Gravitational Curvature: An Introduction to Einstein's Theory

by Theodore Frankel

Overview

This classic text and reference monograph applies modern differential geometry to general relativity. A brief mathematical introduction to gravitational curvature, it emphasizes the subject's geometric essence and stresses the global aspects of cosmology. Suitable for independent study as well as for courses in differential geometry, relativity, and cosmology. 1979 edition.

Product Details

ISBN-13:	9780486289151
Publisher:	Dover Publications
Publication date:	04/10/2013
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	11 MB
Note:	This product may take a few minutes to download.

About the Author

A Professor of Mathematics at UC San Diego, Theodore Frankel was a longtime member of Princeton's Institute for Advanced Study and is known for his work in global differential geometry, Morse theory, and relativity theory.

Table of Contents

Preface to the Dover Edition xi

Preface xiii

Notation xvii

1 Special Relativity 1

The Lorentz Transformations as Viewed by Einstein 1

Minkowski Space 7

The Minkowski Norm 11

2 Clocks and Gravitational Potential 13

Gravitation, Acceleration, and the Principle of Equivalence 13

The Pseudo-Riemannian Structure of Space-Time 16

Gravitational Potential 20

Is Gravitation Governed by a Single Potential? 25

3 A Heuristic Derivation of Einstein's Equations 27

Poisson's Equation 27

The Density ρ^* 29

Einstein's Equations 31

4 The Geometry of Einstein's Equations 35

Curvature in a Pseudo-Riemannian M⁴ 35

The Einstein Tensor G_ij 39

The Gauss Equations in M⁴ 41

A Geometric Form of Einstein's Equations 42

5 The Schwarzschild Solution 46

Schwarzschild Coordinates 46

Embedding the Spatial Section 47

The Gravitational Potential and g₀₀ 50

The Schwarzschild Singularity 52

Concluding Remarks 53

6 The Classical Motion of a Continuum 56

Lie Derivatives, Interior Products, and the Variation of Integrals 56

The Cauchy Stress Tensor in Classical Mechanics 64

The Stress-Energy-Momentum Tensor 69

7 The Relativistic Equations of Motion 71

Fermi Transport and the Relative Velocity Vector 71

Vorticity, Strain, and Expansion 74

Shear and the Stress Tensor for a Viscous Fluid 78

Divergence of the Einstein Tensor: Gravitational "Force" 80

The Equations of Motion 82

Geodesies and Constants of Motion 85

Tidal Forces 88

8 Light Rays and Fermat's Principle 90

Fermat's Principle of Stationary Time 90

Geodesies in Conformally Related Metrics 93

The Deflection of Light 95

9 Electromagnetism in Three-Space and Minkowski Space 99

Twisted Forms and the Vector Product 99

E, B, and the (Heaviside-) Lorentz Force in Three-Space 100

Electromagnetism in Minkowski Space 102

Integration of Twisted Forms 103

The Charge-Current Three-Form in Minkowski Space 105

The Hodge *-Operator 106

The Laws of Gauss and Ampère-Maxwell 108

Faraday's Law and the Absence of Magnetic Monopoles 112

10 Electromagnetism in General Relativity 115

Maxwell's Equations 115

The Electromagnetic Stress-Energy-Momentum Tensor 118

The Reissner Solution 121

Conformal Invariance of Maxwell's Equations 124

Poisson's Equation in a Static Universe 125

Nonstatic Fields 127

11 The Interior Solution 128

Curvature of World Lines and Gravitational Force Potential 128

The Schwarzschild Interior Solution 130

The Oppenheimer-Volkoff Equation 134

12 Cosmology 139

The Einstein Static Universe 139

The Friedmann Cosmology: Assumptions 141

The Friedmann Cosmology: The Solution 144

The (Landau-) Raychaudhuri Equation 149

The Geometry of a Vorticity-Free Flow 152

A Generalized Poisson Equation for Vorticity Free Flows 152

General Vorticity-Free Cosmologies: The Influence of Curvature on Expansion 155

General Vorticity-Free Cosmologies: Singularities 157

General Vorticity-Free Cosmologies: Closed Spatial Universes 159

References 167

Index 169