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

TL;DR

This paper demonstrates that Ladder symmetry is both necessary and sufficient for black holes to have zero static tidal Love numbers, by analyzing parametric deformations in Ladder-symmetric spacetimes within the KRZ framework.

Contribution

It establishes Ladder symmetry as the key condition for vanishing static TLNs in black holes, extending previous results to deformed and rotating cases.

Findings

Deviations from Ladder symmetry induce non-zero static scalar TLNs.

Ladder symmetry is necessary and sufficient for zero static TLNs in static and rotating black holes.

Parametric deformations break the vanishing TLN property, confirming the symmetry's critical role.

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.