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.
Sprache
Verlagsort
ISBN-13
978-1-4831-5184-7 (9781483151847)
Schweitzer Klassifikation
Preface to the English EditionForewordNotationChapter 1. Basic Tensor Analysis 1. Riemann Manifolds 2. Tensor Algebra 3. Covariant Differentiation 4. Parallel Displacement in a Vn Space 5. Curvature Tensor of a Vn Space 6. Geodesies 7. Special Systems of Coordinates in Vn 8. Riemannian Curvature of Vn. Spaces of Constant Curvature 9. The Principal Axes Theorem for a Tensor 10. Lie Groups in VnChapter 2. Einstein Spaces 11. The Basis of the Special Theory of Relativity. Lorentz Transformations 12. Field Equations in the Relativistic Theory of Gravitation 13. Einstein Spaces 14. Some Solutions of the Gravitational Field EquationsChapter 3. General Classification of Gravitational Fields 15. Bivector Spaces 16. Classification of Einstein Spaces 17. Principal Curvatures 18. The Classification of Einstein Spaces for n = 4 19. The Canonical Form of the Matrix (Rab) for Ti and Ti Spaces 20. Classification of General Gravitational Fields 21. Complex Representation of Minkowski Space Tensors 22. Basis of the Complete System of Second Order Invariants of a VA SpaceChapter 4. Motions in Empty Space 23. Classification of Ti by Groups of Motions 24. Non-Isomorphic Structures of Groups of Motions Admitted by Empty Spaces 25. Spaces of Maximum Mobility T1, T2 and T8 26. T1 Spaces Admitting Motions 27. T2 and T3 Spaces Admitting Motions 28. Summary of Results. Survey of Well-known Solutions of the Field EquationsChapter 5. Classification of General Gravitational Fields by Groups of Motions 29. Gravitational Fields Admitting a Gr Group (r = 2) 30. Gravitational Fields Admitting a G3 Group of Motions Acting on a V2 or V2 31. Gravitational Fields Admitting a G3 Group of Motions Acting on a V3 or V3 32. Gravitational Fields Admitting a Simply-Transitive or Intransitive G4 Group of Motions 33. Gravitational Fields Admitting Groups of Motions Gr (r = 5)Chapter 6. Conformal Mapping of Einstein Spaces 34. Conformal Mapping of Riemann Spaces 35. Conformal Mapping of Riemann Spaces on Einstein Spaces 36. Conformal Mapping of Einstein Spaces on Einstein Spaces; Non-isotropic Case 37. Conformal Mapping of Einstein Spaces; Isotropic Case Chapter 7. Geodesic Mapping of Gravitational Fields 38. Historical Survey 39. Algebraic Classification of the Possible Cases 40. The Invariant Equations for gij in a Non-Holonomic Orthonormal Tetrad 41. The Canonical Forms of the Metrics of V4 and K4 in a Holonomic Coordinate System 42. The Projective Mapping of Einstein SpacesChapter 8. The Cauchy Problem for the Einstein Field Equations 43. The Einstein Field Equations 44. The Exterior Cauchy Problem 45. Freedom Available in Choosing Field Potentials for an Einstein Space 46. Characteristic and Bicharacteristic Manifolds 47. The Energy-Momentum Tensor 48. The Conservation Law for the Energy-Momentum Tensor 49. The Interior Cauchy Problem for the Flow of Matter 50. The Interior Cauchy Problem for a Perfect FluidChapter 9. Special Types of Gravitational Field 51. Reducible and Conformal-Reducible Einstein Spaces 52. Symmetric Gravitational Fields 53. Static Einstein Spaces 54. Centro-Symmetric Gravitational Fields 55. Gravitational Fields with Axial Symmetry 56. Harmonic Gravitational Fields 57. Spaces Admitting Cylindrical Waves 58. Spaces and their Boundary ConditionsReferencesIndex
