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Oppenheimer-Snyder type collapse for a collisionless gas

Håkan Andréasson, Gerhard Rein

Abstract

In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf.~ \cite{OS}. In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein-Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlevé-Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework.

Oppenheimer-Snyder type collapse for a collisionless gas

Abstract

In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf.~ \cite{OS}. In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein-Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlevé-Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework.

Paper Structure

This paper contains 37 sections, 23 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. The Einstein-Vlasov system
  3. The Einstein-Vlasov system in general coordinates
  4. The system in Painlevé-Gullstrand coordinates
  5. The local existence result
  6. A suitable reduced system
  7. Initial data
  8. The solution concept and the local existence result
  9. Recovering the full system
  10. Proof of the local existence result
  11. The Vlasov equation and the characteristic system
  12. The iteration scheme---the set-up
  13. The $0\,$th iterates
  14. Induction assumption on the $(n-1)\,$st iterates
  15. $f_n$ and its characteristics
  16. ...and 22 more sections

Key Result

Theorem 1.1

For regular initial data which approximate Oppenheimer-Snyder data in a suitable way the corresponding solution to the Einstein-Vlasov system approximates the Oppenheimer-Snyder dust solution arbitrarily well. In particular, it forms a trapped surface before it can form a spacetime singularity so th

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Remark 3.7
  • ...and 37 more