Stability of Higher-Dimensional Schwarzschild Black Holes

By Akihiro Ishibashi, Hideo Kodama. 2003

277 citations recorded by the source corpus.

We investigate the classical stability of higher-dimensional Schwarzschild black holes with respect to linear perturbations in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes. This formalism was recently developed by the present authors. The perturbations are classified into three types, those of tensor, vector and scalar modes, according to their tensorial behaviour on the spherical section of the background metric. The vector- and scalar-type modes correspond, respectively, to the axial and polar modes in the four-dimensional case. We show that for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the L 2 -Hilbert space, and hence that the master equation for each type of perturbation has no normalisable negative modes that would correspond to unstable solutions. In the same Schwarzschild background, we also analyse static perturbations of the scalar mode and show that there exists no static perturbation that is regular everywhere outside the event horizon and is well behaved at the spatial infinity. This confirms the uniqueness of the higher-dimensional spherically symmetric, static vacuum black hole, within the perturbation framework. Our strategy for treating the stability problem is also applicable to other higherdimensional maximally symmetric black holes with a non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (including the higherdimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable with respect to tensor and vector perturbations.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/78a51cf5e99c1a3c28f46c94ece228a679bf6a4b

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In Astronomy and Astrophysics

1. Jorge E Santos
2. V. Cardoso
3. Paolo Pani
4. Ó. J. Dias
5. R. Brito
6. Roberto Emparan
7. Benson Way
8. Harvey Reall

In Mathematical Physics

1. P. Hintz
2. B. Kol
3. V. Cardoso
4. R. Schiappa
5. José Natário
6. Roberto Emparan
7. András Vasy
8. L. Hui

In Nuclear and High Energy Physics

1. Jorge E Santos
2. V. Cardoso
3. Paolo Pani
4. Ó. J. Dias
5. R. Brito
6. Roberto Emparan
7. Benson Way
8. Harvey Reall

Citations in the corpus

• Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics (2015, background): “2003 In a series of papers [86, 87, 88], Kodama and Ishibashi extend the Regge-Wheeler-Zerilli formalism to higher dimensions.”
• Static response and Love numbers of Schwarzschild black holes (2020, methodology): “As was noticed in [69, 70], in the zero-frequency limit (4.”
• Black hole stereotyping: induced gravito-static polarization (2011, background): “1 Gravitational polarization Our starting point is the master equation for gravitational perturbations of scalar type in d dimensions [20] which are the d dimensional generalization of the Zerilli equation [22] and describe perturbations associated with the Newtonian potential.”
• Superradiance (2015, background): “]. 2001Emparan and Reall provide the rst example of a stationary asymptotically at vacuum solution with an event horizon of nonspherical topology the \black ring" [80]. 2003In a series of papers [81,82,83], Kodama and Ishibashi extend the Regge-Wheeler-Zerilli formalism to higher dimensions. 20”
• The global non-linear stability of the Kerr-de Sitter family of black holes (2016, methodology): “For us, the most convenient formulation, which we use heavily in §7, is due to Ishibashi, Kodama and Seto [93], [92], [86], building on earlier work by Kodama–Sasaki [94].”
• Numerical methods for finding stationary gravitational solutions (2015, background): “[389] A.”
• GRAVITATIONAL PERTURBATIONS OF HIGHER DIMENSIONAL ROTATING BLACK HOLES (2006, background): “Demanding agreement between our results and those of [6] yields λ = l(l + 2N) + 4N + 4σ −m(2) + 2ǫ(1 + σ)m.”
• On the classification of asymptotic quasinormal frequencies for $d$-dimensional black holes and quantum gravity (2004, methodology): “The IK master equations were used in [9, 10] to study the stability of d–dimensional black holes, and although many known solutions were shown to be stable, stability of some other solutions is still an open problem.”
• Black Holes in Higher Dimensions (2008, methodology): “The formalism of [151] allows one to show that the only regular stationary perturbations of a Schwarzschild black hole lead into the MP family of black holes [162].”
• An instability of higher-dimensional rotating black holes (2010, background): “In higher dimensions, the stability of the Schwarzschild solution against linearized gravitational perturbations has been established [11, 22, 20].”