A First Course in General Relativity (Second Edition)
A First Course in General Relativity (Second Edition)
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广义相对论第一课程（第二版）
DOI：
10.1088/0264-9381/27/10/109001
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发表时间：
2010
影响因子：
3.5
通讯作者：
Eric Poisson
中科院分区：
文献类型：
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作者：
Eric Poisson
几年前，在我对 Sean Carroll 在《古典与量子引力》一书中的评论中 [1]，我写道，虽然 1970 年代是 Weinberg [2] 和 Misner、Thorne 和 Wheeler [3] 的十年，而 80 年代是 Schutz [4] 和 Wald [5] 的十年，但 2000 年代显然是 Hartle [6] 和 Carroll [7] 的十年。在我看来，这些书在众多广义相对论入门教科书中继续脱颖而出。在新十年伊始，我期待着看到接下来会产生哪些新的教学见解，以及谁将成为 2010 年代的获胜者。当然，现在下结论还为时过早，但舒茨回来了，他将像 1985 年那样设定标准。这是期待已久的他的“第一课程”的第二版，这是对广义相对论的简短、易懂且非常成功的介绍。与第一版相比的变化不大：舒茨明智地避免用新主题来使文本膨胀，而只更新对引力波源和探测器、中子星和黑洞天体物理学的讨论以及进一步阅读的建议。最重要的是，他完全重写了宇宙学这一章，这个主题自第一版以来已经发生了巨大的变化。本书从第一章开始，对狭义相对论进行了精彩的回顾，强调时空几何，并远离基于洛伦兹变换的代数方法，后者仅在本章后面出现。接下来的第 2 章和第 3 章介绍了平坦时空中的矢量和张量分析。观点是现代的（张量被定义为向量和一式到实数的线性映射），但表达方式非常容易理解，并且避免了过多的数学细则。本书在第四章介绍了流体的时空描述；能量动量张量正是在这里首次出现。接下来解决向弯曲时空的转变。第五章使用等效原理来激发引力是时空弯曲的表现这一概念。首先通过对曲线坐标的推广来温和地处理弯曲时空中的张量微积分。第 6 章系统介绍了微分几何；在这里，读者将了解黎曼流形、协变微分、平行传输、测地线、曲率张量和比安奇恒等式。这是一个令人敬畏的章节，但学生得到了可靠的指导，而且演示既美观又易于理解。接下来的两章将微分几何引入物理学。在第七章中，读者学习如何在弯曲时空中制定物理定律，并在第八章中最终制定爱因斯坦场方程。本章最后对洛伦兹规范中的弱场极限进行了彻底的处理。以下章节介绍了该理论的应用。第 9 章专门介绍引力波：传播、探测、产生、能量平衡和天体物理源。与往常一样，这里的讨论是可访问的并且完全是最新的。我可以找出一个弱点，我在许多其他教科书上都注意到了这一点（这是我的一个小烦恼，它似乎正在变成一种痴迷）：引力波场的四极公式是在线性化理论的基础上推导出来的，没有警告读者该推导不适用于自引力系统。然而，这得到了一个主要优势的补偿：舒茨对引力波带走的能量的推导是基于一个美丽的物理论证，绕过了引力波场的能量动量张量的构造；与这种结构相关的复杂性是众所周知的，很高兴看到舒茨找到了一种很好的解决方法。第 10 章将精确理论应用于恒星结构，第 11 章向学生介绍了黑洞。本章的大部分内容致力于研究史瓦西时空中的测地运动，这使得舒茨能够接触到广义相对论的经典测试：近日点前进和光偏转。详细描述了事件视界处史瓦西坐标的奇异行为。这揭示了本书的另一个弱点：克鲁斯卡尔坐标只是简单地写下来，没有推导，也没有什么动机；遗憾的是，舒茨没有选择引入 Eddington-Finkelstein 坐标或 Painlevé-Gullstand 坐标作为更简单的替代方案。本章最后对黑洞进行了一般性讨论（包括它们在天体物理学中的地位和霍金效应的描述），并详细介绍了克尔解。最后一章（第12章）专门讨论宇宙学，这是本书修订最彻底的部分。演讲首先阐述了宇宙学原理和弗里德曼-勒梅特模型的推导。它继续讨论存在无压物质、辐射和宇宙学常数（在第一版时没有人想被提醒）的宇宙学动力学。最后对宇宙学测量进行了最新回顾，并简要介绍了宇宙的历史，从大爆炸到暴胀，再到重组，再到结构形成。本书中包含的广义相对论及其应用的介绍适合那些更喜欢标准“数学优先方法”而不是哈特尔“物理优先方法”的本科生。学生将以温和的方式学习微分几何的基础知识，然后将这些工具应用于弯曲时空的物理学；所有这些都可以在一个轻松的一学期课程中完成。这本书遗漏了许多在更高级的文本中可以找到的主题，例如李微分、微分形式、杀向量、微分几何的更抽象的公式（在图表和微分同胚方面）以及广义相对论的拉格朗日公式。这种范围限制是明智的：舒茨巧妙地在一个高效而小的包中涵盖了要点，并将所有改进放在其他教科书中进一步阅读；这是一个合理的学习策略。最后我想说我就是喜欢这本书。我今天就像我作为一名本科生第一次接触它时一样喜欢它。这些修订使本书保持最新状态，并确保舒茨的文本将在未来许多年中保留在广义相对论入门书籍的万神殿中。参考文献 [1] Poisson E 2005 年时空与几何评论：广义相对论简介，S M Carroll Class 着。量子重力22 4385–4386 [2] Weinberg S 1972 引力与宇宙学：广义相对论的原理和应用（纽约：Wiley）[3] Misner C W、Thorne K S 和 Wheeler J A 1973 引力（旧金山，加利福尼亚州：弗里曼）[4] Schutz B F 1985 A 第一门广义相对论课程（剑桥：剑桥大学出版社） [5] Wald R M 1984 广义相对论（芝加哥：芝加哥大学出版社） [6] Hartle J B 2003 引力：爱因斯坦广义相对论简介（加利福尼亚州旧金山：Addison-Wesley） [7] Carroll S 2003 时空与几何：广义相对论简介（加利福尼亚州旧金山：本杰明卡明斯）
A few years ago, in my review of Sean Carroll's book in Classical and Quantum Gravity [1], I wrote that while the 1970s was the decade of Weinberg [2] and Misner, Thorne and Wheeler [3], and while the eighties was the decade of Schutz [4] and Wald [5], the 2000s was clearly the decade of Hartle [6] and Carroll [7]. In my opinion, these books continue to stand out in the surprisingly dense crowd of introductory textbooks on general relativity. At the dawn of this new decade I look forward to see what fresh pedagogical insights will be produced next, and who will be revealed as the winners of the 2010s. It is, of course, much too early to tell, but Schutz is back, and he will set the standard just as he did back in 1985. This is the long-awaited second edition of his `First Course', a short, accessible, and very successful introduction to general relativity. The changes from the first edition are modest: Schutz wisely refrained from bloating the text with new topics, and limited himself to updating his discussion of gravitational-wave sources and detectors, neutron-star and black-hole astrophysics, and suggestions for further reading. Most importantly, he completely rewrote the chapter on cosmology, a topic that has evolved enormously since the first edition. The book begins in chapter 1 with a beautiful review of special relativity that emphasizes spacetime geometry and stays away from an algebraic approach based on the Lorentz transformation, which appears only later in the chapter. This is followed up in chapters 2 and 3 with an introduction to vector and tensor analysis in flat spacetime. The point of view is modern (tensors are defined as linear mapping of vectors and one-forms into real numbers) but the presentation is very accessible and avoids an overload of mathematical fine print. In chapter 4 the book introduces the spacetime description of fluids; it is here that the energy–momentum tensor makes its first appearance. The move to curved spacetime is tackled next. In chapter 5 the principle of equivalence is used to motivate the notion that gravity is a manifestation of spacetime curvature. Tensor calculus in curved spacetime is approached gently, by first working through a generalization to curvilinear coordinates. A systematic introduction to differential geometry is provided in chapter 6; here the reader is initiated in Riemannian manifolds, covariant differentiation, parallel transport, geodesics, the curvature tensors, and the Bianchi identities. This is a formidable chapter, but the student is guided by a sure hand, and the presentation is both beautiful and accessible. The next two chapters bring differential geometry to physics. In chapter 7 the reader learns how to formulate the laws of physics in a curved spacetime, and in chapter 8 the Einstein field equations are finally formulated. The chapter ends with a thorough treatment of the weak-field limit in the Lorenz gauge. The following chapters present applications of the theory. Chapter 9 is devoted to gravitational waves: propagation, detection, generation, energy balance, and astrophysical sources. Here, as always, the discussion is accessible and fully up-to-date. I could identify one weakness, which I have noted in many other textbooks (this is a pet peeve of mine, which seems to be turning into an obsession): the quadrupole formula for the gravitational-wave field is derived on the basis of the linearized theory, without warning the reader that the derivation does not apply to self-gravitating systems. This is, however, compensated by a major strength: Schutz's derivation of the energy carried off by gravitational waves is based on a beautiful physical argument that bypasses the construction of an energy–momentum tensor for the gravitational-wave field; the complexities associated with such a construction are well known, and it is nice to see that Schutz has found a nice way around. In chapter 10 the exact theory is applied to stellar structure, and in chapter 11 the student is introduced to black holes. A large part of the chapter is devoted to the study of geodesic motion in Schwarzschild spacetime, and this allows Schutz to make contact with the classical tests of general relativity: perihelion advance and light deflection. The singular behaviour of the Schwarzschild coordinates at the event horizon is described in detail. This reveals another weakness of the book: the Kruskal coordinates are simply written down, with no derivation and little motivation; it is a pity that Schutz did not choose to introduce the Eddington--Finkelstein coordinates, or the Painlevé–Gullstand coordinates, as easier alternatives. The chapter ends with a general discussion of black holes (including their place in astrophysics and a description of the Hawking effect) and a detailed presentation of the Kerr solution. The last chapter (chapter 12) is devoted to cosmology, and this is the part of the book that was the most thoroughly revised. The presentation begins with the enunciation of the cosmological principle and a derivation of the Friedmann–Lemaitre models. It continues with a discussion of cosmological dynamics in the presence of pressureless matter, radiation, and a cosmological constant (of which nobody wanted to be reminded at the time of the first edition). It concludes with an up-to-date review of cosmological measurements and a (very) brief history of the Universe, from the big bang to inflation, to recombination, to structure formation. The presentation of general relativity and its applications contained in this book is suitable for undergraduate students who would prefer the standard `math-first approach' to Hartle's `physics-first approach'. The student will learn the essentials of differential geometry in a gentle way, and will then apply these tools to physics in curved spacetime; all of this can be accomplished in a brisk one-semester course. The book leaves out many topics than can be found in more advanced texts, such as Lie differentiation, differential forms, Killing vectors, the more abstract formulation of differential geometry (in terms of charts and diffeomorphisms), and the Lagrangian formulation of general relativity. This limitation of scope is wise: Schutz masterly covers the essentials in an efficient and small package, and relegates all refinements to further reading in other textbooks; this is a sound learning strategy. To conclude I will state that I just love this book. I love it today as much as I did when I first came across it as an undergraduate student. The revisions bring the book up-to-date, and they ensure that Schutz's text will remain in the pantheon of introductory general relativity books for many years to come. References [1] Poisson E 2005 Review of Spacetime and Geometry: An Introduction to General Relativity, by S M Carroll Class. Quantum Grav. 22 4385–4386 [2] Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley) [3] Misner C W, Thorne K S, and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman) [4] Schutz B F 1985 A First Course in General Relativity (Cambridge: Cambridge University Press) [5] Wald R M 1984 General Relativity (Chicago : Chicago University Press) [6] Hartle J B 2003 Gravity: An Introduction to Einstein's General Relativity (San Francisco, CA: Addison-Wesley) [7] Carroll S 2003 Spacetime and Geometry: An Introduction to General Relativity (San Francisco, CA: Benjamin Cummings)