Ladder Symmetry: The Necessary and Sufficient Condition for Vanishing Love Numbers

TL;DR

The paper addresses why static tidal Love numbers vanish for black holes and whether Ladder symmetry is merely sufficient or also necessary. It employs a theory-agnostic, parametrized TLN formalism with small corrections $\alpha_j$ to the effective potential, analyzing Ladder-symmetric backgrounds in the KRZ class and perturbations $\epsilon,\beta_n$ to track running TLNs. The authors demonstrate that any deviation from Ladder symmetry induces nonzero static scalar TLNs for some multipole order $\ell$, and that enforcing vanishing TLNs for all $\ell$ leads to an infinite hierarchy of coupled constraints whose only consistent solution is the exact Ladder-symmetric background. This establishes Ladder symmetry as a fundamental criterion behind the no-Love property, with implications for GR tests and potential generalizations beyond KRZ spacetimes.

Abstract

Black holes in four-dimensional, asymptotically flat general relativity have vanishing static tidal Love numbers (TLNs), a property tied to a hidden symmetry of the perturbation equations. Within the Konoplya-Rezzolla-Zhidenko (KRZ) parametrization, a subclass of spacetimes was previously shown to admit such Ladder symmetry, which enforces the absence of static scalar TLNs. In this work, we introduce parametric deformations to such Ladder-symmetric spacetimes and analyze the resultant linear tidal response. Using the parametrized formalism for TLNs, we show that any deviation from a Ladder-symmetric background leads to non-zero static scalar TLNs. This establishes Ladder symmetry as a necessary, as well as sufficient condition, for the vanishing of static TLNs in static, spherically symmetric black holes and in rotating black holes of the KRZ class.