ANZIAM J. 49 (2007), no. 1, pp. 53–73.

Initial value formalism for Lemaitre–Tolman–Bondi collapse

P. D. Lasky† A. W. C. Lun R. B. Burston
Centre for Stellar and Planetary Astrophysics
School of Mathematical Sciences
Monash University
Wellington Rd
Melbourne 3800
Australia
Paul.Lasky@sci.monash.edu.au		 Centre for Stellar and Planetary Astrophysics
School of Mathematical Sciences
Monash University
Wellington Rd
Melbourne 3800
Australia		 Max Planck Institute for Solar System Research
37191 Katlenburg-Lindau
Germany
Received January 10, 2007

Abstract

Formulating a dust-filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the matter distribution throughout the spacetime. Furthermore, the metric describes both inhomogeneous dust regions and also vacuum regions in a single coordinate patch, thus alleviating the need for complicated matching schemes at the interfaces. In this way, the system is established as an initial boundary value problem, which has many benefits for its numerical evolution. We show the dust part of the metric is equivalent to the class of Lemaitre–Tolman–Bondi (LTB) metrics under a coordinate transformation. In this coordinate system, shell crossing singularities (SCS) are exhibited as fluid shock waves, and we therefore discuss possibilities for the dynamical extension of shell crossings through the initial point of formation by borrowing methods from classical fluid dynamics. This paper fills a void in the present literature associated with these collapse models by fully developing the formalism in great detail. Furthermore, the applications provide examples of the benefits of the present model.

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2000 Mathematics Subject Classification: primary 83C05, 83C75; secondary 83C57
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2378149 Z'blatt-MATH: pre05243891
†indicates author for correspondence

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