Perturbations of slowly rotating black holes: Massive vector fields in the Kerr metric

Abstract

We discuss a general method to study linear perturbations of slowly rotating black holes which is valid for any perturbation field, and particularly advantageous when the field equations are not separable. As an illustration of the method we investigate massive vector (Proca) perturbations in the Kerr metric, which do not appear to be separable in the standard Teukolsky formalism. Working in a perturbative scheme, we discuss two important effects induced by rotation: a Zeemanlike shift of nonaxisymmetric quasinormal modes and bound states with different azimuthal number m, and the coupling between axial and polar modes with different multipolar index ℓ. We explicitly compute the perturbation equations up to second order in rotation, but in principle the method can be extended to any order. Working at first order in rotation we show that polar and axial Proca modes can be computed by solving two decoupled sets of equations, and we derive a single master equation describing axial perturbations of spin s = 0 and s = ±1. By extending the calculation to second order we can study the superradiant regime of Proca perturbations in a self-consistent way. For the first time we show that Proca fields around Kerr black holes exhibit a superradiant instability, which is significantly stronger than for massive scalar fields. Because of this instability, astrophysical observations of spinning black holes provide the tightest upper limit on the mass of the photon: mγ 4 × 10 −20 eV under our most conservative assumptions. Spin measurements for the largest black holes could reduce this bound to mγ 10 −22 eV or lower.

Black hole bombs and explosions: from astrophysics to particle physics

“…Because there are stable orbits of timelike particles in Kerr, this analogy would imply that the Kerr spacetime is unstable against massive field perturbations. The instability has been checked analytically and numerically for scalar [68,69,70,71,72,73,75], vector [76,16,25,74] and tensor fields [77] 2 , and the first rigorous construction of a superradiant instability appeared very recently [80].…”

“…Some examples of rotating solutions with scalar hair were explicitly constructed in Refs. [66,67].2.5 Massive fields, soft bombs and particle physics Massive fields are another realization of a natural reflecting wall, opening up the exciting possibility to test particle physics via black hole physics [22,16,25]. The idea is very simple and consists in thinking about these massive states orbiting the black hole in bound stable orbits while extracting energy from the ergoregion in an exponentially growing way [68].…”

“…The observation of rotating black holes allows (in principle, see below) to impose interesting constraints on the mass of long-range bosonic fields. The idea is as follows: the instability is active on a timescale M τ ∼ (M µ) −α , where α = 9 for scalar fields for instance [16,25,74]. The only stationary vacuum solution of Einstein equations is the Kerr family; accordingly, the end-state of such mechanism must be a slowly spinning black hole, on a timescale equal to the instability timescale.…”

“…(iv) our ability to numerically evolve black hole binaries at the full nonlinear level [9,10,11] and its immediate importance for gravitational-wave searches and high-energy physics [12]; (v) improvement of perturbative schemes, either by an understanding of regularization schemes to handle the self-force [13,14], or by faster and more powerful methods to deal with the full ladder of perturbation equations [8,15,16,17] (describing wave phenomena, extreme-mass ratio inspirals, etc);…”

“…For further details, I refer the reader to Refs. [16,25,74].There are two potential shortcomings of this approach. The first is that it assumes that the growth rate of the scalar field is independent of nonlinear effects and that it grows until rotation is efficiently extracted out of the rotating black hole.…”