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 $R-αC^2+βR^2$. 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 $λ_E/r_h^5\simeq-0.1224928$ 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.