The Mathematical Theory Of Cosmic Strings – M. R. Anderson – 1st Edition

Introduction ix
1 Cosmic strings and broken gauge symmetries 1
1.1 Electromagnetism as a local gauge theory 3
1.2 Electroweak unification 8
1.3 The Nielsen–Olesen vortex string 15
1.4 Strings as relics of the Big Bang 24
1.5 The Nambu action 27

2 The elements of string dynamics 35
2.1 Describing a zero-thickness cosmic string 35
2.2 The equation of motion 38
2.3 Gauge conditions, periodicity and causal structure 41
2.4 Conservation laws in symmetric spacetimes 44
2.5 Invariant length 48
2.6 Cusps and curvature singularities 49
2.7 Intercommuting and kinks 54

3 String dynamics in flat space 59
3.1 The aligned standard gauge 59
3.2 The GGRT gauge 61
3.3 Conservation laws in flat space 63
3.4 Initial-value formulation for a string loop 68
3.5 Periodic solutions in the spinor representation 70
3.6 The Kibble–Turok sphere and cusps and kinks in flat space 73
3.7 Field reconnection at a cusp 80
3.8 Self-intersection of a string loop 85
3.9 Secular evolution of a string loop 92

4 A bestiary of exact solutions 99
4.1 Infinite strings 99
4.1.1 The infinite straight string 99
4.1.2 Travelling-wave solutions 100
4.1.3 Strings with paired kinks 102
4.1.4 Helical strings 103.
4.2 Some simple planar loops 105
4.2.1 The collapsing circular loop 105
4.2.2 The doubled rotating rod 106
4.2.3 The degenerate kinked cuspless loop 107
4.2.4 Cat’s-eye strings 108
4.3 Balloon strings 112
4.4 Harmonic loop solutions 114
4.4.1 Loops with one harmonic 114
4.4.2 Loops with two unmixed harmonics 117
4.4.3 Loops with two mixed harmonics 122
4.4.4 Loops with three or more harmonics 127
4.5 Stationary rotating solutions 130
4.6 Three toy solutions 135
4.6.1 The teardrop string 135
4.6.2 The cardioid string 137
4.6.3 The figure-of-eight string 141

5 String dynamics in non-flat backgrounds 144
5.1 Strings in Robertson–Walker spacetimes 144
5.1.1 Straight string solutions 145
5.1.2 Ring solutions 147
5.2 Strings near a Schwarzschild black hole 152
5.2.1 Ring solutions 153
5.2.2 Static equilibrium solutions 157
5.3 Scattering and capture of a straight string by a Schwarzschild hole 159
5.4 Ring solutions in the Kerr metric 167
5.5 Static equilibrium configurations in the Kerr metric 170
5.6 Strings in plane-fronted-wave spacetimes 177

6 Cosmic strings in the weak-field approximation 181
6.1 The weak-field formalism 182
6.2 Cusps in the weak-field approximation 185
6.3 Kinks in the weak-field approximation 189
6.4 Radiation of gravitational energy from a loop 191
6.5 Calculations of radiated power 196
6.5.1 Power from cuspless loops 197
6.5.2 Power from the Vachaspati–Vilenkin loops 199
6.5.3 Power from the p/q harmonic solutions 202
6.6 Power radiated by a helical string 204
6.7 Radiation from long strings 208
6.8 Radiation of linear and angular momentum 211
6.8.1 Linear momentum 211
6.8.2 Angular momentum 213
6.9 Radiative efficiencies from piecewise-linear loops 219
6.9.1 The piecewise-linear approximation 219
6.9.2 A minimum radiative efficiency? 223
6.10 The field of a collapsing circular loop 226
6.11 The back-reaction problem 231
6.11.1 General features of the problem 231
6.11.2 Self-acceleration of a cosmic string 234
6.11.3 Back-reaction and cusp displacement 240
6.11.4 Numerical results 242

7 The gravitational field of an infinite straight string 246
7.1 The metric due to an infinite straight string 246
7.2 Properties of the straight-string metric 250
7.3 The Geroch–Traschen critique 252
7.4 Is the straight-string metric unstable to changes in the equation of state? 255
7.5 A distributional description of the straight-string metric 259
7.6 The self-force on a massive particle near a straight string 263
7.7 The straight-string metric in ‘asymptotically-flat’ form 267

8 Multiple straight strings and closed timelike curves 271
8.1 Straight strings and 2 + 1 gravity 271
8.2 Boosts and rotations of systems of straight strings 273
8.3 The Gott construction 274
8.4 String holonomy and closed timelike curves 278
8.5 The Letelier–Gal’tsov spacetime 282

9 Other exact string metrics 286
9.1 Strings and travelling waves 286
9.2 Strings from axisymmetric spacetimes 291
9.2.1 Strings in a Robertson–Walker universe 292
9.2.2 A string through a Schwarzschild black hole 297
9.2.3 Strings coupled to a cosmological constant 301
9.3 Strings in radiating cylindrical spacetimes 303
9.3.1 The cylindrical formalism 303
9.3.2 Separable solutions 305
9.3.3 Strings in closed universes 307
9.3.4 Radiating strings from axisymmetric spacetimes 310
9.3.5 Einstein–Rosen soliton waves 316
9.3.6 Two-mode soliton solutions 321
9.4 Snapping cosmic strings 324
9.4.1 Snapping strings in flat spacetimes 324
9.4.2 Other spacetimes containing snapping strings 329

10 Strong-field effects of zero-thickness strings 332
10.1 Spatial geometry outside a stationary loop 334
10.2 Black-hole formation from a collapsing loop 340
10.3 Properties of the near gravitational field of a cosmic string 343
10.4 A 3 + 1 split of the metric near a cosmic string 346
10.4.1 General formalism 346
10.4.2 Some sample near-field expansions 349
10.4.3 Series solutions of the near-field vacuum Einstein equations 352
10.4.4 Distributional stress–energy of the world sheet 355

Bibliography 359
Index