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Master Equations for Gravitational Perturbations of Static Lovelock Black Holes in Higher Dimensions

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Tomohiro Takahashi, Jiro Soda

General Relativity and Quantum Cosmology · Submitted 8 Aug 2010 · 36 references

Pith's verdict: A solid, useful derivation that completes the Lovelock perturbation set, but the scalar potential ships without the limit checks that would make it trustworthy at a glance.

T0 · AI review

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arXiv ID1008.1385 (version 3)
Pith IDEH6XGS3O, Pith's permanent record of this paper and its review
CitationJiro Soda, Tomohiro Takahashi. "Master Equations for Gravitational Perturbations of Static Lovelock Black Holes in Higher Dimensions." arXiv:1008.1385 (2010). https://pith.science/paper/EH6XGS3O
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The paper

The paper claims

The paper derives the last two missing wave equations needed to describe how a static Lovelock black hole rings, completing a set that already covered tensor-type distortions, so that all three classes of gravitational perturbation in any维度

The reading

Einstein's gravity is not the only theory whose field equations stay second order; Lovelock gravity adds higher powers of curvature while keeping the equations well behaved, and it reduces to Einstein theory in four dimensions. Static black holes exist in this theory in every dimension, and if such objects are produced in high-energy collisions or used as holographic tools, one needs to know how they respond to small gravitational disturbances. This paper derives the wave equations for the vector-type and scalar-type disturbances, complementing the tensor-type case the authors treated earlier, so that all three symmetry classes now have a master equation. Each equation reduces to a one-dimensional Schrödinger-like problem with an explicit potential built from the metric function, the Lovelock couplings, and the harmonic eigenvalue on the internal sphere.

What carries the argument

The load-bearing device is the function $T(r) = r^{n-1} partial_psi W[psi]$, defined from the polynomial $W[psi]$ whose root gives the black hole: $W[psi] = sum_{m>=2} (a_m/m) prod_{p=1}^{2m-2}(n-p) psi^m + psi - 2Lambda/(n(n+1)) = mu r^{n+1}$. Because the background Riemann tensor has only a few nonzero components, the linearized Lovelock tensor arranges itself into terms proportional to $T$, $T'$ and $T''$ multiplying the perturbed Riemann tensor, and a single identity, $[r f' + 2(kappa - f)]T = (n+1)mu$, removes the residual dependence on the couplings. That reduction is what makes one master equation per perturbation type possible at arbitrary Lovelock order, rather than a fresh calculation for each new curvature term.

Extended reading notes

Core claim

The paper's claim is completeness: for a static vacuum Lovelock black hole with metric $ds^2=-f(r)dt^2+dr^2/f(r)+r^2 dSigma_n^2$, the linearized Lovelock equations for every perturbation type collapse to a single radial wave equation. The authors first show that the linearized Lovelock tensor can be written entirely in terms of the perturbed Riemann tensor with three coefficient functions, $T(r)$, $T'(r)$ and $T''(r)$, where $T(r)=r^{n-1}W'[psi]$ packages every Lovelock coefficient into one scalar. They then evaluate this tensor for vector-type perturbations in Regge-Wheeler gauge and for scalar-type perturbations in Zerilli gauge, eliminate the auxiliary fields, and obtain master equations that become $-d^2 Psi/dr_*^2 + V(r) Psi = omega^2 Psi$ in the tortoise coordinate. The resulting potentials are given explicitly in Eqs. (4.7), (5.8) and (6.19), and the authors stress that nothing is assumed about $f(r)$ or the couplings $a_m$ beyond positivity of $T'$, so the equations are as general as the static vacuum solutions themselves.

Load-bearing premise

The derivation assumes that $T'(r)$, the derivative of the coefficient function built from the Lovelock polynomial, stays positive everywhere outside the horizon; this is what licenses the field redefinitions that turn each master equation into Schrödinger form and what is taken to rule out ghost instabilities, and the paper states it but does not prove it for general couplings or masses.

Editorial extensions

If this is right

  • Any static vacuum Lovelock black hole's linear gravitational response, in any dimension and at any Lovelock order, is computable from three explicit potentials rather than from a case-by-case perturbation analysis.
  • Quasinormal frequencies, ringdown waveforms and stability thresholds for these black holes become a one-dimensional scattering problem once $f(r)$ and the couplings are fixed.
  • Because the potentials depend on the couplings only through $T(r)$ and its derivatives, the effect of adding higher-curvature terms can be tracked by how $T$ changes, giving a compact diagnostic of how Lovelock order shifts the spectrum.
  • The framework extends directly to charged Lovelock black holes and to asymptotically anti-de Sitter cases, which the authors flag as natural next steps.
  • The tensor, vector and scalar sectors can be compared term by term against the known Einstein-Gauss-Bonnet results, providing a consistency check at second Lovelock order.

Reading between the lines

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

  • If $T'(r)$ is the quantity that controls both ghost freedom and the existence of the Schrödinger form, then a scan of $T'$ over the exterior of a given solution is effectively a cheap stability map: parameter regions where $T'$ approaches zero are where the linear analysis is most likely to signal a real instability rather than a coordinate artifact.
  • The companion claim that small-mass asymptotically flat Lovelock black holes are unstable should show up as a negative region in one of these potentials; a reader could test this by plotting $V_s$ and $V_v$ for small $mu$ and looking for a bound state, which would be an independent check of the instability result.
  • Because the derivation never uses the specific root of $W[psi]$, the same algebra should carry over to Lovelock theories with non-spherical horizon topology and to solutions with a cosmological constant, with only the harmonic eigenvalues and the asymptotic behaviour of $f$ changing.
  • A useful cross-check that the paper does not perform explicitly would be to truncate to second-order Lovelock theory and verify that the potentials reduce to the published Einstein-Gauss-Bonnet expressions; a mismatch there would point to an error in the elimination steps rather than in the general framework.
Original abstract

In this paper, we derive the master equations for gravitational perturbations of vector and scalar type for static vacuum Lovelock black holes. Together with our previous work on the tensor type perturbation, we now provide the full set of master equations that governs all types of gravitational perturbations of static vacuum Lovelock black holes in any dimensions.

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, 2,311 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In this paper, we derive the master equations for gravitational perturbations of vector and scalar type for static vacuum Lovelock black holes. Together with our previous work on the tensor type perturbation, we now provide the full set of master equations that governs all types of gravitational perturbations of static vacuum Lovelock black holes in any dimensions.

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 In this paper, we derive the master equations for gravitational perturbations of vector and scalar type for static vacuum Lovelock black holes. contribution 0.95 0.97 0.59 asserted 1.00
s1 Together with our previous work on the tensor type perturbation, we now provide the full set of master equations that governs all types of gravitational perturbations of static vacuum Lovelock black holes in any dimensions. contribution 0.81 0.87 0.69 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
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • 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

References
How later papers used it

46 later papers cite it

Evidence: substantial. 42 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 96 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 4 of 6 papers with this label
How the literature uses it

46 arXiv papers cite this in 96 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 · 68 sentences without a confident label

Uses its method or tool 6 papers
  • Based on the master equation, especially the effective potential in [ref] , Reall has provided a smart way to study the hyperbolicity and causality of the perturbation equations in Einstein-Gauss-Bonnet theory [ref] .

    Hyperbolicity and Causality of Einstein-Gauss-Bonnet Gravity in Warped Product Spacetimes · Summary and discussion · 0.94

  • The equations of motion for these dynamical degrees of freedom, $ϕ_h(r)$, the subindex $h=0, 1, 2$ indicating the corresponding helicity, can be recast as Schrödinger type equations [ref] , that in the large momentum limit they reduce to a very simple form[In the notation of [ref] , we must identify $γ_i ≡γ= q^2 L^2$, and our potentials are related to theirs, $U_i →- V_i/γ Λ$, as $γ→∞$.]

    Lovelock gravity, black holes and holography · Graviton potentials · 0.87

  • Following [ref] , we shall define a new function $T(r)$ as: [equation] [equation]

    Are eikonal quasinormal modes linked to the unstable circular null geodesics? · Gravitational perturbations of the Einstein-Lovelock black hole · 0.92

Uses its data or values 1 paper
Compares with it 2 papers
Cites it as background 17 papers

Discussion

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