Hidden Symmetry of Vanishing Love

TL;DR

This paper uncovers a hidden $SL(2,R) imes U(1)$ symmetry in the near zone of Kerr black hole perturbations, providing a new understanding of static tidal responses and the vanishing of Love numbers through representation theory.

Contribution

It reveals a hidden symmetry structure in black hole perturbations that explains the behavior of Love numbers and static responses using $SL(2,R)$ representation theory.

Findings

Static Love numbers vanish for Schwarzschild black holes.

Perturbations form infinite-dimensional $SL(2,R)$ representations.

Symmetry explains the finite-dimensionality of static solutions.

Abstract

We show that perturbations of massless fields in the Kerr black hole background enjoy a hidden $SL(2,\mathbb{R})\times {U}(1)$ ("Love") symmetry in the properly defined near zone approximation. Love symmetry mixes IR and UV modes. Still, this approximate symmetry allows us to derive exact results about static tidal responses. Generators of the Love symmetry are globally well defined and have a smooth Schwarzschild limit. Generic regular solutions of the near zone Teukolsky equation form infinite-dimensional $SL(2,\mathbb{R})$ representations. In some special cases ($\hat{\ell}$ parameter is an integer), these are highest weight representations. This is the situation that corresponds to vanishing Love numbers. In particular, static perturbations of four-dimensional Schwarzschild black holes belong to finite-dimensional representations. Other known facts about static Love numbers also acquire an elegant explanation in terms of the $SL(2,\mathbb{R})$ representation theory.