Weak null singularities in general relativity
HTML articles powered by AMS MathViewer
- by Jonathan Luk
- J. Amer. Math. Soc. 31 (2018), 1-63
- DOI: https://doi.org/10.1090/jams/888
- Published electronically: September 27, 2017
- HTML | PDF | Request permission
Abstract:
We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk and Rodnianski, and conjecturally they are present in the interior of generic black holes arising from gravitational collapse.References
- A. Bonanno, S. Droz, W. Israel, and S. M. Morsink, Structure of the charged spherical black hole interior, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940, 553–567. MR 1356176, DOI 10.1098/rspa.1995.0100
- Patrick R. Brady and John D. Smith, Black hole singularities: a numerical approach, Phys. Rev. Lett. 75 (1995), no. 7, 1256–1259. MR 1343439, DOI 10.1103/PhysRevLett.75.1256
- Lior M. Burko, Structure of the black hole’s Cauchy-horizon singularity, Phys. Rev. Lett. 79 (1997), no. 25, 4958–4961. MR 1487881, DOI 10.1103/PhysRevLett.79.4958
- S. Chandrasekhar and J. B. Hartle, On crossing the Cauchy horizon of a Reissner-Nordström black-hole, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 301–315. MR 684313, DOI 10.1098/rspa.1982.0160
- Demetrios Christodoulou, The formation of black holes in general relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009. MR 2488976, DOI 10.4171/068
- Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR 1316662
- Mihalis Dafermos, Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations, Ann. of Math. (2) 158 (2003), no. 3, 875–928. MR 2031855, DOI 10.4007/annals.2003.158.875
- Mihalis Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math. 58 (2005), no. 4, 445–504. MR 2119866, DOI 10.1002/cpa.20071
- Mihalis Dafermos, Black holes without spacelike singularities, Comm. Math. Phys. 332 (2014), no. 2, 729–757. MR 3257661, DOI 10.1007/s00220-014-2063-4
- Mihalis Dafermos and Igor Rodnianski, A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math. 162 (2005), no. 2, 381–457. MR 2199010, DOI 10.1007/s00222-005-0450-3
- M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity (T. Damour et al., Eds.), World Scientific, Singapore, 2011, pp. 132–189.
- William A. Hiscock, Evolution of the interior of a charged black hole, Phys. Lett. A 83 (1981), no. 3, 110–112. MR 617171, DOI 10.1016/0375-9601(81)90508-9
- S. Hod and T. Piran, Mass inflation in dynamical gravitational collapse of a charged scalar field, Phys. Rev. Lett. 81 (1998), 1554–1557.
- K. A. Khan and R. Penrose, Scattering of two impulsive gravitational plane waves, Nature 229 (1971), 185–186.
- Sergiu Klainerman and Francesco Nicolò, The evolution problem in general relativity, Progress in Mathematical Physics, vol. 25, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946854, DOI 10.1007/978-1-4612-2084-8
- Sergiu Klainerman and Igor Rodnianski, On the formation of trapped surfaces, Acta Math. 208 (2012), no. 2, 211–333. MR 2931382, DOI 10.1007/s11511-012-0077-3
- Jonathan Luk, On the local existence for the characteristic initial value problem in general relativity, Int. Math. Res. Not. IMRN 20 (2012), 4625–4678. MR 2989616, DOI 10.1093/imrn/rnr201
- J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Cambridge J. Math., to appear.
- Jonathan Luk and Igor Rodnianski, Local propagation of impulsive gravitational waves, Comm. Pure Appl. Math. 68 (2015), no. 4, 511–624. MR 3318018, DOI 10.1002/cpa.21531
- J. M. McNamara, Instability of black hole inner horizons, Proc. Roy. Soc. London Ser. A 358 (1978), no. 1695, 499–517. MR 489678, DOI 10.1098/rspa.1978.0024
- H. Müller zum Hagen, Characteristic initial value problem for hyperbolic systems of second order differential equations, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 2, 159–216 (English, with French summary). MR 1079777
- Amos Ori and Éanna É. Flanagan, How generic are null spacetime singularities?, Phys. Rev. D (3) 53 (1996), no. 4, R1754–R1758. MR 1380002, DOI 10.1103/PhysRevD.53.R1754
- Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965), 57–59. MR 172678, DOI 10.1103/PhysRevLett.14.57
- Roger Penrose, The geometry of impulsive gravitational waves, General relativity (papers in honour of J. L. Synge), Clarendon Press, Oxford, 1972, pp. 101–115. MR 0503490
- E. Poisson and W. Israel, Inner-horizon instability and mass inflation in black holes, Phys. Rev. Lett. 63 (1989), no. 16, 1663–1666. MR 1018317, DOI 10.1103/PhysRevLett.63.1663
- Eric Poisson and Werner Israel, Internal structure of black holes, Phys. Rev. D (3) 41 (1990), no. 6, 1796–1809. MR 1048877, DOI 10.1103/PhysRevD.41.1796
- M. Simpson and R. Penrose, Internal instability in a Reissner-Nordström black hole, Internat. J. Theoret. Phys. 7 (1973), 183–197.
- P. Szekeres, Colliding gravitational waves, Nature 228 (1970), 1183–1184.
Bibliographic Information
- Jonathan Luk
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
- MR Author ID: 916843
- Email: jluk@stanford.edu
- Received by editor(s): February 23, 2014
- Received by editor(s) in revised form: May 27, 2015
- Published electronically: September 27, 2017
- Additional Notes: This work is supported by the NSF Postdoctoral Fellowship DMS-1204493 and the NSF grant DMS-1709458.
- © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc. 31 (2018), 1-63
- MSC (2010): Primary 83C75, 35L67
- DOI: https://doi.org/10.1090/jams/888
- MathSciNet review: 3718450