Mathematical Implications of Einstein-Weyl Causality

Contents......Page 9
1 Introduction......Page 13
1.1 Causality as a Physical Principle......Page 16
1.2 Causal Structures as Primary Objects......Page 17
1.3 Space-time at the Planck Scale......Page 18
2.1 Global Structures on R[sup(n)]......Page 19
2.2.1 Remark on Terminology......Page 21
2.2.3 The Conformal Structure......Page 22
2.2.4 The Weyl Projective Structure......Page 23
2.3 Nature of the Present Work......Page 24
2.4 Weyl on the Geometry of Space-Time......Page 25
3.1 Light Rays and Order......Page 27
3.1.1 Light Rays and the Order Axiom......Page 28
3.1.2 l-Completeness and l-Connectedness......Page 30
3.1.3 The Identification Axiom......Page 33
3.2 Construction of Cones......Page 34
3.2.1 Timelike Points......Page 38
4.1 Preliminary Remarks......Page 43
4.2 D-Sets and their Properties......Page 44
4.3 Timelike Order and D-Subsets......Page 54
4.4 Separating Points by D-sets......Page 56
4.5 Local Structure and Topology......Page 57
4.6 Regularity and Complete Regularity......Page 60
4.7 Order Equivalence......Page 61
5.1 Light Rays and D-sets......Page 63
5.1.1 Homeomorphism of Light Ray Segments......Page 64
5.2 Topological Preliminaries......Page 70
5.3 Segments of Light Rays......Page 72
5.4 Spacelike Hyperspheres......Page 73
6.1 General Discussion and Main Results......Page 78
6.1.2 Completeness and Complete Uniformizability......Page 79
6.2.1 The Order Uniformity on D-sets......Page 80
6.3 Uniform Completions of Ordered Spaces......Page 82
6.4 The Concept of Order Completion......Page 83
6.5.1 Symmetry Properties......Page 87
6.5.2 Separation Theorems......Page 89
6.5.3 Density Lemmas......Page 90
6.6.1 New Light Rays in M......Page 91
6.6.2 Verifications......Page 101
6.7 Remarks on the Assumptions 6.4.1 and 6.5.1......Page 104
7.1 An Infinite-dimensional Space......Page 106
7.3 Some Global Results......Page 107
7.3.1 Local Properties that Extend Globally......Page 108
7.3.2 Global Properties Without Local Counterparts......Page 110
8.1 Timelike Curves......Page 113
8.1.1 Topology of Θ[a, b]......Page 117
8.2 Parametrization of D-Intervals......Page 118
8.2.1 Boundaries of D-intervals. The Jordan-Brouwer Separation Property......Page 119
8.2.2 2-cells in D-intervals......Page 120
8.2.3 Cylindrical Coordinates on D-intervals......Page 124
8.2.4 Homogeneity and Homotopy of D-intervals......Page 126
8.3 Locally Compact Spaces......Page 127
8.3.1 Reconstruction of a Local Minkowski Structure......Page 129
8.4 The Differentiable Structure......Page 130
8.4.1 Remarks on the Global Differentiable Structure......Page 131
8.4.2 Monotone Functions......Page 132
8.4.3 Isotropy......Page 133
8.5.1 The Coordinate Transformations......Page 134
8.5.2 The Total Space......Page 135
8.5.3 The Order Structure......Page 136
9.2 Minkowski and de Sitter Spaces......Page 138
9.3 Timelike Curves in Cushion-Free Spaces......Page 142
10 Related Works......Page 145
10.1 The Work of Kronheimer and Penrose......Page 147
10.1.2 Comparison with the Present Work......Page 148
10.2 The Work of Ehlers, Pirani and Schild......Page 151
10.2.2 The EPS Axioms......Page 152
10.3.2 Works of Soviet Scholars......Page 154
11.1 Concerning Physics......Page 155
11.2 Concerning Mathematics......Page 160
11.3 Concerning Cantor, Wigner and Popper......Page 162
A: Uniformities and Uniform Completion......Page 165
A.1.1 Pseudometric Uniformities......Page 166
A.1.3 Covering Uniformities......Page 167
A.2 Equivalence Theorems......Page 168
A.3 The Uniform Topology......Page 169
A.4 Uniform Continuity and Equivalence......Page 170
A.6 Uniform Completion......Page 171
A.6.1 Complete Uniformizability: Shirota’s Theorem......Page 172
A.7 Properties of Hausdorff Uniformities......Page 173
A.8 Inequivalent Uniformities......Page 174
B.1.1 Coordinate Transformations on the Base Space......Page 176
B.1.2 Fibre Bundles......Page 177
B.1.4 Tangent Bundles......Page 178
B.2 G-structures on Differentiable Manifolds......Page 179
C: The Axioms and Special Assumptions......Page 181
References......Page 184
List of Symbols......Page 190
G......Page 192
R......Page 193
Z......Page 194