Master equations and stability of Einstein-Maxwell-scalar black holes
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Springer
Received: September 28, 2019 Accepted: November 14, 2019 Published: December 4, 2019
Aron Jansen,a Andrzej Rostworowskib and Mieszko Rutkowskib a
Departement de F´ısica Quantica i Astrof´ısica, Institut de Ci´encies del Cosmos, Universitat de Barcelona, Mart´ı i Franqu´es 1, E-08028 Barcelona, Spain b M. Smoluchowski Institute of Physics, Jagiellonian University, Lojasiewicza 11, 30-348 Krakow, Poland
E-mail: [email protected], [email protected], [email protected] Abstract: We derive master equations for linear perturbations in Einstein-Maxwell scalar theory, for any spacetime dimension D and any background with a maximally symmetric n = (D − 2)-dimensional spatial component. This is done by expressing all fluctuations analytically in terms of several master scalars. The resulting master equations are KleinGordon equations, with non-derivative couplings given by a potential matrix of size 3, 2 and 1 for the scalar, vector and tensor sectors respectively. Furthermore, these potential matrices turn out to be symmetric, and positivity of the eigenvalues is sufficient (though not necessary) for linear stability of the background under consideration. In general these equations cannot be fully decoupled, only in specific cases such as Reissner-Nordstr¨ om, where we reproduce the Kodama-Ishibashi master equations. Finally we use this to prove stability in the vector sector of the GMGHS black hole and of Einstein-scalar theories in general. Keywords: AdS-CFT Correspondence, Black Holes, Classical Theories of Gravity ArXiv ePrint: 1909.04049
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)036
JHEP12(2019)036
Master equations and stability of Einstein-Maxwell-scalar black holes
Contents 1 Introduction
2
2 Setup
3 5 5 7 9 10 10 11 13 14
4 Stability 4.1 S-deformation 4.2 Stability of tensor perturbations 4.3 Stability of vector perturbations in Einstein-scalar theory 4.4 Stability of vector perturbations of the GMGHS black hole
15 16 17 17 18
5 Discussion
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A Spherical case
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B Special cases B.1 Spherical l = 1 in vector sector B.2 Spherical l = 0 and l = 1 in scalar sector B.3 Planar k = 0
22 23 23 25
C Scalar sector master scalars
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D Transformations D.1 Regge-Wheeler gauge invariants for the scalar sector D.2 Transformation between Fefferman-Graham and Eddington-Finkelstein coordinates
27 27
E Quasinormal modes of AdS planar black holes
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–1–
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3 Master equations 3.1 Sectors 3.2 Gauge invariant fluctuations 3.3 Master equations 3.3.1 Tensor sector 3.3.2 Vector sector 3.3.3 Scalar sector 3.4 Full decoupling 3.5 Comparison to Kodama-Ishibashi
1
Introduction
1
More precisely, in higher dimensions, D > 4, the master scalars in vector and tensor sectors come in a number of copies corresponding to different polarizations of gravitational waves in these sectors.
–2–
JHEP12(2019)036
General Relativity admits a wide variety of Black Hole solutions, especially when coupled to matter. For any su
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