Abstract
We investigate the classical stability of supersymmetric, asymptotically flat, microstate geometries with five non-compact dimensions. Such geometries admit an “evanescent ergosurface”: a timelike hypersurface of infinite redshift. On such a surface, there are null geodesics with zero energy relative to infinity. These geodesics are stably trapped in the potential well near the ergosurface. We present a heuristic argument indicating that this feature is likely to lead to a nonlinear instability of these solutions. We argue that the precursor of such an instability can be seen in the behaviour of linear perturbations: nonlinear stability would require that all linear perturbations decay sufficiently rapidly but the stable trapping implies that some linear perturbation decay very slowly. We study this in detail for the most symmetric microstate geometries. By constructing quasinormal modes of these geometries we show that generic linear perturbations decay slower than any inverse power of time.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
J.M. Maldacena and L. Maoz, Desingularization by rotation, JHEP 12 (2002) 055 [hep-th/0012025] [INSPIRE].
V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
O. Lunin, Adding momentum to D1-D5 system, JHEP 04 (2004) 054 [hep-th/0404006] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates, Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, Nucl. Phys. B 710 (2005) 425 [hep-th/0406103] [INSPIRE].
S. Giusto and S.D. Mathur, Geometry of D1-D5-P bound states, Nucl. Phys. B 729 (2005) 203 [hep-th/0409067] [INSPIRE].
I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [INSPIRE].
P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].
G.W. Gibbons and N.P. Warner, Global structure of five-dimensional fuzzballs, Class. Quant. Grav. 31 (2014) 025016 [arXiv:1305.0957] [INSPIRE].
V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [INSPIRE].
V. Cardoso, Ó.J.C. Dias, J.L. Hovdebo and R.C. Myers, Instability of non-supersymmetric smooth geometries, Phys. Rev. D 73 (2006) 064031 [hep-th/0512277] [INSPIRE].
J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
H. Elvang, R. Emparan, D. Mateos and H.S. Reall, A supersymmetric black ring, Phys. Rev. Lett. 93 (2004) 211302 [hep-th/0407065] [INSPIRE].
D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton U.S.A. (1993) [INSPIRE].
M. Dafermos and G. Holzegel, Dynamic instability of solitons in 4+1 dimensional gravity with negative cosmological constant, http://www.dpmms.cam.ac.uk/~md384/ADSinstability.pdf (2006).
P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].
V. Cardoso, Ó.J.C. Dias and R.C. Myers, On the gravitational stability of D1-D5-P black holes, Phys. Rev. D 76 (2007) 105015 [arXiv:0707.3406] [INSPIRE].
J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes, Anal. Part. Diff. Eq. 8 (2015) 1379 [arXiv:1311.2477] [INSPIRE].
G. Holzegel and J. Smulevici, Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes, arXiv:1303.5944 [INSPIRE].
J. Keir, Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars, Class. Quant. Grav. 33 (2016) 135009 [arXiv:1404.7036] [INSPIRE].
F. John, Blow-up for quasi-linear wave equations in three space dimensions, Commun. Pure Appl. Math. 34 (1981) 29.
S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear systems of partial differential equations in applied mathematics. Part 1, American Mathematical Society, Providence U.S.A., Lect. Appl. Math. 23 (1986) 293.
H. Lindblad and I. Rodnianski, The global stability of the Minkowski space-time in harmonic gauge, math.AP/0411109 [INSPIRE].
H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math. 130 (2008) 115 [math.AP/0511461].
Ó.J.C. Dias, G.T. Horowitz, D. Marolf and J.E. Santos, On the nonlinear stability of asymptotically anti-de Sitter solutions, Class. Quant. Grav. 29 (2012) 235019 [arXiv:1208.5772] [INSPIRE].
G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221 [arXiv:0811.1033] [INSPIRE].
O. Gannot, Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis, Commun. Math. Phys. 330 (2014) 771 [arXiv:1212.1907] [INSPIRE].
V. Cardoso, L.C.B. Crispino, C.F.B. Macedo, H. Okawa and P. Pani, Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects, Phys. Rev. D 90 (2014) 044069 [arXiv:1406.5510] [INSPIRE].
J. Keir, Wave propagation on microstate geometries, arXiv:1609.01733 [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortschr. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
S. Aretakis, Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I, Commun. Math. Phys. 307 (2011) 17 [arXiv:1110.2007] [INSPIRE].
S. Aretakis, Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II, Annales Henri Poincaré 12 (2011) 1491 [arXiv:1110.2009] [INSPIRE].
H.K. Kunduri and J. Lucietti, Supersymmetric black holes with lens-space topology, Phys. Rev. Lett. 113 (2014) 211101 [arXiv:1408.6083] [INSPIRE].
S. Tomizawa and M. Nozawa, Supersymmetric black lenses in five dimensions, Phys. Rev. D 94 (2016) 044037 [arXiv:1606.06643] [INSPIRE].
H.K. Kunduri and J. Lucietti, Black hole non-uniqueness via spacetime topology in five dimensions, JHEP 10 (2014) 082 [arXiv:1407.8002] [INSPIRE].
S. Giusto, L. Martucci, M. Petrini and R. Russo, 6D microstate geometries from 10D structures, Nucl. Phys. B 876 (2013) 509 [arXiv:1306.1745] [INSPIRE].
I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, arXiv:1607.03908 [INSPIRE].
G.W. Gibbons, D. Kastor, L.A.J. London, P.K. Townsend and J.H. Traschen, Supersymmetric selfgravitating solitons, Nucl. Phys. B 416 (1994) 850 [hep-th/9310118] [INSPIRE].
B.E. Niehoff and H.S. Reall, Evanescent ergosurfaces and ambipolar hyperkähler metrics, JHEP 04 (2016) 130 [arXiv:1601.01898] [INSPIRE].
J.B. Gutowski, D. Martelli and H.S. Reall, All supersymmetric solutions of minimal supergravity in six-dimensions, Class. Quant. Grav. 20 (2003) 5049 [hep-th/0306235] [INSPIRE].
R.M. Wald, General relativity, University of Chicago Press, Chicago U.S.A. (1984) [INSPIRE].
M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in Proceedings of the XVI International Congress on Mathematical Physics, P. Exner ed., World Scientific, London U.K. (2009), pp. 421–433 [arXiv:0910.4957] [INSPIRE].
M. Dafermos, G. Holzegel and I. Rodnianski, A scattering theory construction of dynamical vacuum black holes, arXiv:1306.5364 [INSPIRE].
M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Commun. Pure Appl. Math. 62 (2009) 859 [gr-qc/0512119] [INSPIRE].
M. Dafermos and I. Rodnianski, Lectures on black holes and linear waves, Clay Math. Proc. 17 (2013) 97 [arXiv:0811.0354] [INSPIRE].
S. Aretakis, Horizon instability of extremal black holes, Adv. Theor. Math. Phys. 19 (2015) 507 [arXiv:1206.6598] [INSPIRE].
J. Lucietti and H.S. Reall, Gravitational instability of an extreme Kerr black hole, Phys. Rev. D 86 (2012) 104030 [arXiv:1208.1437] [INSPIRE].
M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| ≪ M or axisymmetry, arXiv:1010.5132 [INSPIRE].
B. Chakrabarty, D. Turton and A. Virmani, Holographic description of non-supersymmetric orbifolded D1-D5-P solutions, JHEP 11 (2015) 063 [arXiv:1508.01231] [INSPIRE].
V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30 (1984) 295 [INSPIRE].
H. Yang et al., Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation, Phys. Rev. D 86 (2012) 104006 [arXiv:1207.4253] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, United States Department of Commerce, National Bureau of Standards (1964).
S. Giusto, O. Lunin, S.D. Mathur and D. Turton, D1-D5-P microstates at the cap, JHEP 02 (2013) 050 [arXiv:1211.0306] [INSPIRE].
V. Cardoso, Ó.J.C. Dias, G.S. Hartnett, L. Lehner and J.E. Santos, Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS, JHEP 04 (2014) 183 [arXiv:1312.5323] [INSPIRE].
O. Lunin and S.D. Mathur, The slowly rotating near extremal D1-D5 system as a ‘hot tube’, Nucl. Phys. B 615 (2001) 285 [hep-th/0107113] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1607.06828
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Eperon, F.C., Reall, H.S. & Santos, J.E. Instability of supersymmetric microstate geometries. J. High Energ. Phys. 2016, 31 (2016). https://doi.org/10.1007/JHEP10(2016)031
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2016)031