Near-zone symmetries of Kerr black holes

Abstract

We study the near-zone symmetries of a massless scalar field on four-dimensional black hole backgrounds. We provide a geometric understanding that unifies various recently discovered symmetries as part of an SO(4, 2) group. Of these, a subset are exact symmetries of the static sector and give rise to the ladder symmetries responsible for the vanishing of Love numbers. In the Kerr case, we compare different near-zone approximations in the literature, and focus on the implementation that retains the symmetries of the static limit. We also describe the relation to spin-1 and 2 perturbations.

Love symmetry

“…[35,36,91]. The effective Kerr near zone geometry presented in [88] actually corresponds to our Starobinsky near zone approximation (2.30), i.e. the s = 0 Starobinsky near zone SL (2, R) generators (5.1) are Killing vectors of that effective near zone geometry.…”

“…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.…”

“…This geometry is what [88] refers to as the "effective near zone geometry" of the Kerr-Newman black hole, here connected to the earlier notion of subtracted geometries ( [35,36]). Let us see if the spin weighted Lie derivative (7.12) captures the correct s = 0 extensions in (5.1).…”

Painlevé-II approach to binary black hole merger dynamics: universality from integrability

“…From the initial data u(x, t = 0) of ( 51) we determine the operator L(t = 0) at time t = 0. For instance, in the KdV case (77) this fixes the potential. From L(0) one then determines the scattering data at t = 0: {a(λ, t = 0), b(λ, t = 0), .…”

“…A related but alternative approach to this isospectrality in terms of Darboux transformations is presented [57,71] (see also [72] for an approach in terms of "intertwining operators"). However, it is in the recent work [58,59] where the connection to integrability, specifically through inverse scattering theory, becomes apparent, as well as shedding light on the link between the KdV equation and Darboux transformations (see also [73][74][75][76][77][78] for other hints on integrability and hidden symmetries in this perturbative setting). Specifically, in [58] an infinite branch of new admissible of odd/even effective potentials (with their associated master functions) is identified, all related by Darboux transformations preserving the spectral properties, leading to the notion of "Darboux covariance".…”