Edited by Ray d'Inverno
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:December 2009
Online ISBN:9780511524639
Hardback ISBN:9780521439763
Paperback ISBN:9780521017350
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511524639.009
Subjects: Cosmology, Relativity and Gravitation, Astrophysics
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Abstract. Much of physics concerns temporal dynamics, which describes a spatial world (or Cauchy surface) evolving in time. In Relativity, the causal structure suggests that null dynamics is more relevant. This article sketches Lagrangian and Hamiltonian formalisms for dual-null dynamics, which describes the evolution of initial data prescribed on two intersecting null surfaces. The application to the Einstein gravitational field yields variables with recognisable geometrical meaning, initial data which divide naturally into gravitational and coordinate parts, and evolution equations which are covariant on the intersection surface and free of constraints.
INTRODUCTION
The ADM or “3+1” formalism [1,2] is a natural approach to the Cauchy problem in General Relativity, and has been used widely both analytically and numerically. By comparison, null (or characteristic) evolution problems are more appropriate to the study of problems involving radiation, whether gravitational or otherwise, since radiation propagates in null directions. Null surfaces also have a central place in the causal structure of General Relativity which spatial surfaces do not.
A distinction should be drawn between the null-cone problem discussed elsewhere in this volume, in which the initial surface is a null cone, and the dual-null problem, in which there are two intersecting null initial surfaces. The latter problem was originally described by Sachs [3], with existence and uniqueness proofs being given by Müller zum Hagen and Seifert [4], Friedrich [5] and Rendall [6], and a general “2+2” formalism being developed by d'Inverno, Smallwood and Stachel [7–9].
pp. i-vi
pp. vii-ix
pp. x-xii
pp. xiii-xiv
pp. xv-xx
PART A - THEORETICAL APPROACHES : Read PDF
pp. 1-2
Numerical relativity on a transputer array : Read PDF
pp. 3-19
Some aspects of the characteristic initial value problem in numerical relativity : Read PDF
pp. 20-33
The characteristic initial value problem in general relativity : Read PDF
pp. 34-40
Algebraic approach to the characteristic initial value problem in general relativity : Read PDF
pp. 41-49
On hyperboloidal hypersurfaces : Read PDF
pp. 50-58
The initial value problem on null cones : Read PDF
pp. 59-68
Introduction to dual-null dynamics : Read PDF
pp. 69-78
On colliding plane wave space-times : Read PDF
pp. 79-82
Boundary conditions for the momentum constraint : Read PDF
pp. 83-93
On the choice of matter model in general relativity : Read PDF
pp. 94-102
A mathematical approach to numerical relativity : Read PDF
pp. 103-113
Making sense of the effects of rotation in general relativity : Read PDF
pp. 114-129
Stability of charged boson stars and catastrophe theory : Read PDF
pp. 130-140
PART B - PRACTICAL APPROACHES : Read PDF
pp. 141-142
Numerical asymptotics : Read PDF
pp. 143-162
Instabilities in rapidly rotating polytropes : Read PDF
pp. 163-181
Gravitational radiation from coalescing binary neutron stars : Read PDF
pp. 182-201
“Critical” behaviour in massless scalar field collapse : Read PDF
pp. 202-222
Godunov-type methods applied to general relativistic stellar collapse : Read PDF
pp. 223-229
Astrophysical sources of gravitational waves and neutrinos : Read PDF
pp. 230-246
Gravitational radiation from 3D gravitational stellar core collapse : Read PDF
pp. 247-257
A vacuum fully relativistic 3D numerical code : Read PDF
pp. 258-264
Solution of elliptic equations in numerical relativity using multiquadrics : Read PDF
pp. 265-280
Self-gravitating thin discs around rotating black holes : Read PDF
pp. 281-291
An ADI scheme for a black hole problem : Read PDF
pp. 292-296
Time-symmetric ADI and causal reconnection : Read PDF
pp. 297-307
The numerical study of topological defects : Read PDF
pp. 308-334
Computations of bubble growth during the cosmological quark-hadron transition : Read PDF
pp. 335-344
Initial data of axisymmetric gravitational waves with a cosmological constant : Read PDF
pp. 345-352
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pp. 355-378