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

TL;DR

The work uncoveres a hidden ladder of static perturbation symmetries for black holes, tying horizon regularity to asymptotic behavior and thereby explaining the vanishing Love numbers and no-hair theorems for spin-0,1,2 fields. It constructs raising/lowering operators that generate a tower of solutions in Schwarzschild and Kerr spacetimes and identifies conserved charges that constrain horizon-to-infinity matching, extending the ladder to spin via the Teukolsky framework. An infrared EFT interpretation shows that the ultraviolet ladder imposes a large-distance symmetry which forbids static hair unless explicit symmetry breaking occurs (e.g., Gauss–Bonnet couplings), connecting to a geometric AdS realization and hinting at a deeper algebraic structure. The results have broad implications for black hole phenomenology and effective field theory descriptions of their interactions with external fields.

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.