Mathematics – Differential Geometry
Scientific paper
Apr 2004
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2004cqgra..21.2219f&link_type=abstract
Classical and Quantum Gravity, Volume 21, Issue 8, pp. 2219-2220 (2004).
Mathematics
Differential Geometry
Scientific paper
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?
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