Frankel T. Gravitational Curvature: An Introduction to Einstein's Theory

San Francisco: W. H. Freeman and Company, 1979. — 191 p.I try to present, in this book, a way to view the basic aspects of general relativity, a way that is in harmony with my own feelings about geometry and its relationship with pre-quantum physics. The method of development is "geometric" in the sense that I prefer to compute curvatures or second fundamental forms rather than components of the Riemann tensor or Christoffel symbols. Using this method, simple situations (that is, involving many symmetries) can be handled completely without leaving numerous long and dreary computations "for the reader," as is often done. Differential geometry enters naturally, not only in the very formulation of Einstein's theory but also in the solutions of specific technical problems. I assume that the reader knows some basic differential geometry. Chapter 1 deals with those aspects of special relativity needed for motivational reasons in general relativity and for an understanding of Maxwell's equations. Chapters 2 and 3 are concerned with the formulation of Einstein's theory and with a heuristic derivation of his equations. Chapter 4 translates the tensor form of Einstein's equations into several more geometric forms. Chapter 5 is concerned with the Schwarzschild solution. Chapter 6 reviews background material helpful for the discussion of several topics to be considered. It is assumed here that the reader is familiar with elementary aspects of the exterior differential calculus, as this is clearly the natural tool for several of the topics. Chapter 7 deals with the general relativistic equations of motion and the stress-energy-momentum tensor. Chapter 8 involves light signals and Fermat's principle in general relativity. Chapter 9 reviews electromagnetism in three-space and in Minkowski space. Chapter 10 treats electromagnetism in general relativity. One minor point of departure from the usual treatments is the use of the "Fermat metric" for discussing potentials in a static universe. Chapter 11 investigates Einstein's equations inside matter. Chapter 12 is devoted to cosmology.Notation
Special Relativity
Clocks and Gravitational Potential
A Heuristic Derivation of Einstein's Equations
The Geometry of Einstein's Equations
The Schwarzschild Solution
The Classical Motion of a Continuum
The Relativistic Equations of Motion
Light Rays and Fermat's Principle
Electromagnetism in Three-Space and Minkowski Space
Electromagnetism in General Relativity
The Interior Solution
Cosmology