Hidden Symmetry of Vanishing Love Numbers

Abstract

We show that perturbations of massless fields in a black hole background enjoy a hidden SL(2,R)×U(1) ("Love") symmetry in the properly defined near zone approximation. Love symmetry mixes low- and high-frequency modes. Still, this approximate symmetry allows us to derive exact results about static tidal responses. Generators of the Love symmetry are globally well defined for any value of black hole spin. Generic regular solutions of the near zone equation for linearized perturbations form infinite-dimensional SL(2,R) representations. In some special cases, these are highest weight representations. This situation corresponds to vanishing Love numbers. Other known facts about static Love numbers also acquire an elegant explanation in terms of the SL(2,R) representation theory.