A Mathematical Introduction to General Relativity

Mathematicians and mathematics students who want to learn something of general relativity can find themselves in a quandary. Some of the best books on the subject mix the mathematics in with the physics in such a way that it can be hard to distinguish physical reasoning from rigorous mathematical results. This book aims to provide a rigorous mathematical introduction to the subject with proofs of all the primary results before getting deeply into the physics. This leads to a treatment that is roughly two-thirds mathematics and one-third physics.

 

The mathematical content consists primarily of an extensive introduction to the differential geometry of manifolds, but one limited to what is needed to describe the basic results of general relativity. The first part develops the background needed to understand what is meant by a four-dimensional Lorentzian manifold with a time orientation and to provide the analytical apparatus that goes with that. This begins with the definition of a manifold and moves on to develop the necessary concepts from differential geometry. The sequence of topics goes from tangent and cotangent bundles to tensor fields, Levi-Civita connections, covariant derivatives, parallel transport, geodesics and curvature. Differential forms are not given a full treatment, but enough to create a “volume form field” on a Lorentzian manifold, and so enable coordinate-free integration on a manifold. Integration on manifolds follows; it takes a limited form that does not include submanifolds or manifolds-with-boundary. Proof of a correspondingly restricted Stokes theorem is assigned as an exercise.

 

Throughout the chapters on mathematical background, the author intersperses examples and exercises that apply individual topics to physics. For example, gravitational redshift is described once the Lorentzian manifold with a metric is introduced. Similarly, the bending of a light ray as it passes near the sun (the first experimental test of general relativity) is described when the geodesics in Schwartschild spacetime are introduced. This means that readers get at least a glimpse of the physics as they work their way through a lot of differential geometry.

 

Once we get to the physics the first topic is Minkowski spacetime physics, which is basically special relativity. Full general relativity starts with the introduction of matter and the field equation that equates two tensor fields. The left hand side of the field equation is geometrical; it involves the Ricci and curvature tensors, the cosmological constant, and a metric on a semi-Riemannian manifold. The right hand side represents the matter in spacetime and depends only on the energy-momentum tensor. Einstein introduced the first version of the field equation (without the cosmological constant), and Hilbert followed quickly with a version he found using a variational principle.

 

The field equation is not mathematically derivable but is a powerful model consistently supported by experimental and observational results. Solving the field equation means finding the metric, and that means solving the ten independent second-order PDEs that express the field equation in component form. Fortunately there are a lot of symmetries for simple configurations.

 

The last chapters of the book explore some consequences of the field equation, including black holes and cosmology. The author also describes the relationship between classical physics and general relativity, and illustrates the way that the geometric perspective of general relativity illuminates other classical applications like the description of a perfect fluid. Two of the highlights are demonstrations of how general relativity predicts the perihelion shift in the orbit of Mercury and a model of the big bang.

 

The book is aimed at readers with at least basic analysis and linear algebra. A student new to the field of differential geometry might find the first part of the book intimidating. For pedagogical reasons it might have been desirable to incorporate more exposition describing the whys and wherefores of new constructions and structures even at the expense of leaving out some of the proofs. As a book structured primarily for self-study, this may be particularly important. 

 

This is a solid but challenging introduction to the subject. Many good exercises are provided throughout the book, and very detailed solutions for all of them appear in an appendix.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.