Hidden Symmetry of Vanishing Love

TL;DR

This work reveals a hidden $SL(2,R) \times U(1)$ Love symmetry governing near-zone perturbations of Kerr black holes, providing an algebraic explanation for the vanishing of static Love numbers in four dimensions and organizing static and nonstatic tidal responses into $SL(2,R)$ representations. The authors show that, for integer multipoles in four dimensions, static perturbations live in finite-dimensional representations (highest weights), enforcing polynomial radial behavior and hence zero Love numbers; they extend the framework to higher dimensions and spins, where vanishing results persist only for special integer parameters. They further propose an infinite extension, $SL(2,R) \ltimes U(1)_V$, capturing a broader structure that includes both horizon regularity and UV/IR mixing, and discuss potential holographic interpretations in relation to extreme Kerr physics. The findings offer a new symmetry-based organizing principle for black-hole tidal responses with potential implications for gravitational-wave physics and effective field theory descriptions of compact binaries.

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.