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Stability of Higher-Dimensional Schwarzschild Black Holes

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Akihiro Ishibashi (Univ. of Cambridge), Hideo Kodama (Kyoto Univ.)

High Energy Physics - Theory · Submitted 20 May 2003 · 34 references

Pith's verdict: A clean, checkable completion of the linear stability proof for Schwarzschild-Tangherlini: vector and scalar modes now covered, with explicit factorizations a reader can verify by hand.

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arXiv IDhep-th/0305185 (version 3)
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CitationAkihiro Ishibashi (Univ. of Cambridge), Hideo Kodama (Kyoto Univ.). "Stability of Higher-Dimensional Schwarzschild Black Holes." arXiv:hep-th/0305185 (2003). https://pith.science/paper/SLXXAKQE
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The paper

The paper claims

Higher-dimensional Schwarzschild black holes do not fall apart: every type of linear metric perturbation reduces to a positive operator with no unstable bound state.

The reading

A black hole is only a useful endpoint of gravitational collapse if it survives being slightly disturbed. For the four-dimensional Schwarzschild solution this was settled decades ago, but in more than four dimensions only one of the three possible families of metric perturbations had been analysed. This paper closes the gap: reducing the linearised Einstein equations, in a gauge-invariant way, to a single Schrödinger-type equation for each mode, the authors show that the spatial operator in every case is positive and self-adjoint on the space of square-integrable functions, so no normalisable runaway solution exists. They also show that no static scalar deformation of the higher-dimensional Schwarzschild geometry is regular everywhere outside the horizon while falling off at infinity, which supports the uniqueness of the spherically symmetric static vacuum black hole within perturbation theory. The same operator argument covers Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter backgrounds for the tensor and vector modes.

What carries the argument

The load-bearing device is the factorisation of the spatial operator, $A = -d^2/dr_*^2 + V = (d/dr_* + S)^*(d/dr_* + S) + V + f(dS/dr) - S^2$, for a suitably chosen function $S(r)$. On functions of compact support the boundary term in the inner product vanishes, so positivity of the remainder $V + f S' - S^2$ gives positivity of $A$, and the Friedrichs extension then carries that positivity to the self-adjoint operator that generates the time evolution. A second device handles the anti-de Sitter case, where $r_*$ ranges over $(-l, 0)$ rather than the whole line: the potentials are shown to be of limit-point type at $r_* = 0$ for $n >= 3$, so the self-adjoint extension is unique without extra boundary data, while in four dimensions a Dirichlet condition is imposed by hand.

Extended reading notes

Core claim

The claim is that the $(n+2)$-dimensional Schwarzschild–Tangherlini metric is linearly stable against all three tensorial types of perturbation — tensor, vector and scalar — where the last two are the higher-dimensional counterparts of the axial and polar modes. The master equation for each type has the form $[ -d^2/dr_*^2 + V ]$ acting on a mode function, and stability follows if that operator is positive and self-adjoint on $L^2(r_*, dr_*)$. The obstacle is that for $n > 2$ the potentials $V_V$ and $V_S$ are not positive: the vector potential dips below zero near the horizon for sufficiently large $n$, and the scalar potential is not positive definite in eight or more spacetime dimensions. The authors get around this by writing the operator as a sum of a perfect square and a remainder, $A = (d/dr_* + S)^*(d/dr_* + S) + V + f S' - S^2$, choosing $S$ so that the remainder is manifestly non-negative; for vectors $S = nf/(2r)$ leaves $(f/r^2)[k_V^2 - (n-1)K]$, and for scalars $S = f h'/h$ with $h = r^{n/2+ l -1}[(l-1)(l+n) + n(n+1)x/2]$ leaves the explicitly positive expression of Eqs. (2.34)–(2.35). Separately, the static scalar sector is solved in closed form with hypergeometric functions, and a positivity argument on the transformed potential $V = [n(n-2) + 4l(l+n-1)(1-x) + 2nx(1-x) + x^2]/(4r^2)$ shows that any solution falling off at infinity must blow up at the horizon.

Load-bearing premise

The proof takes the master equations and their potentials from the authors' own earlier papers rather than re-deriving them, so the positivity check is only as strong as the claim that this reduction captures every physical degree of freedom of the linearised Einstein equations.

Editorial extensions

If this is right

  • The Schwarzschild–Tangherlini solution, the standard higher-dimensional generalisation of Schwarzschild, can be treated as a stable equilibrium candidate for classical evolution, closing the gap left by earlier work that covered only tensor modes.
  • Within the perturbation framework, no asymptotically flat static vacuum black hole with spherical horizon topology can be non-spherical: the static scalar analysis rules out such deformations, matching the uniqueness expectations for this class.
  • All maximally symmetric black holes with non-vanishing cosmological constant — including Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter — are stable against tensor and vector perturbations, for any sign of spatial curvature $K$.
  • For black holes whose horizon is a general Einstein manifold rather than a sphere, the same master equations apply with the sphere eigenvalues replaced, giving explicit sufficient conditions on the Lichnerowicz eigenvalue, $\lambda_L >= (3n-2)K$ and $\lambda_L >= 2(n-1)K$, for stability against tensor modes.
  • Completing the stability programme for non-asymptotically-flat maximally symmetric black holes now requires only the scalar sector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The anti-de Sitter conclusion is conditional on a convention: the authors impose $\Phi = 0$ at $r_* = 0$ and explicitly defer other boundary conditions to later work, so a different admissible choice could in principle change which modes are declared stable.
  • The stability statement is proved for initial data of compact support away from the horizon; the authors argue that the four-dimensional Kay–Wald argument should carry over dimension by dimension, but that extension is asserted rather than demonstrated here.
  • Because the factorisation trick only needs a single function $S$ making the remainder positive, the same recipe is likely to apply to any static background whose perturbation potential is bounded below — a route to stability proofs for charged and rotating higher-dimensional black holes that the paper does not attempt.
  • The negative dip in the vector potential for large $n$ is a real feature, not an artefact of the factorisation; it means the spectrum still has a region of small or negative local potential, so the interesting question shifts from 'is there a negative mode' to 'how close does the spectrum come to zero', which is what quasinormal-mode calculations would probe.
Original abstract

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. 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, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,948 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. 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, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. contribution 0.95 0.87 0.05 unclear 0.59
s1 The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. method 0.94 0.72 0.03 asserted 0.99
s2 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, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. result 1.00 0.96 0.75 asserted 1.00
s3 On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. result 0.96 0.96 0.07 asserted 1.00
s4 This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. interpretation 0.99 0.87 0.09 asserted 0.89
s5 Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. future work 0.61 0.55 0.05 asserted 0.75
s6 We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations. result 0.98 0.93 0.39 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
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  • references and citing sentences
  • relations to other papers
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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The potential barriers VT surrounding (n + 2)-dimensional Schwarzschild black holes for tensor perturbations of the mode l = 2. Here, rh denotes the horizon radius. 0 1 2 3 4 r/rh VT K=1, l=2 λ=0.05 n=6 n=5 n=4 n=3 n=2
Figure 2. The potential barriers VT surrounding (n + 2)-dimensional Schwarzschild-de Sitter black holes for tensor perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh
Figure 3. The potential barriers VT surrounding (n+2)-dimensional Schwarzschild-anti-de Sitter black holes for tensor perturbations of the mode l = 2.The horizontal axis is normalised with respect to the black hole horizon radius rh. 2.2. Vector perturbations Next, consider vector perturbations, which are given by δgab = 0 , δgai = rfaVi , δgij = − 2 kV r 2HT Dˆ (iVj) , (2.13) where the vector harmonics are introduced as the solutions of (△ˆ n + k 2 V )Vi = 0 , Dˆ iV i = 0 , (2.14) with eigenvalues k 2 V that are given by k 2 V = l(l + n − 1) − 1 with l ≥ 1 for K = 1 and form
Figure 4. The potential barriers VV surrounding (n + 2)-dimensional Schwarzschild black holes for vector perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh
Figure 5. The potential barriers VV surrounding (n + 2)-dimensional Schwarzschild-de Sitter black holes for vector perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh. Now, to show the positivity of A, let us introduce the differential operator D˜ ≡ d dr∗ + S , (2.18) with S being some function of r. Then, we have (Φ, AΦ)L2 = − h Φ ∗DΦ˜ i boundary + Z dr∗  |DΦ˜ | 2 + V˜ |Φ| 2  , (2.19) where V˜ ≡ V + f dS dr − S 2 . (2.20) For Φ ∈ C∞ 0 (r∗), the boundary term vanishes. Inspecting the form of (2.17), we fin
Figure 6. The potential barriers VV surrounding (n+2)-dimensional Schwarzschild-anti-de Sitter black holes for vector perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh. Here, we note that for K = 1, k 2 V −(n−1)K = (l−1)(l+n) > 0. Hence, a symmetric operator A with domain C∞ 0 (r∗) is positive definite for any K and λ (as long as the Schwarzschild wedge exits), so is the self-adjoint extension. For the case λ < 0, the range of r∗ is incomplete. However, for the higher dimensional case (n ≥ 3), V = VV is shown to be in the limi
Figure 7. The potential barriers VS surrounding (n + 2)-dimensional Schwarzschild black holes for scalar perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh. In 8-dimensional spacetimes, VS is not positive definite. the potential VS is not positive definite in general, as shown in
Figure 8. The potential barriers VS surrounding (n + 2)-dimensional Schwarzschild-de Sitter black holes for scalar perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh. In 8-dimensional spacetimes, VS is not positive definite. §3. Static perturbations In this section, we treat “static” scalar perturbations in the background of the Schwarzschild metric, with K = 1, λ = 0 and n ≥ 2. Because scalar perturbations represent deviations from a spherical form, if there exist any regular static pertur￾bations that fall off at large…
Figure 9. The potential barriers VS surrounding (n+2)-dimensional Schwarzschild-anti-de Sitter black holes for scalar perturbations of the mode l = 2. The horizontal axis is normalised with respect to the black hole horizon radius rh. When the number of spacetime dimensions is greater than 4, VS is not positive definite. + C2 rh r l+n F  n − 1 + l n − 1 , − 2(n − 1) + l n − 1 , − 2(n + l − 1) n − 1 ; x  . (3.2) From this expression, we find that the original gauge-invariant variables F and F a b behave at the spatial infinity as pC1r l+qC2/rn+l−1 , where p and q are non-va

Figures reproduced from arXiv:hep-th/0305185 by the authors.

References
How later papers used it

103 later papers cite it

Evidence: substantial. 99 of the citing papers share no author with it, and 32 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 219 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 6 of 30 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

91 arXiv papers cite this in 188 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 111 sentences without a confident label

Builds on it 2 papers
  • The wave equation can be solved analytically, but for general modes inside the wall – and for tensor modes outside – one may also show stability by extending the argument of [ref] and keeping track of the new boundary terms at $r_D$ that arise when integrating by parts.

    On the stability of gravity with Dirichlet walls · Minkowski space · 0.99

  • Furthermore, the gauge-invariant and coordinate independent formalism for perturbations of spherically symmetric spectimes developed in the 70's by Gerlach and Sengupta in [ref] , has been recently extended to higher-dimensional spacetimes with a maximally symmetric subspace in Refs. [ref] , for the study of perturbations in brane-world models.

    Extraction of Gravitational Waves in Numerical Relativity · Gravitational Waves in the Cauchy-Perturbative Approach · 0.87

Uses its method or tool 25 papers
  • Using the positivity of gravitational potentials and $S$-deformed technique [ref] allows us to prove the stability of $f(R)$-AdS black hole if the scalaron mass squared ($m_A^2$) satisfies the Breitenlohner-Freedman (BF) bound ($m_A^2≥m^2_BF=-9/4ℓ^2$) [ref] .

    Stability analysis of f(R)-AdS black holes · Introduction · 0.80

  • Defining the differential operator [ref] [equation] where $S$ is a function of $r$ such that $S(1)=0$, we get from the expression ([ref]) [equation] with $Ṽ$ given by [equation] Thus for a field such that its effective potential $V$ is not strictly positive, if we find a new potential $Ṽ$ such that $Ṽ ≥0$, then we show the stability of the corresponding de Sitter QNM.

    On the quasinormal modes of the de Sitter spacetime · Stability of the de Sitter QNM · 0.89

  • To show that the operators $A_±$ of the formula ([ref]) can be extended to positive self-adjoint operators we use the $S$-deformation method [ref] – [ref] .

    Classical stability of black holes under massless Dirac perturbations · Stability analysis · 1.00

Compares with it 5 papers
  • In contrast, it is known that for asymptotically de Sitter spherically symmetric black holes, their classical stability depends on the spacetime dimension in a non trivial way [ref] - [ref] .

    On the quasinormal modes of the de Sitter spacetime · Stability of the de Sitter QNM · 0.98

  • Collecting the above results, we conclude that (with respect to a choice of Dirichlet boundary conditions at infinity) the massless topological black hole is stable in four dimensions, in agreement with the analysis in [ref] .

    Stability of Topological Black Holes · Stability of the Massless Black Hole · 0.98

  • This should be compared with the Ishibashi-Kodama modified potential $Ṽ$ [ref] which differs from ours by a multiplicative factor $f$ (similarly to the potentials $V_…$ entering the master equations).

    On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics · Vanishing using the bottom of the spectrum · 1.00

Cites it as support 1 paper
  • Nevertheless, it can be shown that the operator appearing on the LHS of the Schrödinger equation is positive, hence $ω^2>0$ and there is no instability [ref] .

    Black Holes in Higher Dimensions · Stability of the static black hole · 0.99

Cites it as background 25 papers
  • For the scalar (even) modes, reconsidered in [ref] , the problem is far more subtle, because the behaviour of the Zerilli potential $V_z$ [ref] at $x=0$ (which corresponds to the $r=0$ Schwarzschild singularity) implies a one parameter ambiguity [ref] in boundary conditions at this point (parameterized by $θ∈S^1$, see equation ([ref]) below), and also because $V_z$ has a second order pole at $r=r_s:=-6M/(ℓ-1)(ℓ+2)$ which falls within the domain of interest for negative $M$.

    The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity · Introduction · 0.91

  • In [ref] the authors formulated the correlated stability conjecture (CSC): A black brane thermodynamic (in)stability correlates with its dynamical (in)stability The conjecture is applicable only to horizons with translational invariance, simple higher-dimensional Schwarzschild black holes in Einstein gravity with a cosmological constant are stable [ref] .

    Stabilization of the extended horizons · Introduction · 0.84

  • The exact BH solution in the Einstein-Gauss-Bonnet gravity is known as the Boulware-Deser solution [ref] and one can study perturbations of it due to external bosonic fields [ref] .

    Tidal Response of Compact Objects · Black holes in Lovelock gravity · 0.81

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