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Tables of Contents for Aspects of Gravitational Interactions
Chapter/Section Title
Page #
Page Count
Foreword
xi
2
Preface
xiii
2
Conventions and Abbreviations
xv
CHAPTER 1 INTRODUCTION TO THE THEORY OF MANIFOLDS
1
22
1.1 Differential manifolds
1
5
1.2 Tangent and cotangent spaces
6
3
1.3 Lie Groups and Lie Algebra
9
8
1.4 Fibre bundle
17
1
1.5 Tangent and cotangent bundles
18
1
1.6 Principal Bundles
19
1
1.7 Projective Spaces
20
3
CHAPTER 2 TENSORS AND RIEMANNIAN GEOMETRY
23
36
2.1 Tensor Space
23
4
2.2 Metric tensors
27
3
2.3 Levi-Civita Tensors
30
2
2.4 Parallel Transport and Co-variant Derivatives
32
6
2.5 Levi-Civita Connection and Christoffel's Symbols
38
1
2.6 Geodesics and Auto-parallel Curves
39
3
2.7 Geodetic Interval
42
2
2.8 Holonomy
44
3
2.9 Curvature
47
3
2.10 Geodesic Deviations
50
1
2.11 Killing vectors and Isometry
51
5
2.12 Local Lorentz Transformations
56
3
CHAPTER 3 EINSTEIN'S THEORY OF GRAVITY
59
22
3.1 Background
59
3
3.2 Principle of Equivalence
62
1
3.3 Principle of general covariance
63
1
3.4 A passage from special relativity to general relativity
63
1
3.5 Einstein's field equations for gravity
64
5
3.6 Reduction of Einstein's equations to Poisson's equations
69
2
3.7 Cauchy problem
71
2
3.8 Energy conditions
73
1
3.9 Singularity theorem
73
1
3.10 Solution of Einstein's field equations
73
6
3.11 Experimental tests of Einstein's theory of gravity
79
2
CHAPTER 4 SOME IMPORTANT MODIFICATIONS OF EINSTEIN'S THEORY OF GRAVITY
81
16
4.1 Theory of varying constant of gravitation (Brans-Dicke theory)
82
4
4.2 Higher-derivative gravity
86
4
4.3 Strong gravity
90
3
4.4 Einstein-Cartan-Sciama-Kibble theory
93
4
CHAPTER 5 INTERACTION OF QUANTUM FIELDS WITH CLASSICAL GRAVITY
97
42
5.1 Preliminaries of quantum mechanics
97
1
5.2 Quantum fields in Minkowski space-time
98
3
5.3 Quantum fields in quasi-Riemannian space-time
111
7
5.4 Green's functions
118
2
5.5 Adiabatic expansion of Green's functions
120
5
5.6 Calculation of one-loop effective action in curved space-time
125
14
CHAPTER 6 GAUGE THEORY OF GRAVITY
139
14
6.1 Motivation and development of gauge theory of gravity
139
3
6.2 Conservation laws in general relativity and certain problems
142
3
6.3 Carmeli's theory of gravity based on SL(2,C) groups
145
8
CHAPTER 7 KALUZA-KLEIN THEORY
153
28
7.1 Historical background of Kaluza-Klein Theory
153
1
7.2 Compactification in Kaluza-Klein theory and its needs
154
1
7.3 Kaluza-Klein theory of electromagnetism
155
4
7.4 Generalizations of Kaluza-Klein theory to non-Abelian gauge fields
159
3
7.5 Geometrical analysis of compact manifold
162
6
7.6 Scalar, vector and spinor fields in higher-dimensional theories
168
8
7.7 Kaluza-Klein Supergravity
176
5
APPENDIX
181
18
Appendix 1A Mathematical Preliminaries
181
3
Appendix 5A1 Ket and Bra Vectors
184
3
Appendix 5A2 Grassmann Algebra
187
2
Appendix 5A3 Calculation of Path integrals
189
1
Appendix 5A4 Euclideanization of Minkowski and Pseudo-Riemannian Geometry
190
1
Appendix 7A1 Five-dimensional space-time
191
3
Appendix 7A2 d-dimensional space-time
194
2
Appendix 7A3 Spontaneous Symmetry Breaking
196
3
Bibliography
199
6
Index
205