Hidden symmetry of the static response of black holes: Applications to Love numbers

TL;DR

This paper reveals a hidden Schr"odinger algebra symmetry in static perturbations of Schwarzschild black holes, explaining the vanishing of Love numbers through symmetry breaking and extending the understanding of black hole responses.

Contribution

It uncovers a conserved Schr"odinger algebra symmetry in black hole perturbations and links it to the vanishing Love numbers via symmetry breaking mechanisms.

Findings

Identifies a Schr"odinger algebra symmetry in static black hole perturbations.

Shows the HJPSS symmetry coincides with an sl(2,R) generator.

Explains vanishing Love numbers as a consequence of symmetry protection.

Abstract

We show that any static linear perturbations around Schwarzschild black holes enjoy a set of conserved charges which forms a centrally extended Schr\"{o}dinger algebra sh(1) = sl$(2,\mathbb{R}) \ltimes \mathcal{H}$. The central charge is given by the black hole mass, echoing results on black hole entropy from near-horizon diffeomorphism symmetry. The finite symmetry transformations generated by these conserved charges correspond to Galilean and conformal transformations of the static field and of the coordinates. This new structure allows one to discuss the static response of a Schwarzschild black hole in the test field approximation from a symmetry-based approach. First we show that the (horizontal) symmetry protecting the vanishing of the Love numbers recently exhibited by Hui et al, dubbed the HJPSS symmetry, coincides with one of the sl$(2,\mathbb{R})$ generators of the Schr\"{o}dinger group. Then, it is demonstrated that the HJPSS symmetry is selected thanks to the spontaneous breaking of the full Schr\"{o}dinger symmetry at the horizon down to a simple abelian sub-group. The latter can be understood as the symmetry protecting the regularity of the test field at the horizon. In the 4-dimensional case, this provides a symmetry protection for the vanishing of the Schwarzschild Love numbers. Our results trivially extend to the Kerr case.