Einstein Spaces presents the mathematical basis of the theory of gravitation and discusses the various spaces that form the basis of the theory of relativity. This book examines the contemporary development of the theory of relativity, leading to the study of such problems as gravitational radiation, the interaction of fields, and the behavior of elementary particles in a gravitational field.
Organized into nine chapters, this book starts with an overview of the principles of the special theory of relativity, with emphasis on the mathematical aspects. This text then discusses the need for a general classification of all potential gravitational fields, and in particular, Einstein spaces. Other chapters consider the gravitational fields in empty space, such as in a region where the energy-momentum tensor is zero. The final chapter deals with the problem of the limiting conditions in integrating the gravitational field equations.
Physicists and mathematicians will find this book useful.

Contenu

Preface to the English Edition

Foreword

Notation

Chapter 1. Basic Tensor Analysis

Riemann Manifolds
Tensor Algebra
Covariant Differentiation
Parallel Displacement in a Vn Space
Curvature Tensor of a Vn Space
Geodesies
Special Systems of Coordinates in Vn
Riemannian Curvature of Vn. Spaces of Constant Curvature
The Principal Axes Theorem for a Tensor
Lie Groups in Vn

Chapter 2. Einstein Spaces

The Basis of the Special Theory of Relativity. Lorentz Transformations
Field Equations in the Relativistic Theory of Gravitation
Einstein Spaces
Some Solutions of the Gravitational Field Equations

Chapter 3. General Classification of Gravitational Fields

Bivector Spaces
Classification of Einstein Spaces
Principal Curvatures
The Classification of Einstein Spaces for n = 4
The Canonical Form of the Matrix (Rab) for Ti and Ti Spaces
Classification of General Gravitational Fields
Complex Representation of Minkowski Space Tensors
Basis of the Complete System of Second Order Invariants of a VA Space

Chapter 4. Motions in Empty Space

Classification of Ti by Groups of Motions
Non-Isomorphic Structures of Groups of Motions Admitted by Empty Spaces
Spaces of Maximum Mobility T1, T2 and T8
T1 Spaces Admitting Motions
T2 and T3 Spaces Admitting Motions
Summary of Results. Survey of Well-known Solutions of the Field Equations

Chapter 5. Classification of General Gravitational Fields by Groups of Motions

Gravitational Fields Admitting a Gr Group (r = 2)
Gravitational Fields Admitting a G3 Group of Motions Acting on a V2 or V2
Gravitational Fields Admitting a G3 Group of Motions Acting on a V3 or V3
Gravitational Fields Admitting a Simply-Transitive or Intransitive G4 Group of Motions
Gravitational Fields Admitting Groups of Motions Gr (r = 5)

Chapter 6. Conformal Mapping of Einstein Spaces

Conformal Mapping of Riemann Spaces
Conformal Mapping of Riemann Spaces on Einstein Spaces
Conformal Mapping of Einstein Spaces on Einstein Spaces; Non-isotropic Case
Conformal Mapping of Einstein Spaces; Isotropic Case

Chapter 7. Geodesic Mapping of Gravitational Fields

Historical Survey
Algebraic Classification of the Possible Cases
The Invariant Equations for gij in a Non-Holonomic Orthonormal Tetrad
The Canonical Forms of the Metrics of V4 and K4 in a Holonomic Coordinate System
The Projective Mapping of Einstein Spaces

Chapter 8. The Cauchy Problem for the Einstein Field Equations