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Gravitational Curvature: An introduction to Einstein's Theory
Gravitational Curvature: An introduction to Einstein's Theory
Date: 25 January 2011, 13:08

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There are many books on special and general relativity available on the market of varying lengths and degrees of difficulty. Some emphasize the mathematics, others the physics behind these two theories. This book is primarily concerned with the mathematical structure of general relativity, and is written for the student of relativity who has preparation in differential geometry, on the level of manifolds and Riemannian geometry. Readers should not expect an in-depth discussion of general relativity, and so will need to consult other books and texts to gain further understanding. Chapter 1 outlines the theory of special relativity, with discussion first on the Lorentz transformations. The Lorentz group is defined, along with its generalization the Poincare group. Both of these groups could be viewed both mathematically and physically as a constraint on the construction of physical theories. Theories not satisfying the constraint of Lorentz or Poincare covariance cannot be physical theories. The famous Minkowski space is constructed, and the author discusses the (indefinite) norm on this space using an interesting formulation due to A.A. Robb. After a brief discussion of the principle of equivalence, the author begins a study of the (pseudo-Riemmanian) geometry of space-time in chapter 2. He constructs, using very simple geometric considerations and elementary physical reasoning, a pseudo-Riemannian metric that yields in chapter 3, after some "heuristic" considerations, the Einstein gravitational field equations. This derivation is interesting in that the Einstein equations are viewed as a generalization of Poisson's equation. The charge density is replaced by a "mass density" and the Laplacian of the scalar potential by an expression involving the Laplacian of the square root of the zeroth component of the metric of space-time. The stress-energy-momentum tensor is related to this Laplacian, and the Einstein equations follow immediately.


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