Stable black holes: in vacuum and beyond
HTML articles powered by AMS MathViewer
- by Elena Giorgi HTML | PDF
- Bull. Amer. Math. Soc. 60 (2023), 1-27 Request permission
Abstract:
Black holes are important objects in our understanding of the universe, as they represent the extreme nature of General Relativity. The mathematics behind them has surprising geometric properties, and their dynamics is governed by hyperbolic partial differential equations. A basic question one may ask is whether these solutions to the Einstein equation are stable under small perturbations, which is a typical requirement to be physically meaningful. We illustrate the main conjectures regarding the stability problem of known black hole solutions and present some recent theorems regarding the fully nonlinear evolution of black holes in the case of vacuum and their interaction with matter fields.References
- B. P. Abbott and et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016), no. 6, 061102, 16. Authors include B. C. Barish, K. S. Thorne and R. Weiss. MR 3707758, DOI 10.1103/PhysRevLett.116.061102
- B. P. Abbott et al., GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119 (2017), 161101.
- Lars Andersson and Pieter Blue, Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math. (2) 182 (2015), no. 3, 787–853. MR 3418531, DOI 10.4007/annals.2015.182.3.1
- L. Andersson, B. Thomas, P. Blue, and S. Ma, Stability for linearized gravity on the Kerr spacetime, arXiv:1903.03859 (2019), 1–99.
- Xinliang An and Jonathan Luk, Trapped surfaces in vacuum arising dynamically from mild incoming radiation, Adv. Theor. Math. Phys. 21 (2017), no. 1, 1–120. MR 3636691, DOI 10.4310/ATMP.2017.v21.n1.a1
- James M. Bardeen and William H. Press, Radiation fields in the Schwarzschild background, J. Mathematical Phys. 14 (1973), 7–19. MR 321475, DOI 10.1063/1.1666175
- Lydia Bieri, An extension of the stability theorem of the Minkowski space in general relativity, J. Differential Geom. 86 (2010), no. 1, 17–70. MR 2772545
- P. Blue and A. Soffer, Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates, Adv. Differential Equations 8 (2003), no. 5, 595–614. MR 1972492
- G. Bozzola and V. Paschalidis, General relativistic simulations of the quasicircular inspiral and merger of charged black holes: GW150914 and fundamental physics implications, Phys. Rev. Lett. 126 (2021), no. 4, Paper No. 041103.
- Gabriele Bozzola and Vasileios Paschalidis, Numerical-relativity simulations of the quasicircular inspiral and merger of nonspinning, charged black holes: methods and comparison with approximate approaches, Phys. Rev. D 104 (2021), no. 4, Paper No. 044004, 26. MR 4318789, DOI 10.1103/physrevd.104.044004
- Subrahmanyan Chandrasekhar, The mathematical theory of black holes, International Series of Monographs on Physics, vol. 69, The Clarendon Press, Oxford University Press, New York, 1983. Oxford Science Publications. MR 700826
- 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
- Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, The Astrophysical Journal 875 (2019).
- M. Dafermos, H. Gustav, R. Igor, and T. Martin, The non-linear stability of the Schwarzschild family of black holes, arXiv:2104.08222 (2021), 1–513.
- Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski, Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: The case $|a|\ll M$, Ann. PDE 5 (2019), no. 1, Paper No. 2, 118. MR 3919495, DOI 10.1007/s40818-018-0058-8
- Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math. 222 (2019), no. 1, 1–214. MR 3941803, DOI 10.4310/ACTA.2019.v222.n1.a1
- Alexandru D. Ionescu and Sergiu Klainerman, Rigidity results in general relativity: a review, Surveys in differential geometry 2015. One hundred years of general relativity, Surv. Differ. Geom., vol. 20, Int. Press, Boston, MA, 2015, pp. 123–156. MR 3467366, DOI 10.4310/SDG.2015.v20.n1.a6
- Mihalis Dafermos and Igor Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math. 62 (2009), no. 7, 859–919. MR 2527808, DOI 10.1002/cpa.20281
- M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 421–432.
- Mihalis Dafermos and Igor Rodnianski, Lectures on black holes and linear waves, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 97–205. MR 3098640
- Mihalis Dafermos, Igor Rodnianski, and Yakov Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a|<M$, Ann. of Math. (2) 183 (2016), no. 3, 787–913. MR 3488738, DOI 10.4007/annals.2016.183.3.2
- Albert Einstein, On a stationary system with spherical symmetry consisting of many gravitating masses, Ann. of Math. (2) 40 (1939), 922–936. MR 363, DOI 10.2307/1968902
- Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), 141–225 (French). MR 53338, DOI 10.1007/BF02392131
- Elena Giorgi, Boundedness and decay for the Teukolsky equation of spin $\pm 1$ on Reissner-Nordström spacetime: the $\ell =1$ spherical mode, Classical Quantum Gravity 36 (2019), no. 20, 205001, 48. MR 4016187, DOI 10.1088/1361-6382/ab3c03
- Elena Giorgi, Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$, Ann. Henri Poincaré 21 (2020), no. 8, 2485–2580. MR 4127375, DOI 10.1007/s00023-020-00923-3
- Elena Giorgi, The linear stability of Reissner-Nordström spacetime for small charge, Ann. PDE 6 (2020), no. 2, Paper No. 8, 145. MR 4121607, DOI 10.1007/s40818-020-00082-y
- Elena Giorgi, The linear stability of Reissner-Nordström spacetime: the full subextremal range $|Q|<M$, Comm. Math. Phys. 380 (2020), no. 3, 1313–1360. MR 4179729, DOI 10.1007/s00220-020-03893-z
- Elena Giorgi, The Carter tensor and the physical-space analysis in perturbations of Kerr-Newman spacetime, arXiv:2105.14379 (2021), 1–53.
- Elena Giorgi, Electromagnetic-gravitational perturbations of Kerr-Newman spacetime: the Teukolsky and Regge-Wheeler equations, J. Hyperbolic Differ. Equ. 19 (2022), no. 1, 1–139. MR 4405807, DOI 10.1142/S0219891622500011
- E. Giorgi, S. Klainerman, and J. Szeftel, A general formalism for the stability of Kerr, arXiv:2002.02740 (2020), 1–139.
- E. Giorgi, S. Klainerman. and J. Szeftel, Wave equation estimates and the nonlinear stability of slowly rotating Kerr black holes, arXiv:2205.14808 (2022), 1–135.
- Dietrich Häfner, Peter Hintz, and András Vasy, Linear stability of slowly rotating Kerr black holes, Invent. Math. 223 (2021), no. 3, 1227–1406. MR 4213773, DOI 10.1007/s00222-020-01002-4
- Pei-Ken Hung, The Linear Stability of the Schwarzschild Spacetime in the Harmonic Gauge: Odd Part, ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–Columbia University. MR 3818833
- Pei-Ken Hung, Jordan Keller, and Mu-Tao Wang, Linear stability of Schwarzschild spacetime: decay of metric coefficients, J. Differential Geom. 116 (2020), no. 3, 481–541. MR 4182895, DOI 10.4310/jdg/1606964416
- Thomas William Johnson, The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge, Ann. PDE 5 (2019), no. 2, Paper No. 13, 92. MR 4015163, DOI 10.1007/s40818-019-0069-0
- Bernard S. Kay and Robert M. Wald, Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation $2$-sphere, Classical Quantum Gravity 4 (1987), no. 4, 893–898. MR 895907, DOI 10.1088/0264-9381/4/4/022
- Roy P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963), 237–238. MR 156674, DOI 10.1103/PhysRevLett.11.237
- Sergiu Klainerman, Mathematical challenges of general relativity, Rend. Mat. Appl. (7) 27 (2007), no. 2, 105–122. MR 2361024
- S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
- S. Klainerman and J. Szeftel, Constructions of GCM spheres in perturbations of Kerr, arXiv:1911.00697 (2019), 1–135.
- S. Klainerman and J. Szeftel, Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr, arXiv:1912.12195 (2019), 1–76.
- Sergiu Klainerman and Jérémie Szeftel, Global nonlinear stability of Schwarzschild spacetime under polarized perturbations, Annals of Mathematics Studies, vol. 210, Princeton University Press, Princeton, NJ, 2020. MR 4298717, DOI 10.2307/j.ctv15r57cw
- S. Klainerman and J. Szeftel, Kerr stability for small angular momentum, arXiv:2104.11857 (2021), 1–801.
- Sergiu Klainerman, Jonathan Luk, and Igor Rodnianski, A fully anisotropic mechanism for formation of trapped surfaces in vacuum, Invent. Math. 198 (2014), no. 1, 1–26. MR 3260856, DOI 10.1007/s00222-013-0496-6
- Hans Lindblad and Igor Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2) 171 (2010), no. 3, 1401–1477. MR 2680391, DOI 10.4007/annals.2010.171.1401
- Siyuan Ma, Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: Linearized gravity, Comm. Math. Phys. 377 (2020), no. 3, 2489–2551. MR 4121625, DOI 10.1007/s00220-020-03777-2
- V. Moncrief, Stability of a Reissner-Nordstrom black holes, Phys. Rev. D 10 (1974), 1057.
- A. Nathanail, E. R. Most, and L. Rezzolla, Gravitational collapse to a Kerr-Newman black hole, Monthly Notices of the Royal Astronomical Society 469 (2017), no. 1, 31–35.
- E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence, Metric of a rotating, charged mass, J. Mathematical Phys. 6 (1965), 918–919. MR 178947, DOI 10.1063/1.1704351
- Frans Pretorius, Evolution of binary black-hole spacetimes, Phys. Rev. Lett. 95 (2005), no. 12, 121101, 4. MR 2169088, DOI 10.1103/PhysRevLett.95.121101
- Tullio Regge and John A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. (2) 108 (1957), 1063–1069. MR 91832, DOI 10.1103/PhysRev.108.1063
- K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitz. Deut. Akad. Wiss. Berlin, Kl. Math.-Phys. Tech. (1916), 189–196.
- D. Shen, Construction of GCM hypersurfaces in perturbations of Kerr, arXiv:2205.12336 (2022), 1–106.
- Yakov Shlapentokh-Rothman, Quantitative mode stability for the wave equation on the Kerr spacetime, Ann. Henri Poincaré 16 (2015), no. 1, 289–345. MR 3296646, DOI 10.1007/s00023-014-0315-7
- Y. Shlapentokh-Rothman and R. Teixeira da Costa, Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|< m$: frequency space analysis, arXiv:2007.07211 (2020), 1–125.
- Daniel Tataru and Mihai Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN 2 (2011), 248–292. MR 2764864, DOI 10.1093/imrn/rnq069
- S. Teukolsky, Perturbations of a rotating black hole. I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations, The Astrophysical Journal 185 (1973), 635–647.
- Bernard F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30 (1989), no. 6, 1301–1305. MR 995773, DOI 10.1063/1.528308
Additional Information
- Elena Giorgi
- Affiliation: Department of Mathematics, Columbia University
- MR Author ID: 1328804
- ORCID: 0000-0003-1675-8468
- Email: elena.giorgi@columbia.edu
- Received by editor(s): July 15, 2022
- Published electronically: September 7, 2022
- Additional Notes: The author acknowledges the support of NSF Grant No. DMS-2128386. This work was supported by a grant from the Simons Foundation (825870, EG)
- © Copyright 2022 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 60 (2023), 1-27
- MSC (2020): Primary 83C05, 83C22, 83C57, 83C50, 35A01
- DOI: https://doi.org/10.1090/bull/1781
- MathSciNet review: 4520774