \(T^{13}\) = \(T^{31}\), \(T^{23}\) = \(T^{32}\)在时间 t 内穿过垂直于 x 方向的区域 A 的数将为 nv x At；由于每个都携带能量 mγ (v) \(c^2\)，因此通过与 x 方向成直角的表面的每单位面积的能量流量除以 c，将为 \(T^{01}\) = nv Atmγ (v) \(c^2\)/Atc = nmv \(\gamma(v)\) c。由于每个粒子的动量 x 分量由 mγ (v) \(v_{x}\) 给出，因此您可以看到动量 x 分量的密度乘以 c，由相同的表达式给出，因此 \(T^{10}\) = nmv \(\gamma(v)\) c。类似的论证表明 \(T^{02}\) = \(T^{20}\) = nmv \(\gamma(v)\) c，而 \(T^{03}\) = \(T^{30}\) = 0，因为我们选择考虑 \(v_{z}\) = 0 的粒子。最后，我们注意到，在时间 t 内，动量 y 分量为 mγ (v) \(v_{y}\) 的粒子以 nv x At/At = nv x 给出的速率穿过垂直于 x 方向的区域 A，因此每单位面积动量 y 分量通过与 x 方向成直角的表面的流速为 \(T^{21}\) = nv mγ (v) v = nmv v \(\gamma(v)\)，这也是 \(T^{12}\) 的值。通过类似的论证，\(T^{11}\) = nmv 2 \(\gamma(v)\) 和 \(T^{22}\) = nmv 2 \(\gamma(v)\)，但 \(T^{13}\) = \(T^{31}\)、\(T^{23}\) = \(T^{32}\)

and \(T^{33}\) are all zero because they involve \(v_{z}\), which is zero in this particular case.和 \(T^{33}\) 都为零，因为它们涉及 \(v_z\)，在这种特殊情况下 \(v_z\) 为零。

Putting all these results together gives将所有这些结果放在一起得出

The precise form of the energy–momentum tensor will depend on what occupies the region concerned. A particularly simple example to consider is that of a region occupied by a cloud of non-interacting particles, each of mass m. This kind of matter is usually described as dust. For present purposes it’s best to think of the dust cloud as a continuous body of matter that may contain internal currents — rather like a fluid but without any internal pressure. The nature of the dust cloud at any spacetime event in the region of interest can be characterized by the three-velocity v of the flow at the event, and by the value of the cloud’s proper mass density \(\rho\), that is, the density measured by an observer moving with the flow at the event of interest.能量-动量张量的精确形式将取决于相关区域的占据情况。一个特别简单的例子是由非相互作用粒子云占据的区域，每个粒子的质量为 m。这种物质通常被描述为灰尘。就目前的目的而言，最好将尘埃云视为可能包含内部电流的连续物质体——更像是流体，但没有任何内部压力。感兴趣区域中任何时空事件的尘埃云的性质都可以通过该事件处的气流的三速度 v 和云的固有质量密度 \(\rho\) 来表征，即观察者在感兴趣的事件处随气流移动所测量的密度。

Of course, we really want to describe the dust cloud in terms of parameters that have well-known transformation properties under changes of reference frame. This is easy to do: the proper mass density \(\rho\) is a scalar invariant, so it already transforms as simply as possible; the three-velocity v is more complicated, but it can be used to determine a four-velocity \([U^{\mu}]\) = (cγ (v), \(\gamma(v)\) v) (where v = | v | and \(\gamma(v)\) = 1/1 − \(v^{2}\)/\(c^2\)) that transforms as a rank 1 contravariant tensor. The components of the energy–momentum tensor of the dust at any spacetime event can then be written down in a covariant way, in accordance with the rules of tensor algebra, as当然，我们真正想用在参考系变化下具有众所周知的变换特性的参数来描述尘埃云。这很容易做到：适当的质量密度 \(\rho\) 是标量不变量，因此它已经尽可能简单地进行变换；三速度 v 更复杂，但它可用于确定转换为 1 阶逆变张量的四速度 \([U^{\mu}]\) = (cγ (v), \(\gamma(v)\) v)(其中 v = | v | 且 \(\gamma(v)\) = 1/1 − \(v^{2}\)/\(c^2\))。然后，根据张量代数规则，可以以协变的方式写出任意时空事件下的尘埃能量-动量张量的分量，如下

This means that if we choose to use the instantaneous rest frame of the dust at the \([U^{\mu}]\) = (c, 0) and the event in question, then at that event and in that frame, energy–momentum tensor can be represented by the matrix这意味着，如果我们选择使用 \([U^{\mu}]\) = (c, 0) 处尘埃的瞬时静止坐标系和所讨论的事件，那么在该事件和该坐标系中，能量-动量张量可以由矩阵表示

So, in its local instantaneous rest frame, the only non-zero component of the energy–momentum tensor of the dust is \(T^{00}\), which represents the energy density, and that is entirely accounted for by the density of mass–energy in the dust.因此，在其局部瞬时静止坐标系中，尘埃能量-动量张量的唯一非零分量是 \(T^{00}\)，它代表能量密度，并且完全由尘埃中的质能密度来解释。

Another simple example of an energy–momentum tensor is that of an ideal fluid. Such a fluid is slightly more complicated than dust, since its nature at any spacetime event is characterized by a mass density \(\rho\), a four-velocity \([U^{\mu}]\) and a pressure p that acts equally in all directions at that point. At an event where the metric is \(g_{\mu\nu}\), the components of the energy–momentum tensor of an ideal fluid are given covariantly by能量-动量张量的另一个简单例子是理想流体。这种流体比尘埃稍微复杂一些，因为它在任何时空事件中的性质都以质量密度 \(\rho\)、四速度 \([U^{\mu}]\) 和在该点在所有方向上作用相同的压力 p 为特征。在度规为 \(g_{\mu\nu}\) 的事件中，理想流体的能量-动量张量的分量由下式协变给出：

If we restrict ourselves to using locally inertial frames with Cartesian coordinates, then at any chosen spacetime event, the metric can be represented by the Minkowski metric, and the components of the energy–momentum tensor will be given by如果我们限制自己使用具有笛卡尔坐标的局部惯性系，那么在任何选定的时空事件中，度规可以用闵可夫斯基度规表示，并且能量-动量张量的分量将由下式给出

If we again take the additional step of considering things from the point of view of an observer using the instantaneous rest frame of the fluid at that point, then, in that frame and at that point, the energy–momentum tensor of the ideal fluid is represented by the matrix如果我们再次采取额外的步骤，使用流体在该点的瞬时静止坐标系从观察者的角度考虑事物，那么，在该坐标系和该点，理想流体的能量-动量张量由矩阵表示

In this case there will generally be thermal effects leading to flows of energy and momentum. However, because we have chosen to use the instantaneous rest frame, those flows will make no net contribution to the flow of energy, so it will still be the case that \(T^{0i}\) = \(T^{i0}\) = 0, and the lack of interactions between the particles will ensure \(T^{ij}=0\) for \(i\ne j\). Consequently, the only non-zero components will be the total energy density \(T^{00}=\rho c^2\) (which will include contributions from the random thermal motion of the particles in the fluid) and the three components \(T^{ii}=p\) for i = 1, 2, 3 (which represent the effect of momentum being transferred with equal magnitude per unit area per unit time in all directions by the thermal motion of the particles).在这种情况下，通常会存在导致能量和动量流动的热效应。然而，由于我们选择使用瞬时静止坐标系，这些流动不会对能量流产生净贡献，因此仍然会出现 \(T^{0i}\) = \(T^{i0}\) = 0 的情况，并且粒子之间缺乏相互作用将确保 \(i\ne j\) 为 \(T^{ij}=0\)。因此，唯一的非零分量将​​是总能量密度 \(T^{00}=\rho c^2\)(其中包括流体中颗粒的随机热运动的贡献)和 i = 1、2、3 时的三个分量 \(T^{ii}=p\)(其表示通过颗粒热运动在所有方向上每单位面积每单位时间以相等大小传递动量的效果)。

• ● Show that for vanishing pressure (p → 0), the energy–momentum tensor of an证明对于消失压力 (p → 0)，能量-动量张量

ideal fluid reduces to that of dust. ❍ For p → 0 we get理想流体可减少为灰尘。 ❍ 对于 p → 0 我们得到

which is Equation 4.23 for the energy–momentum tensor for dust.这是灰尘能量-动量张量的方程 4.23。

• ● Show that the units of pressure (Pa = \(\mathrm{N\,m^{-2}}\)) are equivalent to those of证明压力单位 (Pa = \(\mathrm{N\,m^{-2}}\)) 相当于

energy density (\(\mathrm{J\,m^{-3}}\)), and also equivalent to those used to measure the rate of flow of momentum per unit area.能量密度(\(\mathrm{J\,m^{-3}}\))，也相当于用于测量单位面积动量流速的能量密度。

❍ In SI units, 1 J = 1 N m, so the unit of energy density may be written as❍ 在 SI 单位中，1 J = 1 N m，因此能量密度的单位可写为

\(\mathrm{J\,m^{-3}}\) = N m \(\mathrm{m^{-3}}\) = \(\mathrm{N\,m^{-2}}\) = Pa, which is the unit of pressure. Similarly, the unit of rate of flow of momentum per unit area will be \(\mathrm{kg\,m\,s^{-1}}\) \(\mathrm{m^{-2}}\) \(\mathrm{s^{-1}}\) = \(\mathrm{kg\,m^{-1}\,s^{-2}}\), but 1 N = 1 \(\mathrm{kg\,m\,s^{-2}}\), so the unit of rate of momentum flow per unit area per unit time can be written as\(\mathrm{J\,m^{-3}}\) = N m \(\mathrm{m^{-3}}\) = \(\mathrm{N\,m^{-2}}\) = Pa，这是压力的单位。类似地，单位面积动量流量的单位为 \(\mathrm{kg\,m\,s^{-1}}\) \(\mathrm{m^{-2}}\) \(\mathrm{s^{-1}}\) = \(\mathrm{kg\,m^{-1}\,s^{-2}}\)，但 1 N = 1 \(\mathrm{kg\,m\,s^{-2}}\)，因此单位时间单位面积动量流量的单位可写为

\(\mathrm{kg\,m^{-1}\,s^{-2}}\) = \(\mathrm{N\,m^{-2}}\) = Pa.kg·m−1·s−2=N·m−2=Pa。

Exercise 4.6 Verify the matrix in Equation 4.27 by练习 4.6 通过以下方式验证方程 4.27 中的矩阵

explicitly evaluating \(T^{00}\), \(T^{0i}\) and \(T^{ij}\) for i, j = 1, 2, 3 from Equation 4.26.根据公式 4.26 显式评估 i、j = 1、2、3 的 \(T^{00}\)、\(T^{0i}\) 和 \(T^{ij}\)。

As a final example of an energy–momentum tensor, we note that in the case of a region of space that contains electric and magnetic fields but no matter (a region occupied by electromagnetic radiation, for example), the components of the energy–momentum tensor are作为能量-动量张量的最后一个例子，我们注意到，在包含电场和磁场但无论如何的空间区域(例如，被电磁辐射占据的区域)的情况下，能量-动量张量的分量为

where F \(\mu\)\(\nu\) is the electromagnetic field tensor that was introduced in Chapter 2. We shall not discuss this energy–momentum tensor in detail, but its existence indicates that in general relativity, electromagnetic radiation alone can be a source of gravitation even though the associated particles (photons) have no mass at all.其中 F \(\mu\)\(\nu\) 是第 2 章中介绍的电磁场张量。我们不会详细讨论这个能量-动量张量，但它的存在表明，在广义相对论中，即使相关粒子(光子)根本没有质量，电磁辐射本身也可以成为引力源。

At this stage it’s useful to recall another result from Chapter 2: in electromagnetism, the conservation of electric charge is represented by the equation of continuity在这个阶段，回顾第二章的另一个结果是有用的：在电磁学中，电荷守恒由连续方程表示

This equation describes how any change in the electric charge density must be balanced by a flow of charge due to electric currents. It is often written more compactly in terms of a three-vector divergence as该方程描述了电荷密度的任何变化必须如何通过电流引起的电荷流来平衡。它通常用三向量散度来写得更紧凑：

or more compactly still, using the current four-vector, by the Lorentz-covariant equation或者更紧凑地，使用当前的四向量，通过洛伦兹协变方程

This suggests that we might expect the conservation of relativistic energy and momentum in a locally inertial frame (where special relativity holds true) to be represented by a relation of the form这表明我们可能期望局部惯性系(狭义相对论成立)中的相对论能量和动量守恒由以下形式的关系表示

and this is indeed the case. The tensor relationship has a free index \(\nu\), so it actually represents four different equations, each of which is similar to the equation of continuity. The first (corresponding to \(\nu\) = 0) relates the rate of change of the energy density \(T^{00}\) to the spatial derivatives of the energy flows \(T^{0i}\) in the 1 -, 2 - and 3 -directions. The other three each relate the rate of change of one of the momentum density terms \(T^{0i}\) to the spatial derivatives of the corresponding momentum flows \(T^{ji}\) for j = 1, 2, 3.事实确实如此。张量关系有一个自由指标 \(\nu\)，因此它实际上代表四个不同方程，每个方程都类似于连续性方程。第一个方程（对应 \(\nu=0\)）把能量密度 \(T^{00}\) 的变化率同 1、2、3 方向上能量流 \(T^{0i}\) 的空间导数联系起来。其余三个方程则分别把一个动量密度项 \(T^{0i}\) 的变化率同相应动量流 \(T^{ji}\) 的空间导数（\(j=1,2,3\)）联系起来。

It also turns out that in arbitrary coordinates and in a spacetime that may be flat or curved, the energy–momentum tensor has the more general property事实证明，在任意坐标和可能是平坦或弯曲的时空中，能量-动量张量具有更一般的性质

This is sometimes described by saying that the covariant divergence of \(T_{\mu\nu}\) is zero. In the absence of gravity, in a flat Minkowski spacetime, this result simply allows us to describe the conservation of energy and momentum using general coordinates. However, if the spacetime is curved, then it turns out that Equation 4.30 does not generally describe the conservation of energy and momentum for the contents of spacetime. And that’s a good thing, because in the presence of gravitation (i.e. curvature), the conservation of energy is not expected to apply to matter and radiation alone — we also have to take the gravitational energy into account, and that is not included in the energy–momentum tensor. We shall return to the significance of the covariant divergence in curved spacetime later; for the moment we just need to emphasize the following.有时用 \(T_{\mu\nu}\) 的协变散度为零来描述这一点。在没有引力的情况下，在平坦的闵可夫斯基时空中，这个结果让我们能够使用通用坐标来描述能量和动量守恒。然而，如果时空是弯曲的，那么方程 4.30 并不能概括地描述时空内容的能量和动量守恒。这是一件好事，因为在存在引力(即曲率)的情况下，能量守恒预计不仅仅适用于物质和辐射——我们还必须考虑引力能，而这不包括在能量-动量张量中。稍后我们将回到弯曲时空中协变散度的意义；目前我们只需要强调以下几点。

The energy–momentum tensor能量-动量张量

The energy–momentum tensor \(T_{\mu\nu}\) describes the distribution and flow of energy and momentum due to the presence and motion of matter and radiation in a region of spacetime. It is a rank 2, symmetric tensor with ten independent components. At any event in the region of interest, its components describe the energy density, the flow of energy in various directions, divided by c (or, equivalently, the density of the corresponding momentum component, multiplied by c), and the flow of the various momentum components in the various directions.能量-动量张量 \(T_{\mu\nu}\) 描述了由于时空区域中物质和辐射的存在和运动而导致的能量和动量的分布和流动。它是一个具有 10 个独立分量的 2 阶对称张量。在感兴趣区域中的任何情况下，其分量描述能量密度、各个方向上的能量流除以 c(或者等效地，相应动量分量的密度乘以 c)以及各个方向上各个动量分量的流。

For pressure-free dust, the components of the energy–momentum tensor are given by对于无压灰尘，能量-动量张量的分量由下式给出

for an ideal fluid,对于理想流体，

and for electromagnetic fields,对于电磁场，

An important general property of the energy–momentum tensor is that its covariant divergence is zero; that is,能量-动量张量的一个重要的一般性质是它的协变散度为零；那是，

4.2.2 The Einstein tensor4.2.2 爱因斯坦张量

The equivalence principle led Einstein to propose that gravity should be regarded not as a force in the conventional sense, but as a manifestation of the curvature of spacetime. Einstein was therefore looking for a geometric theory of gravity, so he needed to find a geometric object that could be related to the energy–momentum tensor. Clearly, he needed a rank 2 tensor involving the components of the metric tensor. However, from the example of the electromagnetic field equations, or even from Newtonian gravity formulated as a field theory and based on Poisson’s equation, we should expect the final equations to be differential equations, so the metric should enter through its derivatives. We might also expect that the required geometric tensor will be symmetric and will have a vanishing covariant divergence.等效原理促使爱因斯坦提出，引力不应被视为传统意义上的力，而应被视为时空曲率的表现。因此，爱因斯坦正在寻找引力的几何理论，因此他需要找到一个可以与能量-动量张量相关的几何物体。显然，他需要一个包含度规张量分量的 2 阶张量。然而，从电磁场方程的例子来看，甚至从作为场论形式表达的牛顿引力和基于泊松方程的例子来看，我们应该期望最终的方程是微分方程，因此度规应该通过其导数进入。我们还可能期望所需的几何张量是对称的并且协变散度消失。

Even with so many clues, it took Einstein some time to find the appropriate tensor quantity. What he eventually arrived at involved contractions of the Riemann curvature tensor that was introduced in Chapter 3. Here is the full form of the Riemann tensor for a four-dimensional spacetime:即使有这么多线索，爱因斯坦还是花了一些时间才找到合适的张量。他最终得出的结论涉及第 3 章中介绍的黎曼曲率张量的收缩。以下是四维时空黎曼张量的完整形式：

δ αβ, which are defined in As you can see, it involves the connection coefficients Γ terms of the metric and its derivatives byδ αβ，如您所见，它涉及度规及其导数的联络系数 Γ 项

You will recall from Chapter 3 that the vanishing of all components of the Riemann tensor is the necessary and sufficient condition for a spacetime to be flat.你会记得在第三章中，黎曼张量所有分量的消失是时空平坦的充分必要条件。

The Riemann tensor has four indices, each of which can take four values (in four-dimensional spacetime), so it has 4 4 = 256 components. However, the tensor has various symmetries, so there are just 20 independent components.黎曼张量有四个索引，每个索引可以取四个值(在四维时空中)，因此它有 4 4 = 256 个分量。然而，张量具有各种对称性，因此只有 20 个独立分量。

Although the Riemann tensor is fundamental to the study of curved spaces, there are two other tensors that have been found to be very useful. If we contract the first and last indices on the Riemann tensor, then we get a new rank 2 tensor with components尽管黎曼张量是弯曲空间研究的基础，但还有其他两个张量被发现非常有用。如果我们收缩黎曼张量的第一个和最后一个索引，那么我们会得到一个带有分量的新的 2 阶张量

which is known as the Ricci tensor. It follows from the definition of the Riemann tensor that the Ricci tensor is symmetric with respect to interchanging its indices, i.e. R αβ = R βα. Further, contracting the indices on the Ricci tensor gives这就是所谓的里奇张量。根据黎曼张量的定义可知，里奇张量在交换其指数方面是对称的，即 R αβ = R βα。此外，收缩 Ricci 张量的指数给出

which is known as the curvature scalar (or sometimes the Ricci scalar). Note that all of these curvature-related quantities are ultimately expressed in terms of the components of the metric tensor \(g_{\mu\nu}\) and their derivatives.这称为曲率标量(有时称为 Ricci 标量)。请注意，所有这些与曲率相关的量最终都以度规张量 \(g_{\mu\nu}\) 的分量及其导数来表示。

The quantity that Einstein found to be a basic ingredient of general relativity is defined in terms of the Ricci tensor and the curvature scalar. It is called the Einstein tensor and its components are given by the following equation.爱因斯坦发现广义相对论基本成分的量是用里奇张量和曲率标量定义的。它称为爱因斯坦张量，其分量由以下方程给出。

The Einstein tensor爱因斯坦张量

Since both R \(\mu\)\(\nu\) and \(g_{\mu\nu}\) are symmetric, it follows that G \(\mu\)\(\nu\) must also be symmetric. This means that only 10 of its 16 components will be independent, just like the energy–momentum tensor. Moreover, it! can be shown that B the \(\mu\) ∇ \(\mu\) G \(\mu\)\(\nu\) = 0, again covariant divergence of the Einstein tensor vanishes just like the energy–momentum tensor.由于 R \(\mu\)\(\nu\) 和 \(g_{\mu\nu}\) 都是对称的，因此 G \(\mu\)\(\nu\) 也必须是对称的。这意味着它的 16 个分量中只有 10 个是独立的，就像能量-动量张量一样。更何况，它！可以证明 B \(\mu\) ∇ \(\mu\) G \(\mu\)\(\nu\) = 0，爱因斯坦张量的协变散度再次消失，就像能量-动量张量一样。

We are now in a position to introduce Einstein’s field equations, the mathematical relations that are at the core of general relativity.我们现在可以介绍爱因斯坦的场方程，这是广义相对论核心的数学关系。

4.3 Einstein’s field equations and geodesic motion4.3 爱因斯坦场方程和测地运动

The central ideas of general relativity were famously summed up by the American physicist John Wheeler:美国物理学家约翰·惠勒对广义相对论的中心思想做出了著名的总结：

Matter tells space how to curve. Space tells matter how to move.物质告诉空间如何弯曲。空间告诉物质如何运动。

This is very memorable (and worth remembering!), though not completely accurate. (You should already be asking yourself: ‘Doesn’t he mean spacetime rather than space, and doesn’t he mean matter and radiation rather than matter?’) Unpacking Wheeler’s quote somewhat, to be more accurate, we can say that the central physical ideas of general relativity are that the energy and momentum in a region of spacetime determine the geometry of spacetime in that region. The spacetime geometry then determines a special class of spacetime pathways — the geodesics. Moving under the influence of gravity alone, massive particles travel along time-like geodesics (where \(ds^2\) > 0), while light rays follow null geodesics (with \(ds^2\) = 0). Thus the distribution of energy and momentum in a region determines the motion of freely falling matter and radiation in that region.这是非常令人难忘的(并且值得记住！)，尽管并不完全准确。(你应该已经问自己：“他不是指的是时空而不是空间，他不是指的是物质和辐射而不是物质吗？”)稍微解释一下惠勒的引言，更准确地说，我们可以说广义相对论的核心物理思想是时空区域中的能量和动量决定了该区域中时空的几何形状。然后，时空几何决定了一类特殊的时空路径——测地线。仅在引力影响下移动，大质量粒子沿着类时测地线(其中 \(ds^2\) > 0)行进，而光线则遵循零测地线(其中 \(ds^2\) = 0)。因此，一个区域中能量和动量的分布决定了该区域中自由落体物质和辐射的运动。

Another helpful but overly simple view is that in Newtonian gravitation, matter tells matter how to move, with the gravitational force playing the role of intermediary. This can be contrasted with general relativity where energy and momentum tell matter and radiation how to move, with spacetime geometry playing the role of intermediary.另一个有用但过于简单的观点是，在牛顿引力中，物质告诉物质如何运动，而引力则扮演中介的角色。这可以与广义相对论形成对比，广义相对论中能量和动量告诉物质和辐射如何移动，而时空几何则扮演中介的角色。

The rest of this section is devoted to spelling out these ideas with greater accuracy and improved precision.本节的其余部分致力于以更高的准确性和更高的精度阐明这些想法。

4.3.1 The Einstein field equations4.3.1 爱因斯坦场方程

As we have seen, Einstein’s objective became the formulation of a ‘geometric’ theory of gravity that would naturally act on all kinds of matter in the same way. He identified the energy–momentum tensor as an important quantity for describing the ‘sources’ of gravitation, and found another symmetric rank 2 tensor, the Einstein tensor, containing derivatives of the metric coefficients \(g_{\mu\nu}\), tensor \(T_{\mu\nu}\) and the that he could relate to it. Both the energy–momentum Einstein tensor [G \(\mu\)\(\nu\)] have zero covariant divergence, so it is natural to suggest that the two tensors are proportional. This led Einstein to propose what are now called the Einstein field equations, which are usually written as in terms of tensor components as follows.正如我们所看到的，爱因斯坦的目标变成了引力的“几何”理论的表述，该理论自然会以相同的方式作用于所有种类的物质。他认为能量-动量张量是描述引力“来源”的重要量，并发现了另一个对称的 2 阶张量，即爱因斯坦张量，其中包含度规系数 \(g_{\mu\nu}\)、张量 \(T_{\mu\nu}\) 的导数以及他可以与之关联的。能量-动量爱因斯坦张量 [G \(\mu\)\(\nu\)] 的协变散度为零，因此很自然地认为这两个张量成比例。这导致爱因斯坦提出了现在所谓的爱因斯坦场方程，该方程通常用张量分量写成如下。

The Einstein field equations爱因斯坦场方程

Here \(\kappa\) is a constant, sometimes called the Einstein constant. We shall show later that requiring the consistency of general relativity and Newtonian gravitation forces us to set \(\kappa = 8\pi G/c^4\).这里 \(\kappa\) 是一个常数，有时称为爱因斯坦常数。稍后我们将证明，要求广义相对论和牛顿引力的一致性迫使我们设置 \(\kappa = 8\pi G/c^4\)。

The Einstein field equations are the fundamental field equations of general relativity, analogous to the Poisson equation in Newtonian gravitation. They are the feature of general relativity that Wheeler was referring to when he said (rather loosely) ‘matter tells space how to curve’. The Einstein field equations have two free indices, \(\mu\) and \(\nu\), so they actually represent a set of 16 equations, though due to symmetries only 10 of them are independent. They are usually regarded as differential equations for the 10 independent metric tensor components \(g_{\mu\nu}\). But they are generally very complicated.爱因斯坦场方程是广义相对论的基本场方程，类似于牛顿引力中的泊松方程。它们是惠勒(相当宽松地)“物质告诉空间如何弯曲”时所指的广义相对论的特征。爱因斯坦场方程有两个自由索引，\(\mu\) 和 \(\nu\)，因此它们实际上代表一组 16 个方程，但由于对称性，其中只有 10 个方程是独立的。它们通常被视为 10 个独立度规张量分量 \(g_{\mu\nu}\) 的微分方程。但它们通常非常复杂。

The reason for the complication is not hard to see. The Ricci tensor and the curvature scalar involve combinations of components of the Riemann tensor. Its coefficients Γ \(\mu\) αβ, components R \(\mu\) \(\nu\)αβ are defined in terms of the connection which are in turn defined in terms of the metric tensor components \(g_{\mu\nu}\) and the components of its inverse \(g_{\mu\nu}\). The way in which the connection coefficients appear in R \(\mu\) \(\nu\)αβ means that the Riemann tensor involves second-order derivatives of the metric coefficients with respect to the spacetime coordinates. However, because the connection coefficients involve both the metric tensor and its inverse, the Einstein field equations are non-linear in \(g_{\mu\nu}\). (An equation is said to be non-linear in a variable y if replacing y by αy throughout the equation does not produce an equation that is equivalent to the original equation multiplied by \(\alpha\).) It is the non-linearity that makes the Einstein field equations particularly difficult to solve.造成并发症的原因不难看出。里奇张量和曲率标量涉及黎曼张量分量的组合。它的系数 Γ \(\mu\) αβ，分量 R \(\mu\) \(\nu\)αβ 根据联络定义，而联络又根据度规张量分量 \(g_{\mu\nu}\) 及其逆 \(g_{\mu\nu}\) 的分量定义。R \(\mu\) \(\nu\)αβ 中联络系数的出现方式意味着黎曼张量涉及度规系数相对于时空坐标的二阶导数。然而，由于联络系数同时涉及度规张量及其逆，因此 \(g_{\mu\nu}\) 中的爱因斯坦场方程是非线性的。(如果在整个方程中用 αy 替换 y 不能产生等于原始方程乘以 \(\alpha\) 的方程，则称该方程在变量 y 中是非线性的。)正是这种非线性使得爱因斯坦场方程特别难以求解。

Solving the Einstein field equations means finding the metric tensor \(g_{\mu\nu}\) that corresponds to a given energy–momentum tensor \(T_{\mu\nu}\). As you saw in Chapter 3, the metric tensor, once it is known, will determine the connection coefficients, the curvature tensor, the geodesic pathways and all the other geometric features of the spacetime that it describes. Given that gravitation is ‘built in’ to the geometry of spacetime in general relativity, the metric tensor that corresponds to a given set of source terms (i.e. a given energy–momentum tensor) is the gravitational field, even though it is not the ‘force per unit mass’ of the Newtonian gravitational field.求解爱因斯坦场方程意味着找到对应于给定能量-动量张量 \(T_{\mu\nu}\) 的度规张量 \(g_{\mu\nu}\)。正如您在第 3 章中看到的，度规张量一旦已知，将确定联络系数、曲率张量、测地线路径以及它所描述的时空的所有其他几何特征。鉴于引力是广义相对论中时空几何的“内置”，对应于给定源项集(即给定能量-动量张量)的度规张量就是引力场，尽管它不是牛顿引力场的“每单位质量的力”。

The act of solving the Einstein field equations might sound straightforward, but the ten independent field equations form a set of simultaneous, non-linear, second-order partial differential equations and, depending on the energy–momentum tensor, the task of finding a solution varies between difficult and impossible. In fact, it is remarkable that the first (and probably most important) exact non-trivial solution was announced very soon after Einstein first proposed his equations. We shall describe that solution in the next chapter.求解爱因斯坦场方程的行为可能听起来很简单，但十个独立的场方程形成了一组联立、非线性、二阶偏微分方程，并且根据能量-动量张量的不同，找到解决方案的任务有时是困难的，有时是不可能的。事实上，值得注意的是，在爱因斯坦首次提出他的方程之后不久，第一个(也可能是最重要的)精确的非平凡解就被宣布了。我们将在下一章中描述该解决方案。

In addition to various numerical procedures for finding solutions to the field equations, there are three different ways to approach the search for solutions.除了寻找场方程解的各种数值程序之外，还可以采用三种不同的方法来寻找解。

1. As already suggested, we could specify the energy–momentum tensor and1. 正如已经建议的，我们可以指定能量-动量张量和

then work very hard to solve for the metric components \(g_{\mu\nu}\). This approach has actually been very successful for some energy–momentum tensors.然后非常努力地求解度规分量 \(g_{\mu\nu}\)。这种方法实际上对于某些能量-动量张量非常成功。

2. We could specify the metric tensor and then work out the energy–momentum2.我们可以指定度规张量，然后计算出能量-动量

tensor. This is generally easier since it is more straightforward to differentiate a function than to solve a non-linear partial differential equation. However, it usually turns out that the resulting energy–momentum tensor is non-physical, so this approach is not as useful as might be hoped.张量。这通常更容易，因为微分函数比求解非线性偏微分方程更直接。然而，通常结果表明所得到的能量-动量张量是非物理的，因此这种方法并不像希望的那样有用。

3. We could try to partly determine both the metric tensor and the3. 我们可以尝试部分确定度规张量和

energy–momentum tensor directly from the physics of a particular situation and then use the field equations as constraints to complete the determination of \(g_{\mu\nu}\) and \(T_{\mu\nu}\). This sometimes yields useful results.直接从特定情况的物理场中获取能量-动量张量，然后使用场方程作为约束来完成 \(g_{\mu\nu}\) 和 \(T_{\mu\nu}\) 的确定。这有时会产生有用的结果。

In any case, a significant part of the discovery of any new solution of the Einstein field equations is to check that the solution really is new, and not merely an old solution expressed in a different coordinate system. This is an interesting problem but its consideration would take us well beyond the limits of this book.无论如何，发现爱因斯坦场方程的任何新解的一个重要部分是检查该解是否确实是新的，而不仅仅是在不同坐标系中表达的旧解。这是一个有趣的问题，但对它的考虑将远远超出本书的范围。

• ● Taking the metric tensor components \(g_{\mu\nu}\) to be dimensionless quantities将度规张量分量 \(g_{\mu\nu}\) 视为无量纲量

(i.e. pure numbers), show that the connection coefficients Γ \(\lambda\) \(\mu\)\(\nu\) can be expressed in units of \(\mathrm{m^{-1}}\), while the Ricci tensor [R \(\mu\)\(\nu\)] and the curvature scalar R can both be expressed in units of \(\mathrm{m^{-2}}\). Combine this with your knowledge of the appropriate units for \(T_{\mu\nu}\) to show that 8 πG/\(c^4\) has the right units to be the Einstein constant \(\kappa\).(即纯数)，表明联络系数 Γ \(\lambda\) \(\mu\)\(\nu\) 可以用 \(\mathrm{m^{-1}}\) 为单位表示，而 Ricci 张量 [R \(\mu\)\(\nu\)] 和曲率标量 R 都可以用 \(\mathrm{m^{-2}}\) 为单位表示。将其与您对 \(T_{\mu\nu}\) 的适当单位的了解相结合，以证明8 πG/\(c^4\) 的正确单位是爱因斯坦常数 \(\kappa\)。

❍ Since \(g_{\mu\nu}\) is dimensionless, it follows from❍ 由于 \(g_{\mu\nu}\) 是无量纲的，因此可以得出

that Γ \(\lambda\) \(\mu\)\(\nu\) can be expressed in units of \(\mathrm{m^{-1}}\). It then follows from则 Γ \(\lambda\) \(\mu\)\(\nu\) 可以用 \(\mathrm{m^{-1}}\) 为单位表示。然后可得

that R δ αβγ can be expressed in units of \(\mathrm{m^{-2}}\), but [R \(\mu\)\(\nu\)] and R are sums of components of R δ αβγ, so they too can be expressed in units of \(\mathrm{m^{-2}}\). With this in mind and recalling that the components of the energy–momentum tensor can be expressed in units of \(\mathrm{J\,m^{-3}}\) = \(\mathrm{kg\,m^{-1}\,s^{-2}}\), it can be seen that suitable units for \(\kappa\) are (1/\(m^{2}\))(1/(\(\mathrm{kg\,m^{-1}\,s^{-2}}\))) = \(\mathrm{kg^{-1}\,m^{-1}\,s^2}\), and the units of 8 πG/\(c^4\) are indeed N \(m^{2}\) kg − 2 s 4 m − 4 = \(\mathrm{kg^{-1}\,m^{-1}\,s^2}\).可知 R δ αβγ 可以用 \(\mathrm{m^{-2}}\) 为单位表示，但 [R \(\mu\)\(\nu\)] 和 R 是 R δ αβγ 各分量的和，因此它们也可以用 \(\mathrm{m^{-2}}\) 为单位表示。考虑到这一点，回想一下能量-动量张量的分量可以用 \(\mathrm{J\,m^{-3}}\) = \(\mathrm{kg\,m^{-1}\,s^{-2}}\) 的单位表示，可以看出，\(\kappa\) 的合适单位为 (1/m 2)(1/(\(\mathrm{kg\,m^{-1}\,s^{-2}}\))) = \(\mathrm{kg^{-1}\,m^{-1}\,s^2}\)，而 8 πG/\(c^4\) 的单位为事实上，N m 2 kg − 2 s 4 m − 4 = \(\mathrm{kg^{-1}\,m^{-1}\,s^2}\)。

Exercise 4.7 Show that Equation 4.34 can also be written练习 4.7 证明方程 4.34 也可以写成

as作为

where T ≡ \(\mu\) T \(\mu\) \(\mu\). (Hint: Multiply the Einstein field equations by \(g_{\mu\nu}\), and contract.)其中 T ≠ \(\mu\) T \(\mu\) \(\mu\)。 (提示：将爱因斯坦场方程乘以 \(g_{\mu\nu}\)，然后收缩。)

In some regions of spacetime, it may be that \(T_{\mu\nu}\) = 0. In such regions, spacetime is said to be empty. Equation 4.35 shows that in such a region, the Einstein field equations may be written as在时空的某些区域中，\(T_{\mu\nu}\) = 0。在这些区域中，时空被称为空的。方程 4.35 显示，在这样的区域中，爱因斯坦场方程可写为

Note that this does not necessarily mean that spacetime in the region is flat. The necessary and sufficient condition for flatness is that the components of the Riemann tensor should vanish at all events in the region, but that tensor has 20 components while the Ricci tensor has only 10. The vanishing of R \(\mu\)\(\nu\) in some, nor, therefore, does it region does not necessarily imply the vanishing of R \(\mu\) \(\nu\)αβ \(T_{\mu\nu}\) = 0 does indicate imply that \(g_{\mu\nu}\) describes a flat spacetime. However, setting that there is no matter or radiation in the region concerned, so solutions of Equation 4.36 are said to be vacuum solutions of the field equations. The study of vacuum solutions is an important sub-field of general relativity.请注意，这并不一定意味着该区域的时空是平坦的。平坦的充分必要条件是黎曼张量的分量在该区域的所有事件中都应该消失，但该张量有 20 个分量，而里奇张量只有 10 个。R \(\mu\)\(\nu\) 在某些区域的消失，因此，它并不一定意味着 R \(\mu\) \(\nu\)αβ \(T_{\mu\nu}\) = 0 的消失确实表明暗示 \(g_{\mu\nu}\) 描述了平坦的时空。然而，假设相关区域不存在物质或辐射，因此方程 4.36 的解被称为场方程的真空解。真空解的研究是广义相对论的一个重要子领域。

4.3.2 Geodesic motion4.3.2 测地运动

Einstein completed his long search for the field equations in 1915 and announced the basic principles of general relativity in a talk at the Prussian Academy of Sciences in Berlin in November 1915. The details of the theory were published in爱因斯坦于 1915 年完成了对场方程的长期探索，并于 1915 年 11 月在柏林普鲁士科学院的一次演讲中宣布了广义相对论的基本原理。该理论的详细信息发表于

1916. At that time Einstein clearly understood that in addition1916年。当时爱因斯坦清楚地认识到，除了

to using the field equations to find the spacetime metric, the theory also required that the metric should be used to determine the geodesics of the spacetime via the geodesic equations. These were introduced in Chapter 3. For a four-dimensional spacetime with metric tensor \(g_{\mu\nu}\), they take the form为了使用场方程找到时空度规，该理论还要求使用该度规通过测地线方程确定时空的测地线。这些已在第 3 章中介绍。对于具有度规张量 \(g_{\mu\nu}\) 的四维时空，它们采用以下形式

where \(\lambda\) is an affine parameter and, as usual,其中 \(\lambda\) 是仿射参数，并且像往常一样，

The functions x \(\rho\) (\(\lambda\)) that satisfy the geodesic equation describe parameterized curves through spacetime that represent the most direct routes between events. (A tangent to such a curve, parallel transported along the curve, remains a tangent.) So these geodesic curves are the analogues of straight lines in a curved space.满足测地线方程的函数 x \(\rho\) (\(\lambda\)) 描述了时空参数化曲线，代表事件之间最直接的路线。 (这样一条曲线的切线，沿曲线平行传输，仍然是切线。)因此，这些测地曲线类似于弯曲空间中的直线。