Ladder Symmetries of Black Holes: Implications for Love Numbers and No-Hair Theorems

TL;DR

This paper reveals that symmetries in black hole perturbations explain the vanishing of Love numbers and support the no-hair theorem, providing a geometric and algebraic framework for understanding black hole responses.

Contribution

It introduces a ladder symmetry structure governing static perturbations of black holes, explaining the vanishing Love numbers and the absence of linear hair.

Findings

Static Love numbers vanish due to underlying symmetries.

Solutions respecting symmetries are polynomial and horizon-regular.

Symmetries constrain black hole responses and hair in effective theories.

Abstract

It is well known that asymptotically flat black holes in general relativity have a vanishing static, conservative tidal response. We show that this is a result of linearly realized symmetries governing static (spin 0,1,2) perturbations around black holes. The symmetries have a geometric origin: in the scalar case, they arise from the (E)AdS isometries of a dimensionally reduced black hole spacetime. Underlying the symmetries is a ladder structure which can be used to construct the full tower of solutions, and derive their general properties: (1) solutions that decay with radius spontaneously break the symmetries, and must diverge at the horizon; (2) solutions regular at the horizon respect the symmetries, and take the form of a finite polynomial that grows with radius. Taken together, these two properties imply that static response coefficients -- and in particular Love numbers -- vanish. Moreover, property (1) is consistent with the absence of black holes with linear (perturbative) hair. We also discuss the manifestation of these symmetries in the effective point particle description of a black hole, showing explicitly that for scalar probes the worldline couplings associated with a non-trivial tidal response and scalar hair must vanish in order for the symmetries to be preserved.