REVIEW
Static tidal Love numbers of Schwarzschild and non-Schwarzschild black holes in quadratic gravity
General Relativity and Quantum Cosmology · Submitted 17 Oct 2026 · 64 references
Pith's verdict: First calculation of static tidal Love numbers for non-Schwarzschild black holes in quadratic gravity; the result is nonzero and negative, and the derivation is careful, but the lack of shared code and the finite-endpoint approximation limit reproducibility.
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Referee report
The paper
The paper claims
The static tidal Love number of a non-Schwarzschild black hole in quadratic gravity is nonzero and negative, at about −0.184 for α/r_h^2=1/2.
The reading
What carries the argument
The calculation uses Kodama–Ishibashi gauge-invariant variables to formulate the static polar perturbations, leading to a system of six first-order ordinary differential equations for the metric and curvature invariants. After excluding the scalar curvature perturbation via horizon regularity and the absence of an applied scalar tide, the system reduces to three coupled second-order equations. The coupling prevents separation into independent massless and massive channels. The tidal Love number is extracted by imposing a regular horizon, an applied massless quadrupole, and decaying massive fields, then matching numerical solutions integrated from the horizon and from large radius. The background non-Schwarzschild solution is constructed via a shooting method that enforces asymptotic flatness by removing the growing Yukawa mode.
Extended reading notes
Core claim
The central discovery is that the nonzero background Ricci tensor on the non-Schwarzschild branch couples the gravitational perturbation channels, allowing a nonzero massless quadrupolar response even when no external massive or scalar tide is applied. This contrasts with the Schwarzschild branch, where the equations decouple and the massless response remains zero. The computed Love number $k_2^{(r_h)}$ is negative, meaning the induced quadrupole has the opposite sign relative to the tidal field compared to a positive-deformability object. The paper presents a reference value $k_2^{(r_h)} \simeq -0.1837391$ at $\alpha/r_h^2=1/2$, with an outer-endpoint sensitivity of $1.60\times10^{-7}$.
Load-bearing premise
The numerical background non-Schwarzschild solution is constructed by shooting and is assumed to be asymptotically flat by imposing a finite-radius outer boundary condition that removes the growing Yukawa mode; the systematic errors from this finite-radius approximation are only partially probed by endpoint variation tests.
Editorial extensions
If this is right
- If correct, the nonzero Love number shows that black holes in higher-derivative gravity can have tidal deformabilities qualitatively different from those in general relativity, potentially distinguishing them in gravitational-wave observations.
- The negative sign of the Love number implies that the induced quadrupole is opposite in sign to the tidal field under the stated convention, which could affect the orbital dynamics of binary systems.
- The result confirms that the vanishing Love number of Schwarzschild black holes is not a universal property of all black holes, but depends on the gravitational theory and the specific black hole solution.
- The paper provides a method for computing tidal responses in theories where the perturbation equations are coupled, which can be applied to other modified gravity theories.
- The coupling scan shows a negative response approaching zero near the Schwarzschild intersection, consistent with the expectation that the non-Schwarzschild branch merges with Schwarzschild at a special coupling.
Reading between the lines
- One could test the robustness of the numerical result by computing the Love number for the same background using an alternative method, such as a direct integration of the fourth-order metric equations without reduction to first-order.
- The negative Love number may have implications for the stability of these black holes under tidal perturbations, although the paper notes that the static quadrupole does not directly relate to the known Gregory–Laflamme instability.
- The approach could be extended to compute tidally induced scalar perturbations if a scalar tide were applied, or to higher multipoles, which might reveal a pattern in the response coefficients.
- A careful matching to the worldline effective field theory, as mentioned by the authors, would determine whether the metric Love number coincides with the Wilson coefficient used in gravitational-wave data analysis, and is necessary to connect this result to observations.
Original abstract
We calculate the static electric quadrupolar response of black holes in four-dimensional quadratic gravity, with action $R-\alpha C^2+\beta R^2$. The Schwarzschild solution has zero massless gravitational Love number under the stated boundary conditions. A numerical non-Schwarzschild solution instead has a nonzero negative response. At $\alpha/r_h^2=1/2$, we find $\lambda_E/r_h^5\simeq-0.1224928$ and $k_2^{(r_h)}\simeq-0.1837391$, with a final outer-endpoint change in $k_2^{(r_h)}$ of about $1.60\times10^{-7}$. We explain how the background is obtained by shooting and how the tidal solution is selected by matching at an intermediate radius. The perturbations are written in Kodama--Ishibashi gauge-invariant variables. After the scalar-curvature perturbation is excluded by horizon regularity and the absence of an applied scalar tide, the remaining static sector has three coupled second-order radial equations. We define the curvature amplitudes in these equations and describe why simple attempts to separate them fail. The coupling scan shows a negative response approaching zero near the Schwarzschild intersection. We also explain why this full-theory result does not conflict with the removal of curvature-squared operators in a perturbative low-energy expansion.
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 calculate the static electric quadrupolar response of black holes in four-dimensional quadratic gravity, with action $R-\alpha C^2+\beta R^2$. The Schwarzschild solution has zero massless gravitational Love number under the stated boundary conditions. A numerical non-Schwarzschild solution instead has a nonzero negative response. At $\alpha/r_h^2=1/2$, we find $\lambda_E/r_h^5\simeq-0.1224928$ and $k_2^{(r_h)}\simeq-0.1837391$, with a final outer-endpoint change in $k_2^{(r_h)}$ of about $1.60\times10^{-7}$. We explain how the background is obtained by shooting and how the tidal solution is selected by matching at an intermediate radius. The perturbations are written in Kodama--Ishibashi gauge-invariant variables. After the scalar-curvature perturbation is excluded by horizon regularity and the absence of an applied scalar tide, the remaining static sector has three coupled second-order radial equations. We define the curvature amplitudes in these equations and describe why simple attempts to separate them fail. The coupling scan shows a negative response approaching zero near the Schwarzschild intersection. We also explain why this full-theory result does not conflict with the removal of curvature-squared operators in a perturbative low-energy expansion.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We calculate the static electric quadrupolar response of black holes in four-dimensional quadratic gravity, with action $R-\alpha C^2+\beta R^2$. | contribution 1.00 | 0.95 | 0.07 | asserted 0.99 |
| s1 | The Schwarzschild solution has zero massless gravitational Love number under the stated boundary conditions. | result 0.99 | 0.76 | 0.24 | asserted 1.00 |
| s2 | A numerical non-Schwarzschild solution instead has a nonzero negative response. | result 1.00 | 0.86 | 0.33 | asserted 1.00 |
| s3 | At $\alpha/r_h^2=1/2$, we find $\lambda_E/r_h^5\simeq-0.1224928$ and $k_2^{(r_h)}\simeq-0.1837391$, with a final outer-endpoint change in $k_2^{(r_h)}$ of about $1.60\times10^{-7}$. | result 1.00 | 0.97 | 0.27 | asserted 1.00 |
| s4 | We explain how the background is obtained by shooting and how the tidal solution is selected by matching at an intermediate radius. | method 1.00 | 0.84 | 0.03 | asserted 0.99 |
| s5 | The perturbations are written in Kodama--Ishibashi gauge-invariant variables. | method 1.00 | 0.69 | 0.02 | asserted 1.00 |
| s6 | After the scalar-curvature perturbation is excluded by horizon regularity and the absence of an applied scalar tide, the remaining static sector has three coupled second-order radial equations. | method 0.99 | 0.78 | 0.04 | asserted 1.00 |
| s7 | We define the curvature amplitudes in these equations and describe why simple attempts to separate them fail. | method 0.93 | 0.82 | 0.03 | asserted 0.88 |
| s8 | The coupling scan shows a negative response approaching zero near the Schwarzschild intersection. | result 0.96 | 0.89 | 0.15 | asserted 1.00 |
| s9 | We also explain why this full-theory result does not conflict with the removal of curvature-squared operators in a perturbative low-energy expansion. | interpretation 1.00 | 0.63 | 0.08 | asserted 0.98 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2610.24625 by the authors.
References
Discussion
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