Abstract
Recently, a general analysis has been given of the stability with respect to axisymmetric perturbations of stationary-axisymmetric black holes and black branes in vacuum general relativity in arbitrary dimensions. It was shown that positivity of canonical energy on an appropriate space of perturbations is necessary and sufficient for stability. However, the notions of both “stability” and “instability” in this result are significantly weaker than one would like to obtain. In particular, if there exists a perturbation with negative canonical energy, “instability” has been shown to occur only in the sense that this perturbation cannot asymptotically approach a stationary perturbation at late times. In this paper, we prove that if a perturbation of the form \({\pounds_t \delta g}\)—with \({\delta g}\) a solution to the linearized Einstein equation—has negative canonical energy, then that perturbation must, in fact, grow exponentially in time. The key idea is to make use of the t- or (t-ϕ)-reflection isometry, i, of the background spacetime and decompose the initial data for perturbations into their odd and even parts under i. We then write the canonical energy as \({\mathscr{E} = \mathscr{K} + \mathscr{U}}\), where \({\mathscr{K}}\) and \({\mathscr{U}}\), respectively, denote the canonical energy of the odd part (“kinetic energy”) and even part (“potential energy”). One of the main results of this paper is the proof that \({\mathscr{K}}\) is positive definite for any black hole background. We use \({\mathscr{K}}\) to construct a Hilbert space \({\mathscr{H}}\) on which time evolution is given in terms of a self-adjoint operator \(\tilde{\mathcal{A}}\), whose spectrum includes negative values if and only if \({\mathscr{U}}\) fails to be positive. Negative spectrum of \(\tilde{\mathcal{A}}\) implies exponential growth of the perturbations in \({\mathscr{H}}\) that have nontrivial projection into the negative spectral subspace. This includes all perturbations of the form \({\pounds_t \delta g}\) with negative canonical energy. A “Rayleigh-Ritz” type of variational principle is derived, which can be used to obtain lower bounds on the rate of exponential growth.
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References
Christodoulou D., Klainerman S.: The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, New Jersey (1993)
Wald, R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20(6), 1056–1058 (1979). Erratum. J. Math. Phys. 21(1), 218–218 (1980)
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Clay Math. Proc. 17, 97–205. arXiv:0811.0354
Wald R.M.: On the instability of the n = 1 Einstein Yang-Mills black holes and mathematically related systems. J. Math. Phys. 33, 248–255 (1992)
Kay B.S., Wald R.M.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation two-sphere. Class. Quant. Grav. 4, 893–898 (1987)
Finster, F., Kamran, N.,Smoller, J., Yau, S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264(2), 465–503 (2006). arXiv:gr-qc/0504047
Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009). arXiv:0908.2265
Tataru, D.: Local decay of waves on asymptotically flat stationary space-times. Am. J. Math. 135(2), 361–401 (2013). arXiv:0910.5290
Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a| < M (2014). arXiv:1402.7034
Regge T., Wheeler J.A.: Stability of a Schwarzschild singularity. Phys. Rev. 108, 1063–1069 (1957)
Zerilli F.J.: Effective potential for even parity Regge-Wheeler gravitational perturbation equations. Phys. Rev. Lett. 24, 737–738 (1970)
Ishibashi, A., Kodama, H.: Stability of higher dimensional Schwarzschild black holes. Prog. Theor. Phys. 110, 901–919 (2003). arXiv:hep-th/0305185
Hollands, S., Wald, R.M.: Stability of black holes and black branes. Commun. Math. Phys. 321, 629–680 (2013). arXiv:1201.0463
Figueras, P., Murata, K., Reall, H.S.: Black hole instabilities and local Penrose inequalities. Class. Quant. Grav. 28, 225030 (2011). arXiv:1107.5785
Chrusciel, P.T., Wald, R.M.: Maximal hypersurfaces in stationary asymptotically flat spacetimes. Commun. Math. Phys. 163(3), 561–604 (1994). arXiv:gr-qc/9304009
Schiffrin, J.S., Wald, R.M.: Reflection symmetry in higher dimensional black hole spacetimes. Class. Quant. Grav. 32(10), 105005 (2015). arXiv:1501.02752
Sorkin R.: Kaluza-Klein monopole. Phys. Rev. Lett. 51, 87–90 (1983)
Gross D.J., Perry M.J.: Magnetic monopoles in Kaluza-Klein theories. Nucl. Phys. B226, 29–48 (1983)
Wald R.M.: General relativity. The University of Chicago Press, Chicago (1984)
Laval G., Mercier C., Pellat R.: Necessity of the energy principles for magnetostatic stability. Nucl. Fusion 5(2), 156 (1965)
Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2009)
Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111102 (2005). arXiv:gr-qc/0506013
Galloway, G.J., Schoen, R.: A Generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys. 266, 571–576 (2006). arXiv:gr-qc/0509107
Chrusciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mem. Soc. Math. France 94, 1–103 (2003). arXiv:gr-qc/0301073
Cantor M., Brill D.: The Laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compositio Math. 43(3), 317–330 (1981)
Bray, H.L., Lee, D.A.: On the Riemannian Penrose inequality in dimensions less than 8. Duke Math. J. 148(1), 81–106 (2009). arXiv:0705.1128
Carter B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971)
Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, London-New York (1973)
Riesz F., Nagy B.S.: Functional Analysis. Courier Dover Publications, New York (1990)
Reed M., Simon B.: Functional Analysis. Methods of Modern Mathematical Physics. Elsevier Science, Amsterdam (1981)
Seifert, M.D., Wald, R.M.: General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories. Phys. Rev. D 75, 084029 (2007). arXiv:gr-qc/0612121
Chandrasekhar S.: Dynamical instability of Gaseous masses approaching the Schwarzschild limit in general relativity. Phys. Rev. Lett. 12, 114–116 (1964)
Corvino J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)
Iyer, V., Wald, R.M.: Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D50, 846–864 (1994). arXiv:gr-qc/9403028v1
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Communicated by P. T. Chrusciel
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Prabhu, K., Wald, R.M. Black Hole Instabilities and Exponential Growth. Commun. Math. Phys. 340, 253–290 (2015). https://doi.org/10.1007/s00220-015-2446-1
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DOI: https://doi.org/10.1007/s00220-015-2446-1