General Relativity [Paperback]

The first chapter is a short introduction to special relativity put in by the author for motivation. And, instead of introducing the mathematical formalism "as needed" in the book, the author chooses to outline it in detail in chapters two and three. The approach taken is a "modern" coordinate-free one, at least from the standpoint of differential geometry, but he delegates to an appendix the relevant background in topology. Since he is targeting the physicist reader, he does not hesitate to use diagrams to explain the concepts. The author introduces the idea of a dual vector using the example of a magnetic field. Tensors are then defined with great clarity from the standpoint of mathematical rigor. The physicist reader may have trouble digesting this if seeing tensors defined this way for the first time, instead of via their transformations properties, as is typically done. The abstract index notation is introduced to deal with the plethora of indices involved in manipulating tensors. In the treatment of geodesics, the author shows that it is sufficient to consider curves that are affinely parametrized, and the geodesic equation is derived in a coordinate basis. Riemannian and Gaussian normal coordinates are discussed as consequences of the unique solution of the geodesic equation. Curvature is also characterized in terms of the geodesic equation and two methods for calculating it are discussed: the coordinate component and tetrad methods, with the Newman-Penrose method briefly discussed. The existence of symbolic programming languages such as Mathematica and Maple make tensor manipulation much less laborius than the author contends in the book.

In the next chapter, the principle of general covariance is introduced as one that prohibits the existence of perferred vector fields in the laws of physics. The metric is the only quantity permitted to be related to space in the laws of physics. Thus quantities such as the Christoffel symbols, cannot appear in these laws. The author discusses in detail how general relativity views gravitation in terms of curved spacetime geometry and how Mach's principle is incorporated, the later forcing the spacetime metric to be a dynamical variable. The author discusses the difficulty in solving the Einstein equation, namely that a simultaneous solution for the spacetime metric and matter distribution is required (since the stress-energy tensor, the "source", requires knowledge of the spacetime metric for its interpretation). The linearized theory is discussed in detail along with the Newtonian limit. Gravitational waves are shown to follow from the linearized Einstein equation. The effect of energy loss on the orbital period of the Taylor-McCulloch binary star system is discussed as an experimental verification of general relativity.

Applications to cosmology are given in chapter 5, which is restricted to the case of homogeneous, isotropic cosmologies. The reader gets introduced to the famous Hubble constant, along with Robertson-Walker and Friedman solutions. A fairly lengthy overview of the evolution of the universe is given.

The next chapter is devoted entirely to the Schwarzschild solution, which is used to discuss the four experimental verifications of general relativity, namely the gravitational redshift, the precession of Mercury's orbit, bending of light by the Sun, and the time delay of radar signals. The singularities in the Schwarzschild solution are treated via the Kruskal extension.

Methods for obtaining physically realistic solutions are discussed in chapter 7, most of these being obtained by exploiting stationarity and symmetry properties. Perturbation theory is discussed very briefly with no explicit examples given.

Topics of a more mathematical nature appear in chapter 8, wherein the causal structure of spacetime is discussed. The discussion is qualitative and not based on Einsteins equation, and so is applicable to general spacetimes. One wonders when reading it if the obtained framework can be based on an analytical (or possibly numerical) treatment of the Einstein equation, instead of pure differential geometry. It is shown that null geodesics are The discussion here sets the tone for the next chapter on singularities, wherein the author derives criteria for determining when a timelike geodesic is not a local maximum in proper time between two points, and for when a null geodesic fails to remain on the boundary of the future of a point or two-dimensional surface. By using the local positivity of the stress-energy tensor (this is the only place the Einstein equation gets used) to get an inequality on the Ricci tensor, the author shows that timelike geodesics cannot be maximal length curves and null geodesics cannot remain on past or future boundaries. However, using compactness properties of the space of causal curves allows one to prove the existence of timelike and nullike curves of maximal length in globally hyperbolic spacetimes. The singularity theorems are shown to follow from this contraction, giving the result that spacetime is timelike or nulllike incomplete. A very detailed discussion of the definition of a singularity in physics is given. In all of the author's discussion, it is very interesting to note that the Einstein equation is only used once in obtaining the bound on the Ricci tensor. One naturally wonders if this framework is more general than what is available via general relativity, namely a question to ask is whether the Einstein notion of gravity can be derived from a consideration of singularities. Enforcing the presence (or absence) of singularities may allow the derivation of gravitational theories that are not the same as Einsteins, and yet have the same experimental success.