The Mathematical Theory of Cosmic Strings

Contents......Page 6
Introduction......Page 10
1 Cosmic strings and broken gauge symmetries......Page 14
1.1 Electromagnetism as a local gauge theory......Page 16
1.2 Electroweak unification......Page 21
1.3 The Nielsen–Olesen vortex string......Page 28
1.4 Strings as relics of the Big Bang......Page 37
1.5 The Nambu action......Page 40
2.1 Describing a zero-thickness cosmic string......Page 48
2.2 The equation of motion......Page 51
2.3 Gauge conditions, periodicity and causal structure......Page 54
2.4 Conservation laws in symmetric spacetimes......Page 57
2.5 Invariant length......Page 61
2.6 Cusps and curvature singularities......Page 62
2.7 Intercommuting and kinks......Page 67
3.1 The aligned standard gauge......Page 72
3.2 The GGRT gauge......Page 74
3.3 Conservation laws in flat space......Page 76
3.4 Initial-value formulation for a string loop......Page 81
3.5 Periodic solutions in the spinor representation......Page 83
3.6 The Kibble–Turok sphere and cusps and kinks in flat space......Page 86
3.7 Field reconnection at a cusp......Page 93
3.8 Self-intersection of a string loop......Page 98
3.9 Secular evolution of a string loop......Page 105
4.1.1 The infinite straight string......Page 112
4.1.2 Travelling-wave solutions......Page 113
4.1.3 Strings with paired kinks......Page 115
4.1.4 Helical strings......Page 116
4.2.1 The collapsing circular loop......Page 118
4.2.2 The doubled rotating rod......Page 119
4.2.3 The degenerate kinked cuspless loop......Page 120
4.2.4 Cat's-eye strings......Page 121
4.3 Balloon strings......Page 125
4.4.1 Loops with one harmonic......Page 127
4.4.2 Loops with two unmixed harmonics......Page 130
4.4.3 Loops with two mixed harmonics......Page 135
4.4.4 Loops with three or more harmonics......Page 140
4.5 Stationary rotating solutions......Page 143
4.6.1 The teardrop string......Page 148
4.6.2 The cardioid string......Page 150
4.6.3 The figure-of-eight string......Page 154
5.1 Strings in Robertson–Walker spacetimes......Page 157
5.1.1 Straight string solutions......Page 158
5.1.2 Ring solutions......Page 160
5.2 Strings near a Schwarzschild black hole......Page 165
5.2.1 Ring solutions......Page 166
5.2.2 Static equilibrium solutions......Page 170
5.3 Scattering and capture of a straight string by a Schwarzschild hole......Page 172
5.4 Ring solutions in the Kerr metric......Page 180
5.5 Static equilibrium configurations in the Kerr metric......Page 183
5.6 Strings in plane-fronted-wave spacetimes......Page 190
6 Cosmic strings in the weak-field approximation......Page 194
6.1 The weak-field formalism......Page 195
6.2 Cusps in the weak-field approximation......Page 198
6.3 Kinks in the weak-field approximation......Page 202
6.4 Radiation of gravitational energy from a loop......Page 204
6.5 Calculations of radiated power......Page 209
6.5.1 Power from cuspless loops......Page 210
6.5.2 Power from the Vachaspati–Vilenkin loops......Page 212
6.5.3 Power from the p/q harmonic solutions......Page 215
6.6 Power radiated by a helical string......Page 217
6.7 Radiation from long strings......Page 221
6.8.1 Linear momentum......Page 224
6.8.2 Angular momentum......Page 226
6.9.1 The piecewise-linear approximation......Page 232
6.9.2 A minimum radiative efficiency?......Page 236
6.10 The field of a collapsing circular loop......Page 239
6.11.1 General features of the problem......Page 244
6.11.2 Self-acceleration of a cosmic string......Page 247
6.11.3 Back-reaction and cusp displacement......Page 253
6.11.4 Numerical results......Page 255
7.1 The metric due to an infinite straight string......Page 259
7.2 Properties of the straight-string metric......Page 263
7.3 The Geroch–Traschen critique......Page 265
7.4 Is the straight-string metric unstable to changes in the equation of state?......Page 268
7.5 A distributional description of the straight-string metric......Page 272
7.6 The self-force on a massive particle near a straight string......Page 276
7.7 The straight-string metric in 'asymptotically-flat' form......Page 280
8.1 Straight strings and 2 + 1 gravity......Page 284
8.2 Boosts and rotations of systems of straight strings......Page 286
8.3 The Gott construction......Page 287
8.4 String holonomy and closed timelike curves......Page 291
8.5 The Letelier–Gal'tsov spacetime......Page 295
9.1 Strings and travelling waves......Page 299
9.2 Strings from axisymmetric spacetimes......Page 304
9.2.1 Strings in a Robertson–Walker universe......Page 305
9.2.2 A string through a Schwarzschild black hole......Page 310
9.2.3 Strings coupled to a cosmological constant......Page 314
9.3.1 The cylindrical formalism......Page 316
9.3.2 Separable solutions......Page 318
9.3.3 Strings in closed universes......Page 320
9.3.4 Radiating strings from axisymmetric spacetimes......Page 323
9.3.5 Einstein–Rosen soliton waves......Page 329
9.3.6 Two-mode soliton solutions......Page 334
9.4.1 Snapping strings in flat spacetimes......Page 337
9.4.2 Other spacetimes containing snapping strings......Page 342
10 Strong-field effects of zero-thickness strings......Page 345
10.1 Spatial geometry outside a stationary loop......Page 347
10.2 Black-hole formation from a collapsing loop......Page 353
10.3 Properties of the near gravitational field of a cosmic string......Page 356
10.4.1 General formalism......Page 359
10.4.2 Some sample near-field expansions......Page 362
10.4.3 Series solutions of the near-field vacuum Einstein equations......Page 365
10.4.4 Distributional stress–energy of the world sheet......Page 368
Bibliography......Page 372
B......Page 380
C......Page 381
E......Page 384
G......Page 385
K......Page 387
M......Page 388
N......Page 389
R......Page 390
S......Page 391
T......Page 392
Z......Page 393