This timely and valuable book provides a detailed pedagogical introduction and treatment of the brane-localized gravity program of Randall and Sundrum, in which gravitational signals are able to localize around our four-dimensional world in the event that it is a brane embedded in an infinitely-sized, higher dimensional anti-de Sitter bulk space. A completely self-contained development of the material needed for brane-world studies is provided for both students and workers in the field, with a significant amount of the material being previously unpublished. Particular attention is given to issues not ordinarily treated in the brane-world literature, such as the completeness of tensor gravitational fluctuation modes, the causality of brane-world propagators, and the status of the massless graviton fluctuation mode in brane worlds in which it is not normalizable.
Author(s): Philip D. Mannheim
Publisher: World Scientific Publishing Company
Year: 2005
Language: English
Pages: 354
Contents......Page 12
Foreword......Page 8
Preface......Page 10
1. Why Higher Dimensions are Interesting......Page 18
2.1 The Einstein tensor of pure AdSs......Page 24
2.2 Introducing the brane......Page 26
2.3 Gravitational Gauss’s law in the context of the brane world......Page 27
2.4 Compactified brane world with two branes......Page 29
2.5 Removing the second brane......Page 30
3.1 Topology of anti-de Sitter space......Page 32
3.2 Coordinate system choices for anti-de Sitter space......Page 33
3.3 "Poincare patch” coordinates......Page 35
3.4 Other coordinate sectionings of anti-de Sitter space......Page 37
4.1 Gravitational versus purely geometric embeddings......Page 40
4.2 Gauss embedding formula......Page 41
4.3 Evaluation of the induced Einstein tensor in a specific case......Page 42
5.1 Junction conditions in the Newtonian and M+4 cases......Page 44
5.2 General Israel junction conditions......Page 45
6.1 Covariant analysis of an embedded Newtonian sheet of matter......Page 50
6.2 The embedded sheet Riemann tensor and its weak gravity limit......Page 52
6.3 Analogy to the static Einstein universe......Page 54
7.1 General structure of the Einstein equations in the RW case......Page 56
7.2 Static RW brane in an empty bulk......Page 57
7.3 Static RW brane in a bulk with a cosmological constant......Page 58
8.1 Embedding of a de Sitter brane in AdS5......Page 62
8.2 Embedding of an anti-de Sitter brane in AdS5......Page 63
9.1 Embedding an RW brane in a bulk with a cosmological constant......Page 66
9.2 Embedding an RW brane in an Ads5 bulk......Page 68
9.3 Difference between embedding RW and Schwarzschild branes......Page 70
10.1 Embedding a k > 0 RW brane in M(4, 2)......Page 72
10.2 Embedding a k > 0 RW brane and Ads5 bulk in M(4, 2)......Page 73
10.3 Embedding a Ic > 0 de Sitter brane and Ads5 bulk in M(4, 2)......Page 74
10.4 Analog to the Rindler horizon......Page 76
10.5 Embedding a k < 0 RW brane and Ads5 bulk in M(4, 2)......Page 77
10.6 Embedding a k = 0 RW brane and Ads5 bulk in M(4, 2)......Page 78
11.1 Null geodesics of the k > 0 RW brane world......Page 82
11.2 Null geodesics of the k = 0 and k < 0 RW brane worlds......Page 84
11.3 Alternate derivation of the k = 0 null geodesics......Page 85
11.4 Variation of the metric coefficients along a bulk null geodesic......Page 86
12.1 Derivation of the generalized Einstein equations on the brane......Page 88
12.2 Some exact solutions to the modified Friedmann equation......Page 91
12.3 Localization of gravity around a Robertson-Walker brane......Page 93
13.1 Structure of the brane-world background geometries......Page 98
13.2 General formulation of the gravitational fluctuation equations......Page 99
13.3 Gauge invariance considerations......Page 100
13.4 Imposition of the axial gauge in the brane world......Page 101
13.5 Gravitational fluctuation equations in the axial gauge......Page 102
14.1 NT sector fluctuations......Page 106
14.2 TT sector fluctuations......Page 108
14.3 Physical status of the NT and TT sector fluctuations......Page 110
14.4 Comparison with unembedded M4 fluctuations......Page 111
15.1 Gauge invariance in the dS4 and Ad& brane worlds......Page 116
15.2 NT/TT decomposition of the fluctuation equations......Page 118
15.3 Comparison with unembedded Ads4 fluctuations......Page 123
16.1 Orthogonality measure for TT modes......Page 126
16.2 Energy of gravitational fluctuations around flat spacetime......Page 128
16.3 First order total gravitational energy as a surface term......Page 131
16.4 Second order total gravitational energy as a surface term......Page 132
16.5 Second order gauge transformation properties of the energy......Page 133
16.6 Gauge invariant conservation condition for the total energy......Page 135
16.7 Origin of the gauge invariant energy conservation condition......Page 137
16.8 Abbott and Deser treatment of the curved space case......Page 138
16.9 The needed curved space Killing vectors......Page 140
16.10 The needed brane-world energy-momentum tensor......Page 141
16.11 M+4 total energy, normalization and completeness relations......Page 143
16.12 Construction of the M+4 propagator via completeness......Page 145
17.1 M+4 mode basis......Page 150
17.2 Mode propagation in M+4 and long distance gravity......Page 153
17.3 Causality issues in the M+4 brane world......Page 156
17.4 Causality in flat spacetime......Page 158
17.5 Upper half plane determination of the M+4 propagator......Page 160
17.6 Lower half plane determination of the M+4 propagator......Page 164
17.7 Evaluation of the cut discontinuities of the M+4 propagator......Page 166
17.8 Alternate propagators in M+4......Page 171
17.9 Some exact solutions to the M+4 fluctuation equations......Page 176
17.10 Connection to RW perturbations on the brane......Page 180
18.1 M-4 mode basis......Page 184
18.2 Completeness test for normalizable M-4 basis modes......Page 187
18.3 Completeness test for non-normalizable M-4 basis modes......Page 189
18.4 Upper half plane determination of the M-4 propagator......Page 193
18.5 Lower half plane determination of the M-4 propagator......Page 196
18.6 Negative signature modes in the M-4 propagator......Page 198
18.7 Temporal evolution of localized square steps in M-4......Page 200
18.8 Some exact solutions to the M-4 fluctuation equations......Page 203
19.1 AdS+4 mode basis......Page 206
19.2 Normalizability criterion for AdS+4 modes......Page 212
19.3 Discrete normalizable mode basis for AdS+4......Page 214
19.4 Completeness test for normalizable AdS+4 basis modes......Page 220
19.5 Normalizable Ads: basis mode propagator......Page 221
19.6 Completeness test for non-normalizable AdS+4 basis modes......Page 224
19.7 Some exact solutions to the AdS+4 fluctuation equations......Page 226
19.8 Constructing the causal AdS+4 propagator......Page 232
19.9 Causal propagation in the AdS+4 brane world......Page 241
20.1 Equivalent brane-world Schrodinger equation......Page 246
20.2 Instructive approximate brane-world Schrodinger equation......Page 248
21.1 dS+4 mode basis......Page 252
21.2 Causality in the dS+4 brane world......Page 255
22.1 Kink generated M4 domain wall......Page 266
22.2 Soliton generated M4 domain wall......Page 271
22.3 dS4 domain wall......Page 273
Appendix A Gauge Invariance of the Fluctuation Equation......Page 278
Appendix B Perturbative Weyl Tensor......Page 282
C.l Transverse projection for vector fields......Page 286
C.2 Transverse projection for flat space tensor fields......Page 291
C.3 Transverse-traceless projection for flat space tensor fields......Page 296
C.4 Transverse-traceless projection for curved space tensor fields......Page 300
D.l General second order Einstein-Hilbert action......Page 306
D.2 Second order fluctuation energy-momentum tensor in AdS5......Page 308
D.3 Conservation of the second order energy-momentum tensor......Page 311
D.4 Brane-world second order energy-momentum tensor......Page 313
E.1 Retarded propagator for pure AdS spacetimes......Page 316
E.2 Retarded propagator for pure dS spacetimes......Page 324
E.3 Retarded propagator for pure M4 spacetime......Page 328
F.l M+3 and M-3 brane-world background geometries......Page 330
F.2 Fluctuations in the M+3 and M+3 brane worlds......Page 331
F.3 M+3 brane-world mode basis......Page 332
F.4 Causality in the M+3 brane world......Page 333
F.5 Causality in the M-3 brane world......Page 338
Bibliography......Page 346
Author Index......Page 350
Subject Index......Page 352
