Magic zeroes in the black hole response problem and a Love symmetry resolution

TL;DR

This paper uncovers an $SL(2,R)$ Love symmetry in black hole response problems, explaining the vanishing of static Love numbers and connecting it to near-horizon geometries, with implications for various gravity theories.

Contribution

It introduces a new Love symmetry in black hole physics, linking static Love numbers to symmetry representations and near-horizon geometries, extending understanding across different black hole types and theories.

Findings

Love symmetry is globally defined and preserves physical solutions.

Vanishing static Love numbers follow from highest-weight representations of the Love symmetry.

Love symmetry relates to the near-horizon AdS$_2$ geometry and varies with black hole extremality.

Abstract

In this thesis, we present the emergence of an $SL(2,R)$ ("Love") symmetry in the suitably defined near-zone region, relevant for studying the black hole response problem. This symmetry is globally defined and physical solutions of the black hole linearized field equations are closed under its action. The vanishing of static Love numbers is found to naturally arise as a selection rule following from the fact that the relevant solution belongs to a particular highest-weight representation of the Love symmetry. Interestingly, the Love symmetry appears to be connected to the well-known enhanced $SL(2,R)$ isometry subgroup of the near-horizon extremal geometry. Namely, the Love symmetry exactly reduces to the isometry of the near-horizon AdS$_2$ throat for extremal Reissner-Nordstr\"om black holes, while, for rotating black holes, one is lead to consider an infinite-dimensional $SL(2,R)\ltimes\hat{U}(1)_V$ extension of the Love symmetry, a family of subalgebras of which precisely recovers the Killing vectors of the corresponding AdS$_2$ throat in the extremal limit. Similar results persist when studying perturbations of general-relativistic black holes in higher dimensions. Even though the black hole static Love numbers now vanish only for a discrete set of resonant conditions that depends on the orbital number of the perturbation, Love symmetry exists independently of these details. This hints at a geometric interpretation of the Love symmetry; a statement that becomes more rigorous within the framework of subtracted geometries. However, Love symmetry appears to be theory-dependent. We extract, in particular, a sufficient geometric condition for its existence according to which, for instance, it does not exist in Riemann$^3$ gravity or some stringy-corrected low-energy effective gravitational actions, in accordance with explicit computations of the corresponding black hole Love numbers.