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Initial value formalism for Lemaitre-Tolman-Bondi collapse

Published online by Cambridge University Press:  17 February 2009

P. D. Lasky ,
A. W. C. Lun  and
R. B. Burston
P. D. Lasky
Affiliation:
Centre for Stellar and Planetary Astrophysics School of Mathematical Sciences Monash UniversityWellington Rd Melbourne 3800 Australia; [email protected]
A. W. C. Lun
Affiliation:
Centre for Stellar and Planetary Astrophysics School of Mathematical Sciences Monash UniversityWellington Rd Melbourne 3800 Australia; [email protected]
R. B. Burston
Affiliation:
2Max Planck Institute for Solar System Research37191 Katlenburg-Lindau Germany
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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.

Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 1 , July 2007 , pp. 53 - 73
Copyright
Copyright © Australian Mathematical Society 2007

References

reference

[1] Adler, R. J., Bjorken, J. D., P. Chen and J. S. Liu, “Simple analytic models of gravitational collapse” Am J Phys 73 (12) (2005) 1148–59, arXiv:gr-qc/0502040.CrossRefGoogle Scholar
[2] Arnowitt, R., Deser, S. and C. W. Misner, “The dynamics of general relativity, in Gravitation: An introduction to Current Research (ed. Witten), , (John Wiley and Sons, Inc., New York., 1962).Google Scholar
[3] Bondi, H., “Spherically symmetrical models in general relativity” Mon Not Roy Astron Soc 107 (1947)410–25.CrossRefGoogle Scholar
[4] Christodoulou, D., “Violation of cosmic censorship in the gravitational collapse of a dust cloud”, Commun Math Phys 93 (1984) 171–95.CrossRefGoogle Scholar
[5] Ellis, G. F. R., “Relativistic cosmology”, in General Relativity and Cosmology (ed. B. K. Sachs), , (Academic Press, New York, 1971) 104–82.Google Scholar
[6] Gautreau, R. and Cohen, J. M., “Gravitational collapse in a single coordinate system”, Am J Phys 63(1995)991–9.CrossRefGoogle Scholar
[7] Gullstrand, A., “Allegemeine losung des statischen einkurper-problems in der Einsteinchen gravitations theorie”, Arkiv Mat Astron Fys 16 (1922) 1–15.Google Scholar
[8] Lasky, P. D. and Lun, A. W. C., “Generalized Lemaitre-Tolman-Bondi solutions with pressure”, Phys Rev D 74 (2006) 084013.CrossRefGoogle Scholar
[9] Lasky, P. D. and Lun, A. W. C., “Spherically symmetric collapse of general fluids”, Phys Rev D 75 (2007)024031.CrossRefGoogle Scholar
[10] Lax, P. D. and Wendroff, B., “Systems of conservation laws”, Commun Pure Appl Math 13 (1960) 217–37.CrossRefGoogle Scholar
[11] Lemaitre, G., “L'univers en expansion”, Ann Soc Sci BruxellesA 53 (1933) 51.Google Scholar
[12] Misner, C. W. and Sharp, D. H., “Relativistic equations for adrbatic, spherically symmetric gravitational collapse”, Phys Rev 136 (2) (1964) B571–6.CrossRefGoogle Scholar
[13] Misner, C. W., Thome, K. S. and J. A. Wheeler, Gravitation (Freeman, New York, 1973).Google Scholar
[14] zum Hagen, H. Miiller, Yodzis, P. and H. J. Seifert, “On the occurrence of naked singularities in general relativity II”, Commun Math Phys 37 (1974) 29–40.CrossRefGoogle Scholar
[15] Newman, R. P. A. C., “Strengths of naked singularities in Tolman-Bondi-spacetimes”, Class Quantum Grav 3 (1986) 527–539.CrossRefGoogle Scholar
[16] Nolan, B. C. and Mena, F. C., “Geometry and topology of singularities in spherical dust collapse”, Class Quantum Grav 19 (2002) 2587–605.CrossRefGoogle Scholar
[17] Oppenheimer, J. R. and Snyder, H., “On continued gravitational contraction”, Phys Rev 56 (1939) 455–9.CrossRefGoogle Scholar
[18] Painleve, P., “La mecanique classique et la theorie de la relativite”, C R Acad Sci 173 (1921) 677–80.Google Scholar
[19] Smoller, J., Shock Waves and Reaction-Diffusion Equations (Springer-Verlag New York Inc., New York, 1983).CrossRefGoogle Scholar
[20] Szekeres, P. and Lun, A., “What is a shell crossing singularity?”, J Austral Math Soc Sen B 41 (1999) 167–79.CrossRefGoogle Scholar
[21] Tolman, R. C., “Effect of inhomogeneity on cosmological models”, Proc Nat Acad Sci USA 20 (1934) 169–76.CrossRefGoogle Scholar
[22] Wald, R., General Relativity (U. of Chicago Press, Chicago, 1984).CrossRefGoogle Scholar
[23] Yodzis, P., Seifert, H. J. and H. Miiller zum Hagen, “On the occurrence of naked singularities in general relativity”, Commun Math Phys 34 (1973) 135–48.CrossRefGoogle Scholar