Black hole perturbations of massive and partially massless spin-2 fields in (anti) de Sitter spacetime

Luca Santoni; Rachel A. Rosen

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Black hole perturbations of massive and partially massless spin-2 fields in (anti) de Sitter spacetime

Rachel A. Rosen, Luca Santoni

High Energy Physics - Theory · Submitted 1 Oct 2020

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The authors' main claim We show that isospectrality between partially massless modes of different parity is broken in higher-dimensional Schwarzschild de Sitter spacetimes.

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arXiv ID2010.00595 (version 3)

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CitationLuca Santoni, Rachel A. Rosen. "Black hole perturbations of massive and partially massless spin-2 fields in (anti) de Sitter spacetime." arXiv:2010.00595 (2020). https://pith.science/paper/2WVLRKU4

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We provide a systematic and comprehensive derivation of the linearized dynamics of massive and partially massless spin-2 particles in a Schwarzschild (anti) de Sitter black hole background, in four and higher spacetime dimensions. In particular, we show how to obtain the quadratic actions for the propagating modes and recast the resulting equations of motion in a Schr\"odinger-like form. In the case of partially massless fields in Schwarzschild de Sitter spacetime, we study the isospectrality between modes of different parity. In particular, we prove isospectrality analytically for modes with multipole number $L=1$ in four spacetime dimensions, providing the explicit form of the underlying symmetry. We show that isospectrality between partially massless modes of different parity is broken in higher-dimensional Schwarzschild de Sitter spacetimes.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We provide a systematic and comprehensive derivation of the linearized dynamics of massive and partially massless spin-2 particles in a Schwarzschild (anti) de Sitter black hole background, in four and higher spacetime dimensions.	contribution 1.00	0.97	0.16	asserted 1.00
s1	In particular, we show how to obtain the quadratic actions for the propagating modes and recast the resulting equations of motion in a Schr\"odinger-like form.	method 0.80	0.96	0.09	asserted 1.00
s2	In the case of partially massless fields in Schwarzschild de Sitter spacetime, we study the isospectrality between modes of different parity.	contribution 0.99	0.94	0.10	asserted 0.76
s3	In particular, we prove isospectrality analytically for modes with multipole number $L=1$ in four spacetime dimensions, providing the explicit form of the underlying symmetry.	result 0.93	0.96	0.16	asserted 1.00
s4	We show that isospectrality between partially massless modes of different parity is broken in higher-dimensional Schwarzschild de Sitter spacetimes.	result 1.00	0.95	0.24	asserted 1.00

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Original abstract

We provide a systematic and comprehensive derivation of the linearized dynamics of massive and partially massless spin-2 particles in a Schwarzschild (anti) de Sitter black hole background, in four and higher spacetime dimensions. In particular, we show how to obtain the quadratic actions for the propagating modes and recast the resulting equations of motion in a Schr\"odinger-like form. In the case of partially massless fields in Schwarzschild de Sitter spacetime, we study the isospectrality between modes of different parity. In particular, we prove isospectrality analytically for modes with multipole number $L=1$ in four spacetime dimensions, providing the explicit form of the underlying symmetry. We show that isospectrality between partially massless modes of different parity is broken in higher-dimensional Schwarzschild de Sitter spacetimes.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Symmetry preservation in black hole quasinormal mode spectra
gr-qc 2026-09 accept novelty 7.0 of 10

A general theorem shows that symmetries imply quasinormal mode isospectrality under bijective perturbation maps, and applied to pure Weyl gravity it gives the complete axial spectrum as the union of Regge-Wheeler and ...
Dynamical Tidal Response of Schwarzschild Black Holes
gr-qc 2025-11 conditional novelty 7.0 of 10

The dynamical Love numbers of a Schwarzschild black hole are nonzero at quadratic order in frequency, run logarithmically with a coefficient set by dissipation, and are now matched including their finite, scheme-depen...
The Love for Three Logarithms: Universal Running of Dynamical Love Numbers
hep-th 2026-10 conditional novelty 6.0 of 10

For any static, spherically symmetric black hole, the logarithmic running of dynamical Love numbers is encoded in one traceless 2x2 monodromy matrix with three entries A, B, C.
Ladder Symmetries of Black Holes: Implications for Love Numbers and No-Hair Theorems
hep-th 2021-05 accept novelty 6.0 of 10

Static perturbations of Schwarzschild and Kerr black holes obey a ladder symmetry whose conserved charges force the horizon-regular solution to be a pure growing polynomial, so static Love numbers and linear hair vanish.
Static response and Love numbers of Schwarzschild black holes
hep-th 2020-10 accept novelty 6.0 of 10

Black holes respond to static external scalar, electromagnetic, and gravitational fields in every dimension except four, where all response coefficients vanish.

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11 arXiv papers cite this in 21 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 · 15 sentences without a confident label

Uses its method or tool 1 paper

Concretely, we can write $h_μν$ as follows [ref] :

Dynamical Tidal Response of Schwarzschild Black Holes · General relativistic solution via small-frequency expansion · 0.85

Compares with it 1 paper

Note that the two equations will remain in general coupled even in the eikonal limit, $ℓ→∞$, as opposed to what happens in other situations, e.g., in the case of massive and partially massless higher-spin fields in a Schwarzschild (de Sitter) spacetime [ref] .

Effective Field Theory for the Perturbations of a Slowly Rotating Black Hole · Isospectrality and eikonal limit in the EFT for the perturbations · 0.94

Cites it as support 1 paper

applies even to cases where the potential depends also generically on the frequency, $𝐕≡𝐕(r_⋆,ω)$, and has real eigenvalues for real values of $ω$ — see Refs. [ref] for some explicit examples.

An analytic approach to quasinormal modes for coupled linear systems · Discussion · 0.98

Cites it as background 3 papers

The property is featured by massless fields in four dimensions with flat or de Sitter asymptotics, at least for spins $s=0,1,2$.[In the case of a Schwarzschild-de Sitter BH, isospectrality also holds for partially massless spin-2 perturbations [ref] .] Isospectrality is however known to fail in higher dimensions [ref] , for asymptotically anti-de Sitter BHs [ref] , for massive fields [ref] , and for BHs in non-linear electrodynamics [ref] or in the presence of higher-curvature corrections [ref] .

Schwarzschild quasi-normal modes of non-minimally coupled vector fields · Introduction · 0.94
The Regge-Wheeler gauge is also well understood in AdS [ref] .

Soft graviton exchange and the information paradox · Discussion and conclusions · 0.90
For the generalization to massive and partially massless spin-$2$ fields on a Schwarzschild–(A)dS spacetime in $D$-dimensions, see [ref] .

Static response and Love numbers of Schwarzschild black holes · Massless spin-2 field · 0.87

What it cites · 60 sentences without a confident label

Builds on 3 papers

Here, we extend the results of [ref] to generic spacetime dimensions.

Partially massless gravitons do not destroy general relativity black holes · The extremal limit: breaking of isospectrality and the Gregory-Laflamme instability · 1.00
Furthermore, we use the extremal limit to extend the work of [ref] to arbitrary spacetime dimensions, showing that the helicity-0 mode of the massive spin-2 perturbation is unstable on static Schwarschild or Schwarzschild de Sitter spacetimes.

Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass · The extremal limit: breaking of isospectrality and the Gregory-Laflamme instability · 0.96
Furthermore, we use the extremal limit to extend the work of [ref] to arbitrary spacetime dimensions, showing that the helicity-0 mode of the massive spin-2 perturbation is unstable on static Schwarschild or Schwarzschild de Sitter spacetimes.

Instability of black holes in massive gravity · The extremal limit: breaking of isospectrality and the Gregory-Laflamme instability · 0.96

Uses the method or tool of 4 papers

Plugging [ref] into the equation for $Ψ_+$ and using the $Ψ_-$'s equation of motion, one can derive the following constraints on $β$ and $F$ [ref] :

The Darboux transformation in black hole perturbation theory · Darboux transformations and isospectrality · 0.88
The procedure follows exactly the one above—see also [ref] —and the final result can be read off from Eq. [ref] upon fixing $G=0$ and $m^2=2Λ/3$, [equation]

Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories · Even sector · 0.80
The procedure follows exactly the one above—see also [ref] —and the final result can be read off from Eq. [ref] upon fixing $G=0$ and $m^2=2Λ/3$, [equation]

Stable wormholes in scalar-tensor theories · Even sector · 0.80

Compares with 3 papers

in agreement with [ref] .

Partially massless gravitons do not destroy general relativity black holes · Odd sector · 0.98
which agree with [ref] in the limit $Λ→0$. $r_⋆$ is the tortoise coordinate, defined by [equation]

Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass · Odd sector · 0.99
This parallels what happens for massless spin-$2$ fields in general dimensions [ref] .

Gravitational quasinormal radiation of higher-dimensional black holes · Introduction · 0.84

Cites as background 21 papers

An analysis of massive and partially massless (PM) spin-$2$ fields on a Schwarzschild de Sitter spacetime in four dimensions has been performed in [ref] and [ref] , respectively.

Partially massless gravitons do not destroy general relativity black holes · Introduction · 0.88
An analysis of massive and partially massless (PM) spin-$2$ fields on a Schwarzschild de Sitter spacetime in four dimensions has been performed in [ref] and [ref] , respectively.

Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass · Introduction · 0.88
These are known as partially massless particles [ref] .

Evidence for and Obstructions to Non-Linear Partially Massless Gravity · Introduction · 0.99

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