REVIEW
Stability of Schwarzschild-like solutions in f(R,G) gravity models
General Relativity and Quantum Cosmology · Submitted 8 Feb 2011 · 41 references
Pith's verdict: A clean no-go for f(R,G) on Schwarzschild-like backgrounds: the even sector always carries a ghost, and the sign argument is a sum of squares that holds up.
T0 · AI review
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Referee report
The paper
The paper claims
Gravity theories that add a Gauss–Bonnet term to the Ricci scalar carry an unavoidable ghost around any spherically symmetric static vacuum background, so they survive as dark-energy models only if that ghost is made heavy enough to decuple
The reading
What carries the argument
The load-bearing device is the Regge–Wheeler–Zerilli decomposition applied to the second-order action, followed by a systematic elimination of the non-propagating fields. For the odd sector the authors introduce a Lagrange multiplier $Q$ to linearise the awkward $(dot{h}_1-h_0')^2$ structure, integrate out $h_0$ and $h_1$, and obtain a canonical Lagrangian whose no-ghost condition is $AF-2Bxi'A'geq 0$ and whose gradient condition is $F-4Bxi''-2B'xi'geq 0$. For the even sector the analogous procedure requires two field redefinitions, $delta F=v_1-(a_1/a_2)delta xi$ and $j^2a_4alpha=a_4v_0-a_3H_2-a_2v_1'$, chosen precisely so that the constraint from $H_0$ loses all second radial derivatives and can be solved for $H_2$; substituting back leaves a $3imes 3$ system of the form $sum[K_{ij}dot{v}_idot{v}_j-L_{ij}v_i'v_j'-D_{ij}v_i'v_j-M_{ij}v_iv_j]$. The sign structure of $K_{ij}$ is what carries the no-go, and the mass matrix $M_{ij}$, rewritten through $(U_{,FF},U_{,Fxi};U_{,Fxi},U_{,xi xi})$ as the inverse Hessian of $f$, is what carries the viability analysis.
Extended reading notes
Core claim
The central claim is a no-go statement: on a general spherically symmetric static vacuum solution, the even-parity perturbation sector of any $f(R,G)$ theory with nonvanishing Hessian determinant $f_{,RR}f_{,GG}-f_{,RG}^{2}$ contains a ghost for every value of the multipole index. The proof is an explicit computation of the $3imes 3$ kinetic matrix $K_{ij}$ of the even modes after the non-dynamical fields $H_0$, $H_1$ and $H_2$ have been integrated out; one of its required positivity conditions, $K_{22}K_{33}-K_{23}^{2}$, is proportional to minus the square of the real combination $[4(B-1)G_{,r}-r^{2}F_{,r}]$ divided by a positive factor, so it is non-positive by construction, and the determinant is negative once the odd-sector gradient condition holds. Hence either one ghost (if $K_{33}>0$) or three ghosts (if $K_{33}<0$). The counting of propagating modes is also settled: one odd mode for $lgeq 2$ and none for $l=0,1$; three even modes for $lgeq 2$ and two for $l=0,1$, of which one is the ordinary massless graviton, one is a chameleon-like non-ghost scalar, and one is the ghost. Two of the three even modes share a propagation speed that reduces to the speed of light when $A=B$, while the third matches the odd-mode speed and depends directly on the radial profiles of $F$ and $xi$. Finally, the authors argue that the ghost need not be fatal if it is heavy enough: they derive the mass eigenvalues $m^{2}_{+}$ and $m^{2}_{-}$ from the mass matrix, show that the ghost mass diverges as the background approaches Minkowski (its kinetic term vanishes there), and reduce the viability of a model to the two conditions $|9tilde{G}det(f_{,ij})| ll L_{m exp}^{4}$ and $|3(f_{,RR}-tilde{G}f_{,GG})| ll L_{m exp}^{2}$, where $tilde{G}pprox 16r_{s}^{2}/r^{6}$ and $L_{m exp}$ is the length scale of the experiment being protected.
Load-bearing premise
The mass estimates and the final viability conditions rest on treating the stellar exterior as a near-Schwarzschild vacuum solution with $Fpprox 1$, $F'pprox 0$ and $xi'pprox xi''pprox 0$, even though the authors concede in footnote 2 that matter cannot consistently be included and enters only by shifting the background values of $R$ and $G$.
Editorial extensions
If this is right
- Any $f(R,G)$ model with a genuinely nondegenerate Hessian is excluded as a ghost-free alternative to general relativity on stellar backgrounds, leaving only the degenerate subclass (including $f(R)$) and near-Minkowski or near-de Sitter backgrounds as safe.
- A model can still survive if its ghost mass exceeds the effective cutoff, which requires a hierarchy among $f_{,RR}$, $f_{,RG}$ and $f_{,GG}$; the paper exhibits a toy model $f=R+F_R(R)+F_G(G)$ with $F_Rpropto R^{-p}$ and $F_Gpropto G^{-2n}$ and shows the ghost mass scales as $H_0^{2}[H_0^{-2}r_s/r^{3}]^{4n+2}$, easily made huge, while the non-ghost scalar needs $pgeq 4$ to be heavy enough at sola
- The odd sector yields a definite prediction for the radial propagation speed, $c_{m odd}^{2}=A(AF-2Bxi'A')/(F-4Bxi''-2B'xi')$, which differs from light speed in these theories and is therefore a target for gravitational-wave and pulsar-timing tests.
- The number of propagating degrees of freedom is background-dependent: it drops by one on Minkowski or de Sitter, exactly as previously found on FLRW, so counting modes in one background does not tell you the theory's degrees of freedom in another.
- For the degenerate case $f_{,RR}f_{,GG}-f_{,RG}^{2}=0$ the ghost can be absent, and the authors show that ordinary $f(R)$ gravity with $F=partial f/partial R>0$ satisfies the resulting no-ghost condition.
Reading between the lines
- The authors note in passing that $G$ changes sign inside a star (negative in a Schwarzschild interior, positive outside), and their ghost mass formula scales with powers of $G$; combining the two means the ghost mass must pass through zero at some radius near the stellar surface, which is a sharper statement than 'the ghost is heavy outside the star' and would be worth checking against explicit in
- Because the no-go rests on a sign-definite square, the escape hatch is narrow in a specific sense: any model whose Hessian determinant is nonzero is caught, so the Gauss–Bonnet coupling itself, rather than the choice of function, is the load-bearing ingredient that creates the problem.
- If the ghost must be heavier than the cutoff everywhere the theory is used, then the required hierarchy among $f_{,ij}$ is itself a fine-tuning that radiative corrections could spoil; the paper does not estimate that stability, and doing so would be a natural next step.
- The mode-counting result suggests a cleaner observational handle than the ghost itself: the extra scalar speed in the odd sector is a single function of the background, so binary pulsar timing or future gravitational-wave observations of a compact object could in principle constrain the combination $F-4Bxi''-2B'xi'$ directly.
Original abstract
We study linear metric perturbations around a spherically symmetric static spacetime for general f(R,G) theories, where R is the Ricci scalar and G is the Gauss-Bonnet term. We find that unless the determinant of the Hessian of f(R,G) is zero, even-type perturbations have a ghost for any multi-pole mode. In order for these theories to be plausible alternatives to General Relativity, the theory should satisfy the condition that the ghost is massive enough to effectively decouple from the other fields. We study the requirement on the form of f(R,G) which satisfies this condition. We also classify the number of propagating modes both for the odd-type and the even-type perturbations and derive the propagation speeds for each mode.
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.
The abstract, sentence by sentence
We study linear metric perturbations around a spherically symmetric static spacetime for general f(R,G) theories, where R is the Ricci scalar and G is the Gauss-Bonnet term. We find that unless the determinant of the Hessian of f(R,G) is zero, even-type perturbations have a ghost for any multi-pole mode. In order for these theories to be plausible alternatives to General Relativity, the theory should satisfy the condition that the ghost is massive enough to effectively decouple from the other fields. We study the requirement on the form of f(R,G) which satisfies this condition. We also classify the number of propagating modes both for the odd-type and the even-type perturbations and derive the propagation speeds for each mode.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study linear metric perturbations around a spherically symmetric static spacetime for general f(R,G) theories, where R is the Ricci scalar and G is the Gauss-Bonnet term. | contribution 0.80 | 0.87 | 0.03 | asserted 0.88 |
| s1 | We find that unless the determinant of the Hessian of f(R,G) is zero, even-type perturbations have a ghost for any multi-pole mode. | result 1.00 | 0.95 | 0.64 | asserted 1.00 |
| s2 | In order for these theories to be plausible alternatives to General Relativity, the theory should satisfy the condition that the ghost is massive enough to effectively decouple from the other fields. | interpretation 0.83 | 0.14 | 0.14 | asserted 0.82 |
| s3 | We study the requirement on the form of f(R,G) which satisfies this condition. | contribution 0.99 | 0.87 | 0.06 | unclear 0.61 |
| s4 | We also classify the number of propagating modes both for the odd-type and the even-type perturbations and derive the propagation speeds for each mode. | contribution 0.83 | 0.96 | 0.08 | asserted 0.99 |
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References
How later papers used it
49 later papers cite it
5 later papers by other authors are listed below: 5 used its method or tool. 3 more papers by the same authors are listed: 3 used its method or tool.
Uses its method or tool 8 of 20 papers with this label · 3 by the same authors
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Introduced an additional field by rewriting the Lagrangian, then an auxiliary field q, to obtain the same equations of motion.
Black hole stability under odd-parity perturbations in Horndeski gravity (2017) · 0.98 · paraphrased: the licence does not allow quoting
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Introduced an auxiliary field as in this paper to extract a gauge-invariant variable.
Implications of the Weak Gravity Conjecture for Tidal Love Numbers of Black Holes (2022) · 1.00 · paraphrased: the licence does not allow quoting
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Introduced an auxiliary variable to rewrite this paper's quadratic action in a form with a squared derivative term.
Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations (2019) · 0.95 · paraphrased: the licence does not allow quoting
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We introduce an auxiliary variable $χ$ to write [ref] 2ℓ+12πj^2^(2)_odd= p_1h_0^2+p_2h_1^2+p_3-χ^2+2χ⟨ḣ_1-h'_0|.
Black hole perturbations in DHOST theories: Master variables, gradient instability, and strong coupling (2021) · Multipole · 0.97
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Following the procedure introduced in [ref] , we perform an integration by parts and complete the square of terms containing derivatives to obtain [equation] where we have defined [equation] We then introduce an auxiliary variable $χ$ (whose mass dimension is 2) such that the Lagrangian above can be rewritten as [equation] Notice that the Lagrangian ([ref]) can be recovered from [ref] by integrating out $χ$.
Generalized Regge-Wheeler Equation from Effective Field Theory of Black Hole Perturbations with a Timelike Scalar Profile (2022) · General multipoles · 1.00
By the same authors
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Introduced an auxiliary variable to rewrite this paper's quadratic action in a form with a squared derivative term.
Linear perturbation analysis of hairy black holes in shift-symmetric Horndeski theories: Odd-parity perturbations (2016) (same authors) · 0.93 · paraphrased: the licence does not allow quoting
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Invoked this paper's trick to eliminate the nondynamical degree of freedom.
Black hole perturbation in the most general scalar-tensor theory with second-order field equations I: The odd-parity sector (2012) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
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Introduced new variables and defined a new second-order action, following the same method used in this paper's f(R,G) context.
Gravitational collapse and odd-parity black hole perturbations in Minimal Theory of Bigravity (2023) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
47 arXiv papers cite this in 63 sentences
Papers citing it · 23 sentences without a confident label
Builds on it 1 paper
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We now extend the work of [ref] to admit time-dependent scalar hair, following closely the previous works of [ref] and [ref] .
Instability of hairy black holes in shift-symmetric Horndeski theories · Perturbations of a black hole with time-dependent scalar hair · 1.00
Uses its method or tool 20 papers
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Then, the metric perturbations corresponding to odd-mode perturbations are expressed in the form [ref]
Odd-parity stability of hairy black holes in $U(1)$ gauge-invariant scalar-vector-tensor theories · General BH stability against odd-parity perturbations · 0.91
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Then, we introduce an auxiliary variable $χ$ to write [ref] 2ℓ+12π^(2)=⟨a_1-2(ra_3)'r^2|h_0^2+a_2h_1^2+a_3-χ^2+2χ⟨ḣ_1-h_0'+2rh_0|+a_4h_0h_1.
Linear perturbation analysis of hairy black holes in shift-symmetric Horndeski theories: Odd-parity perturbations · The $ℓ≥2$ modes · 0.93
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The components of metric perturbations $h_μν$ in the odd-parity sector are expressed in the forms [ref] :
Black holes in quartic-order beyond-generalized Proca theories · Odd-parity perturbations · 0.91
Compares with it 2 papers
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It is noted that comparing against the formulas utilized in [ref] .
Gravitational odd-parity perturbation of a Horndeski hairy black hole: quasinormal mode and parameter constraint · Odd-parity perturbation and modified Regge-Wheeler equation · 0.98
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In this appendix we highlight the difference between the odd mode analysis in [ref] and an alternative analysis employing a technique of auxiliary fields, which is commonly used in the literature [ref] .
Self-accelerating Massive Gravity: Hidden Constraints and Characteristics · Alternative odd analysis · 0.93
Cites it as background 14 papers
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These degenerate theories were first noticed to behave differently in Ref. [ref] when studying perturbations on a Friedmann-Robertson-Walker background, and later shown to be in general healthier than generic $f(R,)$ theories [ref] .
Purely metric Horndeski theories and spontaneous curvaturization of black holes · Purely Metric Horndeski Theories · 0.82
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In Horndeski theories, the stabilities of BH solutions against odd- and even-parity perturbations were investigated in Refs. [ref] (see also Refs. [ref] ).
Black hole perturbations in vector-tensor theories: The odd-mode analysis · Introduction · 0.96
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In Horndeski theories, the BH stability against odd- and even-parity perturbations for a time-independent scalar field was originally addressed by Kobayashi et al. [ref] (see also Refs. [ref] ).
Instability of hairy black holes in regularized 4-dimensional Einstein-Gauss-Bonnet gravity · Introduction · 0.93
Discussion
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