Symmetries of Vanishing Nonlinear Love Numbers of Schwarzschild Black Holes

TL;DR

This paper proves that the nonlinear tidal Love numbers of Schwarzschild black holes vanish for all multipoles and orders, revealing underlying symmetries related to ladder operators and the Geroch group.

Contribution

It demonstrates that all static, parity-even nonlinear Love numbers vanish for Schwarzschild black holes, extending previous linear results to all orders in perturbation theory.

Findings

Nonlinear Love numbers vanish for all multipoles and perturbation orders.

Ladder symmetries analogous to linear theory are present in nonlinear black hole perturbations.

Connection established between symmetries and the Geroch group from dimensional reduction.

Abstract

The tidal Love numbers parametrize the conservative induced tidal response of self-gravitating objects. It is well established that asymptotically-flat black holes in four-dimensional general relativity have vanishing Love numbers. In linear perturbation theory, this result was shown to be a consequence of ladder symmetries acting on black hole perturbations. In this work, we show that a black hole's tidal response induced by a static, parity-even tidal field vanishes for all multipoles to all orders in perturbation theory. Our strategy is to focus on static and axisymmetric spacetimes for which the dimensional reduction to the fully nonlinear Weyl solution is well-known. We define the nonlinear Love numbers using the point-particle effective field theory, matching with the Weyl solution to show that an infinite subset of the static, parity-even Love number couplings vanish, to all orders in perturbation theory. This conclusion holds even if the tidal field deviates from axisymmetry. Lastly, we discuss the symmetries underlying the vanishing of the nonlinear Love numbers. An $\mathfrak{sl}(2,\mathbb R)$ algebra acting on a covariantly-defined potential furnishes ladder symmetries analogous to those in linear theory. This is because the dynamics of the potential are isomorphic to those of a static, massless scalar on a Schwarzschild background. We comment on the connection between the ladder symmetries and the Geroch group that is well-known to arise from dimensional reduction.