Exact Solutions of Einstein's Field Equations

by Hans Stephani , Dietrich Kramer , Malcolm MacCallum , Cornelius Hoenselaers , Eduard Herlt

• ISBN13: 9780521467025
• ISBN10: 0521467020
• Edition: 2nd
• Format: Paperback
• Copyright: 2009-10-30
• Publisher: Cambridge University Press

Summary

Now expanded and updated, this book gives a unique survey of the known solutions of Einstein’s field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources. It starts with an introductory part on the foundations of differential geometry and Riemannian geometry and the methods used to characterize, find or construct solutions (Petrov classification, Newman-Penrose formalism, groups of motions or homotheties, prolongations and generation techniques). The solutions are then considered, ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties. A chapter is devoted to the application of generation methods. Following the success of the first edition, this book will be an important source and guide for researchers working in Relativity and in Mathematical and Theoretical Physics and Astrophysics.

Table of Contents

Preface

List of tables

Notation

Introduction

General Methods

Differential geometry without a metric

Some topics in Riemannian geometry

The Petrov classification

Classification of the Ricci tensor and the energy-movement tensor

Vector fields

The Newman-Penrose and related formalisms

Continuous groups of transformations isometry and homothety groups

Invariants and the characterization of geometrics

Generation techniques

Solutions with Groups of Motions

Classification of solutions with isometries or homotheties

Homogeneous space-times

Hypersurface-homogeneous space-times

Spatially-homogeneous perfect fluid cosmologies

Groups G3 on non-null orbits V2. Spherical and plane symmetry

Spherically-symmetric perfect fluid solutions

Groups G2 and G1 on non-null orbits

Stationary gravitational fields

Stationary axisymmetric fields: basic concepts and field equations

Stationary axisymmetiric vacuum solutions

Non-empty stationary axisymmetric solutions

Groups G2I on spacelike orbits: cylindrical symmetry

Inhomogeneous perfect fluid solutions with symmetry

Groups on null orbits. Plane waves

Collision of plane waves

Algebraically Special Solutions

The various classes of algebraically special solutions. Some algebraically general solutions

The line element for metrics with κ=Ïâ=0=R11=R14=R44, Π+iÏëâë 0

Robinson-Trautman solutions

Twisting vacuum solutions

Twisting Einstein-Maxwell and pure radiation fields

Non-diverging solutions (Kundt's class)

Kerr-Schild metrics

Algebraically special perfect fluid solutions

Special Methods

Applications of generation techniques to general relativity

Special vector and tensor fields

Solutions with special subspaces

Local isometric embedding of four-dimensional Riemannian manifolds

Tables

The interconnections between the main classification schemes

References

Index.

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