Gravity: An Introduction to Einstein's General Relativity
Gravity: An Introduction to Einstein's General Relativity
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DOI：
10.1088/0264-9381/21/8/b01
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发表时间：
2004-04
影响因子：
3.5
通讯作者：
J. Fabris
中科院分区：
文献类型：
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作者：
J. Fabris
广义相对论是现代物理学的基石之一。尽管如此，本科阶段的广义相对论教学仍然相当边缘化。造成这种特殊情况的原因是众所周知的。例如，我们可以引用其中两个：广义相对论需要特定的数学工具，而这些工具在某种程度上超出了本科生技术开发的主流；此外，这是物理学的一个分支，直到最近，其观测和实验应用仍然很少，尽管这种情况在过去几年中发生了巨大变化，但新情况尚未被吸收到本科教学中。然而，有许多教科书专门用于本科阶段的广义相对论教学。 J B Hartle 最近的书《引力：爱因斯坦广义相对论简介》就是这个意义上的一个新提议。这也许是最有趣的教学方法之一，旨在克服当人们试图将广义相对论纳入本科教学时出现的困难。在这本新书中，哈特尔试图解决任何在本科阶段教授广义相对论的人必须面临的困难。为了不让学生因为获得广义相对论的基本方程——爱因斯坦方程所需的艰苦技术准备而感到害怕，他干脆一开始就放弃了引入这些方程的想法。相反，他选择在本书的最后部分介绍爱因斯坦方程以及完成这些方程所需的大部分数学知识。这种微妙的（当然也是危险的）选择的优点是首先向读者介绍广义相对论的物理方面。这种方法可能很危险，因为讨论广义相对论物理内容所需的方程的相关解首先在没有形式推导的情况下提出。但作者以一种非常巧妙的方式规避了这个潜在的缺点。可以说，他找到了一种有效的方法来解决这门学科教学中常见的困境：先物理，后数学，但又不牺牲理论的完全一致性。本书分为三个部分。第一部分，涵盖全书24章中的5章，回顾了牛顿物理学和狭义相对论。这篇评论的目的是让读者为随后对广义相对论本身的讨论做好准备。介绍了相对论原理、变分原理、牛顿理论的几何内容以及狭义相对论背后的主要思想。总的来说，第一部分是相当标准的，除了强调几何方面，例如在物理问题中使用不同的坐标，特定几何的后果，例如球体的后果，以及时空流形的概念。这使得作者能够向读者介绍非欧几里得几何的概念及其内在属性。从某种意义上说，本书的核心是第二部分，包含 14 章，几乎涵盖了全书的三分之二。这里充分讨论了等效原理以及引力可以用时空几何表示的思想。重力不再被视为一种力，而是时空的曲率。首先，作者展示了如何获得牛顿引力作为特定时空几何的极限，这毕竟是一般伪黎曼时空的弱场极限。由此，他能够引入代表弱场限制之外的重力的几何形状。作者遵循的一般方法可总结如下。等效原理被编码在弯曲时空中。当引力效应较弱且速度与光速相比较小时，我们可以从这种时空结构中恢复牛顿理论。现在，让我们假设一个代表物理情况的特定弯曲时空，例如由球形质量分布创建的引力场。这导致了史瓦西时空。暂时忘记这个解决方案是如何获得的：将其视为几何结构。现在让我们探索这个几何结构。因此，可以通过测地方程来研究可能的轨道类型，可以讨论观测结果（光偏转、闭合轨道的近日点进动等），可以提出“坐标奇点”的概念，等等。这种方法真正有效的是，基本上可以提取特定物理情况的所有内容，例如静态、球对称问题。对于史瓦西时空，讨论了不同坐标的使用（Eddington--Finkelstein、Kruskal等），详细研究了Kruskal扩展和彭罗斯图，并分析了广义相对论的检验。还介绍和讨论了克尔度量，重点是它带来的新功能（例如能量提取）。宇宙学（由于明显的原因，仅限于各向同性和均匀的情况）和引力波的传播也是如此。通过这种方式，可以探索定性和定量特征。每一章都配有问题，这些问题可以简单地验证数学表达式，甚至可以是特定情况的数值模拟。提供了一个网站，其中包含本书的许多补充内容（例如用于解决某些问题的 Mathematica 程序）。作者成功地实现了广义相对论的理论和观测方面的平衡。例如，当他开始介绍广义相对论时，就展示了对GPS的详细描述。仔细讨论了黑洞的观测状况、均匀和各向同性宇宙的证据以及探测引力波的努力。从这个意义上说，这本书是对广义相对论的完整介绍：提出了有关黑洞识别的问题（有时还提供了它们的解决方案），以及宇宙学中的不同观测项目，如超新星Ia型、星系的2DFRGS映射和宇宙微波背景的各向异性光谱。其中许多讨论都是借助方框来介绍的，这使得作者可以在不破坏文本主要发展的情况下揭示特定主题。第三部分（剩余五章）介绍了微分几何，最终引出了爱因斯坦方程。现在可以通过严格的方式得到前一部分讨论的解决方案。我有一种感觉，这里的文字变得更加密集，在某种意义上变得“沉重”。但广义相对论的所有美妙之处都已经成功地呈现出来了。无论如何，掌握了该理论的完整数学结构，就可以以更定量的方式分析引力波和恒星结构。本书的最后有四个附录，包括公式的通用术语表和可能的教学策略的建议。在本科阶段教授广义相对论不可避免地会带来一个困境：从一开始就严格要求，开发所有必要的工具，但可能会因为困难的新数学而让学生望而却步，或者强调物理方面，但可能会因为定性而导致学生无法掌握理论的全部内容。我认为 J B Hartle 的这本新书以一种相当一致的方式解决了这个困境。保留了基于广义相对论的物理学风味，同时读者最终可以自行计算。在一本关于这个困难而迷人的主题的介绍性书籍中，我们还能要求什么呢？
General relativity is one of the cornerstones of modern physics. In spite of this, the teaching of general relativity at undergraduate level remains quite marginal. The reasons for this particular situation are quite well known. We can quote, for example, two of them: general relativity requires specific mathematical tools that are somehow outside the mainstream of undergraduate technical development; moreover, this is a branch of physics whose observational and experimental applications have remained rare until recent times, and even though this scenario has changed dramatically in the last few years, the new situation has not yet been absorbed into undergraduate teaching. However, there are many textbooks devoted to the teaching of general relativity at undergraduate level. The recent book of J B Hartle, Gravity: An Introduction to Einstein's General Relativity, is a new proposal in this sense. It is perhaps one of the most interesting pedagogical approaches seeking to surmount the difficulties that arise when one tries to include general relativity in undergraduate teaching. In this new book, Hartle attempts to address the difficuties that must be faced by anyone who teaches general relativity at undergraduate level. In order to not scare the student with the hard technical preparation needed to obtain the basic equations of general relativity, Einstein's equations, he simply gives up the idea of introducing these equations at the very beginning. Instead, he chooses to present Einstein's equations, with most of the mathematics needed to do them, in the last part of the book. This delicate (and of course dangerous) choice has the advantage of introducing the reader first to the physical aspects of general relativity. This approach can be dangerous because the relevant solutions of the equations necessary to discuss the physical content of general relativiy are presented first without a formal derivation. But the author circumvents this potential drawback in a very skilful way. We can say that he has found an efficient way to solve the usual dilemma in teaching this subject: physics first, mathematics later, but without sacrificing the full consistency of the theory. The book is divided into three parts. In the first, covering five chapters of the 24 in the whole book, Newtonian physics and special relativity are reviewed. This review is done in a manner that prepares the reader for the subsequent discussion of general relativity itself. The principle of relativity, the variational principle, the geometrical content of Newtonian theory and the main ideas behind special relativity are all presented. In general, this first part is quite standard, except for the emphasis on geometrical aspects, such as the employment of different coordinates in a physical problem, the consequences of specific geometries, such as that of a sphere, and the notion of a spacetime manifold. This allows the author to introduce the reader to the idea of non-Euclidean geometries and their intrinsic properties. The heart of the book is, in some sense, in the second part, containing 14 chapters and covering almost two thirds of the book. Here, the principle of equivalence is fully discussed as well as the idea that gravity can be represented by the geometry of spacetime. Gravity is no longer conceived as a force but instead as the curvature of spacetime. First, the author shows how Newtonian gravity can be obtained as the limit of a specific spacetime geometry, which is after all the weak field limit of a general pseudo-Riemannian spacetime. From this, he is able to introduce geometries that represent gravity outside this weak field limit. The general approach followed by the author could be summarized as follows. The equivalence principle is encoded in curved spacetime. When gravitation effects are weak and the velocity small compared with the velocity of light, we can recover Newtonian theory from this spacetime structure. Now, let us assume a specific curved spacetime representing a physical situation, for example, the gravitational field created by a spherical mass distribution. This leads to the Schwarzschild spacetime. Forget for the moment how this solution is obtained: see it as a geometrical structure. Let us now explore this geometrical structure. Hence, possible kinds of orbits can be studied through the geodesic equation, observational results (light deflection, perihelion precession of closed orbits, etc) can be discussed, the notion of 'coordinate singularity' can be presented, and so on. What is really effective in this approach is that essentially all the content of specific physical situations, such as the static, spherically symmetric problem, can be extracted. For the Schwarzschild spacetime, the employment of different coordinates is discussed (Eddington--Finkelstein, Kruskal, etc), the Kruskal extension and Penrose diagram are studied in detail and tests of general relativity are analysed. The Kerr metric is also presented and discussed, with emphasis on the new features that it brings (energy extraction, for example). The same is done for cosmology (restricted, for obvious reasons, to the isotropic and homogenous cases) and the propagation of gravitational waves. In this way, qualitative and quantitative features can be explored. Each chapter is complemented by problems, which can simply be a verification of a mathematical expression or even the numerical simulation of a specific situation. A Web site with many supplements to the book (e.g.the Mathematica program for solving some problems) is provided. The author successfully achieves an equilibrium between the theoretical and observational aspects of general relativity. For example, when he starts to present general relativity, a detailed description of the GPS is exhibited. The observational status of black holes, the evidence for a homogenous and isotropic universe, and efforts to detect gravitational waves are carefully discussed. In this sense, this book is a complete introduction to general relativity: the problems concerning the identification of black holes are presented (sometimes with their solutions), as well as different observational programmes in cosmology, like the supernova type Ia, the 2DFRGS mapping of galaxies and the anisotropy spectrum of the cosmic microwave background. Many of these discussions are introduced with the aid of boxes, which allows the author to expose specific topics without breaking the main developments of the text. In the third part (the remaining five chapters) differential geometry is presented, leading finally to Einstein's equations. Now the solutions discussed in the previous part can be obtained in a rigorous way. I had the feeling that the text here gets denser, 'heavy' in some sense. But all the beauty of general relativity has already been presented successfully. In any case, having in hand the complete mathematical structure of the theory, gravitational waves and stellar structure can be analysed in a more quantitative way. Four appendices, including a general glossary of formulae and proposals for possible pedagogical strategies, close the book. Teaching general relativity at undergradute level inevitably brings a dilemma: to be rigorous from the beginning, developing all the tools necessary to do it but risk discouraging the student with difficult new mathematics or to emphasize the physical aspects but risk being so qualitative that the full content of the theory cannot be grasped by the student. I think that this new book by J B Hartle solves this dilemma in a quite consistent way. The flavour of the physics which relies on general relativity theory is preserved and, at the same time, the reader can, at the end, perform calculations by himself. What more could we ask for in an introductory book on this difficult and fascinating subject?