Hamiltonian Field Theory in the Radiating Regime

Lecture Notes in Physics: Monographs 70......Page 1
Hamiltonian Field Theory in the Radiating Regime......Page 3
Contents......Page 5
1. Introduction......Page 7
2.1 Hamiltonian dynamics......Page 13
2.2 The role of boundary conditions in Hamiltonian field theory......Page 16
2.3 Tangential translations as a Hamiltonian system......Page 19
2.4 The Hamiltonian description of a mixed Cauchy - characteristic initial value problem......Page 21
2.5 The Trautman-Bondi energy for the scalar field......Page 26
3.1 The framework......Page 29
3.2 Hamiltonian dynamics for Lagrangian theories......Page 35
3.3 Space-time integrals......Page 40
3.4 Changes of Psi and of the Lagrangian......Page 44
4.1 Preliminaries......Page 47
4.2 Energy: convergence of integrals......Page 51
4.3 The phase space P_(-infinity,0]......Page 55
4.4 The phase space P_[-1,0]......Page 57
4.5 The phase space ^P_[-1,0]......Page 58
4.6 The preferred Hamiltonian role of the Trautman-Bondi energy for scalar fields......Page 60
4.7 The Poincaré group......Page 63
4.8 \"Supertranslated\" hyperbolae......Page 67
5.1 Preliminaries......Page 71
5.2 Moving spacelike hypersurfaces......Page 76
5.3 Cosmological space-times......Page 80
5.4 Space-times asymptotically flat in spacelike directions......Page 82
5.5 Space-times with anti-de Sitter asymptotic behaviour......Page 86
5.6 Energy in the radiation regime: convergence of integrals......Page 95
5.7 Phase spaces: the space P......Page 103
5.8 The phase space P_[-1,0]......Page 108
5.9 The phase space ^P_[-1,0]......Page 109
5.10 Preferred role of the Trautman-Bondi energy......Page 110
6.1 The Poincaré group: convergence of integrals......Page 111
6.2 Supertranslations (and space translations): convergence of integrals......Page 117
6.3 The abstract Seri......Page 122
6.4 Lorentz charges......Page 125
6.5 A Hamiltonian definition of angular momentum of sections of I......Page 127
6.6 An example: Schwarzschild space-time......Page 132
6.7 An example: stationary space-times......Page 135
6.8 Lorentz covariance of global charges......Page 137
6.9 BMS invariance of energy-momentum......Page 140
6.10 Polyhomogeneous Scri\'s......Page 142
A. Odd forms (densities)......Page 145
B. Solutions of the wave equation smoothly extendable to I^+......Page 147
C.1.1 Smooth Scri\'s......Page 149
C.1.2 Polyhomogeneous asymptotics......Page 151
C.2 Solutions of the vacuum Einstein equations containing hyperboloidal hypersurfaces......Page 152
C.3 Bondi coordinates vs hyperboloidal initial data......Page 158
C.4 The calculation of the W^(x mu)\'s......Page 163
C.5 Transformation rules of the Bondi functions under supertranslations......Page 165
C.6 Transformation rules of the Bondi functions under boosts......Page 168
C.7 Bondi coordinates in the Kerr space-time......Page 169
C.8 Conformal rescalings of ADM Cauchy data......Page 172
References......Page 173