Static response and Love numbers of Schwarzschild black holes

TL;DR

This paper derives the quadratic action for perturbations on Schwarzschild-(A)dS backgrounds, computes black hole Love numbers across dimensions, and reveals new polarization effects and non-vanishing Love numbers in higher dimensions.

Contribution

It provides the first calculation of Love numbers for vector gravitational perturbations in higher dimensions and clarifies discrepancies in previous static response analyses.

Findings

Love numbers vanish for all perturbations in four dimensions.

Black holes polarize in electromagnetic fields in all dimensions except four.

Non-zero Love numbers for vector gravitational perturbations in higher dimensions.

Abstract

We derive the quadratic action for the physical degrees of freedom of massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild--(A)dS background in arbitrary dimensions. We then use these results to compute the static response of asymptotically flat Schwarzschild black holes to external fields. Our analysis reproduces known facts about black hole Love numbers, in particular that they vanish for all types of perturbation in four spacetime dimensions, but also leads to new results. For instance, we find that neutral Schwarzschild black holes polarize in the presence of an electromagnetic background in any number of spacetime dimensions except four. Moreover, we calculate for the first time black hole Love numbers for vector-type gravitational perturbations in higher dimensions and find that they generically do not vanish. Along the way, we shed some light on an apparent discrepancy between previous results in the literature, and clarify some aspects of the matching between perturbative calculations of static response on a Schwarzschild background and the point-particle effective theory