Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system
Authors:
Håkan Andréasson, Markus Kunze and Gerhard Rein
Journal:
Quart. Appl. Math. 68 (2010), 17-42
MSC (2000):
Primary 35Q75; Secondary 83C75, 85A05
DOI:
https://doi.org/10.1090/S0033-569X-09-01165-9
Published electronically:
October 15, 2009
MathSciNet review:
2598878
Full-text PDF Free Access
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Abstract: We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general context, and we include arguments which show that the spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain a black hole in the sense of suitable mathematical definitions of these concepts which are available in the literature.
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References
- H. Andréasson, The Einstein-Vlasov System/Kinetic Theory, Living Rev. Relativity 5 (2002). MR 1966539 (2004c:83009)
- H. Andréasson, On global existence for the spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates, Indiana Univ. Math. J. 56, 523–552 (2007). MR 2317537 (2008d:83016)
- H. Andréasson, On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system, Commun. Math. Phys. 274, 409–425 (2007) MR 2322910 (2008h:83048)
- H. Andréasson, Sharp bounds on $2m/r$ of general spherically symmetric static objects, J. Differential Equations 245, 2243–2266 (2008). MR 2446191
- H. Andréasson, M. Kunze, G. Rein, Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter, Comm. Partial Differential Eqns. 33, 656–668 (2008). MR 2424372
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- H. Andréasson, G. Rein, A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system, Classical Quantum Gravity 23, 3659–3677 (2006). MR 2235427 (2007d:83073)
- H. Andréasson, G. Rein, On the steady states of the spherically symmetric Einstein-Vlasov system, Classical Quantum Gravity 24, 1809–1832 (2007). MR 2310358 (2008b:83054)
- C. Bardos, P. Degond, Global existence for the Vlasov-Poisson system in 3 space variables with small initial data, Ann. Inst. Henri Poincaré, Analyse non linéaire 2, 101–118 (1985). MR 794002 (86k:35129)
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- J. Binney, S. Tremaine, Galactic Dynamics, Princeton University Press, 1987.
- Y. Choquet-Bruhat, Problème de Cauchy pour le système intégro différentiel d’Einstein-Liouville, Ann. Inst. Fourier 21, 181–201 (1971). MR 0337248 (49:2018)
- D. Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Comm. Math. Phys. 93, 171–195 (1984). MR 742192 (86a:83058)
- D. Christodoulou, The problem of a self-gravitating scalar field, Comm. Math. Phys. 105, 337–361 (1986). MR 848643 (87i:83009)
- D. Christodoulou, A mathematical theory of gravitational collapse, Comm. Math. Phys. 109, 613–647 (1987). MR 885564 (88f:83005b)
- D. Christodoulou, The formation of black holes and singularities in spherically symmetric gravitational collapse, Comm. Pure Appl. Math. 44, 339–373 (1991). MR 1090436 (92d:83044)
- D. Christodoulou, Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Comm. Pure Appl. Math. 46, 1131–1220 (1993). MR 1225895 (95b:35221)
- D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. (2) 140, 607-653 (1994). MR 1307898 (95j:83100)
- D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. (2) 149, 183-217 (1999). MR 1680551 (2000a:83086)
- D. Christodoulou, On the global initial value problem and the issue of singularities, Classical Quantum Gravity 16, A23–A35 (1999). MR 1728432 (2001a:83010)
- M. Dafermos, Spherically symmetric spacetimes with a trapped surface, Classical Quantum Gravity 22, 2221–2232 (2005). MR 2145241 (2006a:83034)
- M. Dafermos, A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyperbolic Differential Equations 3, 589–598 (2006). MR 2289606 (2008f:83019)
- M. Dafermos, A. D. Rendall, An extension principle for the Einstein-Vlasov system in spherical symmetry, Ann. Henri Poincaré 6, 1137–1155 (2005). MR 2189379 (2007c:83011)
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- A. Einstein, Die Feldgleichungen der Gravitation, Sitzungsber. der Preuss. Akad. der Wissenschaften zu Berlin; 1915, 844–847 (1915).
- R. T. Glassey, W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal. 92, 59–90 (1986). MR 816621 (87j:82064)
- R. T. Glassey, W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Commun. Math. Phys. 113, 191–208 (1987). MR 919231 (88k:76034)
- C. Gundlach, Critical phenomena in gravitational collapse, Living Rev. Relativity 2 (1999). MR 1728882 (2001a:83061)
- S. Hawking, G. F. R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press 1975. MR 0424186 (54:12154)
- P.-L. Lions, B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105, 415–430 (1991). MR 1115549 (92e:35160)
- I. Olabarrieta, M. W. Choptuik, Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry, Phys. Rev. D. 65, 024007 (2002). MR 1892138 (2003a:83048)
- J. R. Oppenheimer, H. Snyder, On continued gravitational contraction, Phys. Rev. 56, 455–459 (1939).
- R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57–59 (1965). MR 0172678 (30:2897)
- K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95, 281–303 (1992). MR 1165424 (93d:35170)
- E. Poisson, A Relativist’s Toolkit. The Mathematics of Black-hole Mechanics, Cambridge University Press, 2004. MR 2063287 (2005d:83002)
- G. Rein, Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Camb. Phil. Soc. 115, 559–570 (1994). MR 1269939 (95b:83054)
- G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitationsschrift, München, 1995.
- G. Rein, Collisionless Kinetic Equations from Astrophysics—The Vlasov-Poisson System, Handbook of Differential Equations, Evolutionary Equations. Vol. 3. Eds. C. M. Dafermos and E. Feireisl, Elsevier (2007).
- G. Rein, A. D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys. 150, 561–583 (1992). Erratum: Comm. Math. Phys. 176, 475–478 (1996). MR 1204320 (94c:83028)
- G. Rein, A. D. Rendall, The Newtonian limit of the spherically symmetric Vlasov-Einstein system, Comm. Math. Phys. 150, 585–591 (1992). MR 1204321 (94c:83029)
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- G. Rein, A. D. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000). MR 1735298 (2001k:85001)
- G. Rein, A. D. Rendall, J. Schaeffer, A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system, Comm. Math. Phys. 168, 467–478 (1995). MR 1328249 (96g:58187)
- G. Rein, A. D. Rendall, J. Schaeffer, Critical collapse of collisionless matter—a numerical investigation, Phys. Rev. D 58, 044007 (1998).
- A. D. Rendall, Cosmic censorship and the Vlasov equation, Classical Quantum Gravity 9, L99–L104 (1992). MR 1178792 (93e:83052)
- A. D. Rendall, The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system. Comm. Math. Phys. 163, 89–112 (1994). MR 1277935 (95e:83026)
- A. D. Rendall, An introduction to the Einstein-Vlasov system, Banach Center Publications 41, 35–68 (1997). MR 1466508 (98k:83007)
- A. D. Rendall, Theorems on existence and global dynamics for the Einstein equations, Living Rev. Relativity 6 (2005). MR 1932418 (2003h:83016)
- K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie, Sitzungsber. der Preuss. Akad. der Wissenschaften zu Berlin; 1916, 189–196 (1916).
- S. Shapiro, S. Teukolsky, Formation of naked singularities: The violation of cosmic censorship, Phys. Rev. Letters 66, 994–997 (1991). MR 1095978 (91k:83058)
- R. Wald, General Relativity, Chicago University Press, 1984. MR 757180 (86a:83001)
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Additional Information
Håkan Andréasson
Affiliation:
Mathematical Sciences, Chalmers University of Technology, Göteborg University, S-41296 Göteborg, Sweden
Email:
hand@math.chalmers.se
Markus Kunze
Affiliation:
Fachbereich Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany
Email:
markus.kunze@uni-due.de
Gerhard Rein
Affiliation:
Mathematisches Institut der Universität Bayreuth, D-95440 Bayreuth, Germany
Email:
gerhard.rein@uni-bayreuth.de
Keywords:
General relativity,
Einstein-Vlasov system,
gravitational collapse,
black holes
Received by editor(s):
December 9, 2008
Published electronically:
October 15, 2009
Additional Notes:
Support of the first author by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged.
Dedicated:
Dedicated to Prof. W. A. Strauss on the occasion of his 70th birthday
Article copyright:
© Copyright 2009
Brown University
