A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions

By H. Kodama, A. Ishibashi. 2003

421 citations recorded by the source corpus.

We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single second-order wave equation in a two-dimensional static spacetime, irrespective of the mode of perturbations. Our starting point is the gauge-invariant formalism for perturbations in an arbitrary number of dimensions developed by the present authors, and the variable for the final second-order master equation is given by a simple combination of gauge-invariant variables in this formalism. Our formulation applies to the case of non-vanishing as well as vanishing cosmological constant Λ. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background with Λ = 0 and K = 1, the master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with Λ< 0 and K =0 , 1, our equation coincides with those recently derived by Cardoso and Lemos. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Λ. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimension. Although in the present paper we treat only the case in which the horizon geometry is maximally symmetric, the final master equations are valid even when the horizon geometry is described by a generic Einstein manifold, if we employ an appropriate reinterpretation of the curvature K and the eigenvalues for harmonic tensors.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/cfe73564086d2234863fd6bf404c7bd3d6b1e81a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. V. Cardoso
2. Paolo Pani
3. Jorge E Santos
4. R. Brito
5. Ó. J. Dias
6. Benson Way
7. V. Hubeny
8. M. Rangamani

In Nuclear and High Energy Physics

1. V. Cardoso
2. Paolo Pani
3. Jorge E Santos
4. R. Brito
5. Ó. J. Dias
6. Benson Way
7. V. Hubeny
8. R. Myers

Citations in the corpus

• Black hole stereotyping: induced gravito-static polarization (2011, background): “We do that for several reasons (i) this case is simpler (ii) the gravitational case will turn out to be similar and (iii) this case actually describes the master field for tensor mode perturbations [19], namely perturbations of the spatial metric in d > 4.”
• The large D limit of General Relativity (2013, positive): “Our analysis also provides the solution for several types of linearized gravitational perturbations of this black hole [45]: perturbations that are tensors on SD−2 are governed by the same equation as (6.”
• Causal hydrodynamics of gauge theory plasmas from AdS/CFT duality (2007, background): “As a result, our notations and theirs are related as follows: Kodama and Ishibashi Ours λ −1/R2 M rp+10 /2R 2 f(r) f(r)h(r) dσ2p d~xp 2/R2 k R q 7 where k and q represent wave numbers in their and our notations, respectively.”
• Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS (2013, methodology): “The only exceptions to the above are the trivial perturbations, the “` = 0” and “` = 1” modes, which shift, respectively, the mass and angular momentum of the solution along the original Kerr-AdS family, and to which the Teukolsky formalism is blind [46, 27, 47, 30].”
• The phase transition between caged black holes and black strings (2004, result): “These results were first obtained by [52] by a somewhat different method.”
• 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&quot; [80]. 2003In a series of papers [81,82,83], Kodama and Ishibashi extend the Regge-Wheeler-Zerilli formalism to higher dimensions. 20”
• Sound waves in strongly coupled non-conformal gauge theory plasma (2005, background): “(Such an approach has long been used in cosmology [41], [42] and in studying black hole fluctuations [43].”
• On the nonlinear stability of asymptotically anti-de Sitter solutions (2012, methodology): “The linear differential map h (j) ab = h (j) ab ( Φ ) that reconstructs the metric perturbations (in a given gauge) can be found in [15].”
• GRAVITATIONAL PERTURBATIONS OF HIGHER DIMENSIONAL ROTATING BLACK HOLES (2006, neutral): “If we set a = 0 then we have a Schwarzschild(-AdS) black hole and our tensor perturbations should form a subset of the tensor perturbations considered in [4, 5, 6].”
• DE SITTER MUSINGS (2012, background): “It has been shown in [61] that the Schwarzschildde Sitter geometry is perturbatively stable with respect to gravitational perturbations.”