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

Abstract

We show that any static linear perturbations around Schwarzschild black holes enjoy a set of conserved charges which forms a centrally extended Schrödinger algebra $$ \mathfrak{sh} $$ sh (1) = $$ \mathfrak{sl} $$ sl (2, ℝ) ⋉ $$ \mathcal{H} $$ 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 $$ \mathfrak{sl} $$ sl (2, ℝ) generators of the Schrödinger group. Then, it is demonstrated that the HJPSS symmetry is selected thanks to the spontaneous breaking of the full Schrödinger 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.

Love symmetry

“…As a continuation of that work, [88] appeared while our paper was being prepared (see also [90]). There, the ladder symmetry structure arises from a larger conformal group of an effective conformally flat near zone metric.…”

Near-zone symmetries of Kerr black holes

“…The time translation T and spatial rotations J ij (i, j = 1, 2, 3) are the familiar symmetries of the exact dynamics. In addition, the symmetry generators J 0i (i = 1, 2, 3) have a somewhat privileged status: they generate symmetries of the exact system in the static limit, ω = 0 [4]; see also [19] for a related discussion. The other (C)KVs do not give rise to exact symmetries in this limit.…”

Scalar Love numbers and Love symmetries of 5-dimensional Myers-Perry black hole

“…This hints at the interpretation of the Love symmetry as a remnant of this enhanced isometry for extremal black holes. It should be remarked here that there have been other attempts of explaining the vanishing of black holes Love numbers via symmetry arguments, notably, the "ladder symmetries" proposal [50,51] (see also [52,53]), which bares resemblance to the earlier notion of "mass ladder operators" for spacetimes admitting closed conformal Killing vectors [54].…”