Vanishing of Quadratic Love Numbers of Schwarzschild Black Holes

TL;DR

This paper demonstrates that quadratic Love numbers of Schwarzschild black holes remain zero even when nonlinear effects are included, extending the known linear result to second order in perturbation theory.

Contribution

It proves that quadratic Love numbers vanish for Schwarzschild black holes at all orders, including nonlinearities, using analytic solutions of Einstein equations and effective field theory matching.

Findings

Quadratic Love numbers are zero to all orders.

Analytic solutions for second-order perturbations are derived.

Nonlinearities do not induce non-zero Love numbers.

Abstract

The induced conservative tidal response of self-gravitating objects in general relativity is parametrized in terms of a set of coefficients, which are commonly referred to as Love numbers. For asymptotically-flat black holes in four spacetime dimensions, the Love numbers are notoriously zero in the static regime. In this work, we show that this result continues to hold upon inclusion of nonlinearities in the theory for Schwarzschild black holes. We first solve the quadratic Einstein equations in the static limit to all orders in the multipolar expansion, including both even and odd perturbations. We show that the second-order solutions take simple analytic expressions, generically expressible in the form of finite polynomials. We then define the quadratic Love numbers at the level of the point-particle effective field theory. By performing the matching with the full solution in general relativity, we show that quadratic Love number coefficients are zero to all orders in the derivative expansion, like the linear ones.