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

References

  1. R. J. Adler, J. D. Bjorken, P. Chen and J. S. Liu, “Simple analytic models of gravitational collapse”, Am. J. Phys. 73 (12) (2005) 1148–59, arXiv:gr-qc/0502040.
  2. R. Arnowitt, S. Deser 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). MR143629
  3. H. Bondi, “Spherically symmetrical models in general relativity”, Mon. Not. Roy. Astron. Soc. 107 (1947) 410–25. MR26866
  4. D. Christodoulou, “Violation of cosmic censorship in the gravitational collapse of a dust cloud”, Commun. Math. Phys. 93 (1984) 171–95. MR742192
  5. G. F. R. Ellis, “Relativistic cosmology”, in General Relativity and Cosmology (ed. B. K. Sachs), (Academic Press, New York, 1971) 104–82. MR351371
  6. R. Gautreau and J. M. Cohen, “Gravitational collapse in a single coordinate system”, Am. J. Phys. 63 (1995) 991–9. MR1360538
  7. A. Gullstrand, “Allegemeine losung des statischen einkorper-problems in der Einsteinchen gravitations theorie”, Arkiv. Mat. Astron. Fys 16 (1922) 1–15.
  8. P. D. Lasky and A. W. C. Lun, “Generalized Lemaitre-Tolman-Bondi solutions with pressure”, Phys. Rev. D 74 (2006) 084013. MR2278523
  9. P. D. Lasky and A. W. C. Lun, “Spherically symmetric collapse of general fluids”, Phys. Rev. D 75 (2007) 024031. MR2302106
  10. P. D. Lax and B. Wendroff, “Systems of conservation laws”, Commun. Pure. Appl. Math. 13 (1960) 217–37. MR120774
  11. G. Lemaitre, “L'univers en expansion”, Ann. Soc. Sci. Bruxelles A 53 (1933) 51.
  12. C. W. Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse”, Phys. Rev. 136 (2) (1964) B571–6. MR177783
  13. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, New York, 1973). MR418833
  14. H. Müller zum Hȧgen, P. Yodzis and H. J. Seifert, “On the occurrence of naked singularities in general relativity II”, Commun. Math. Phys. 37 (1974) 29–40. MR345583
  15. R. P. A. C. Newman, “Strengths of naked singularities in Tolman-Bondi-spacetimes”, Class. Quantum Grav. 3 (1986) 527–539. MR851286
  16. B. C. Nolan and F. C. Mena, “Geometry and topology of singularities in spherical dust collapse”, Class. Quantum Grav. 19 (2002) 2587–605. MR1908117
  17. J. R. Oppenheimer and H. Snyder, “On continued gravitational contraction”, Phys. Rev. 56 (1939) 455–9.
  18. P. Painlevé, “La mécanique classique et la théorie de la relativité”, C. R. Acad. Sci. 173 (1921) 677–80.
  19. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag New York Inc., New York, 1983). MR688146
  20. P. Szekeres and A. Lun, “What is a shell crossing singularity?”, J. Austral. Math. Soc. Ser. B. 41 (1999) 167–79. MR1723654
  21. R. C. Tolman, “Effect of inhomogeneity on cosmological models”, Proc. Nat. Acad. Sci. USA 20 (1934) 169–76.
  22. R. Wald, General Relativity (U. of Chicago Press, Chicago, 1984). MR757180
  23. P. Yodzis, H. J. Seifert and H. Müller zum Hȧgen, “On the occurrence of naked singularities in general relativity”, Commun. Math. Phys. 34 (1973) 135–48. MR345582
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