Static tidal Love numbers of Schwarzschild and non-Schwarzschild black holes in quadratic gravity
Abstract
We calculate the static electric quadrupolar response of black holes in four-dimensional quadratic gravity, with action R−αC2+βR2. 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 α/rh2=1/2, we find λE/rh5≃−0.1224928 and k2(rh)≃−0.1837391, with a final outer-endpoint change in k2(rh) of about 1.60×10−7. 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.
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BibTeX
@misc{bhattacharyya2026static,
title = {{Static tidal Love numbers of Schwarzschild and non-Schwarzschild black holes in quadratic gravity}},
author = {Soham Bhattacharyya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24625},
eprint = {2610.24625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24625},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Static tidal Love numbers of Schwarzschild and non-Schwarzschild black holes in quadratic gravity
AU - Soham Bhattacharyya
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24625
UR - https://arxiv.org/abs/2610.24625
AB - We calculate the static electric quadrupolar response of black holes in four-dimensional quadratic gravity, with action R−αC^2+βR^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 α/r_h^2=1/2, we find λ_E/r_h^5≃−0.1224928 and k_2^(r_h)≃−0.1837391, with a final outer-endpoint change in k_2^(r_h) of about 1.60×10^−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.
ER -