Love symmetry

Abstract

Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (``Love'') SL$(2,\mathbb{R})$ symmetry in the suitably defined near zone approximation. We present a detailed study of this symmetry and show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL$(2,\mathbb{R})$ representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL$\left(2,\mathbb{R}\right)$ representations. It is this highest weight property that forces the static Love numbers to vanish. We find that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. This relation is simplest for extremal charged spherically symmetric (Reissner-Nordström) solutions, where the Love symmetry exactly reduces to the isometry of the near horizon AdS$_2$ throat. For rotating (Kerr-Newman) black holes one is lead to consider an infinite-dimensional SL$\left(2,\mathbb{R}\right)\ltimes \hat U(1)_{\mathcal{V}}$ extension of the Love symmetry. It contains three physically distinct subalgebras: the Love algebra, the Starobinsky near zone algebra, and the near horizon algebra that becomes the Bardeen-Horowitz isometry in the extremal limit. We also discuss other aspects of the Love symmetry, such as the geometric meaning of its generators for spin weighted fields, connection to the no-hair theorems, non-renormalization of Love numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects of its existence in modified theories of gravity.