Perturbations of Kerr spacetime are typically studied with the Teukolsky formalism, in which a pair of invariant components of the perturbed Weyl tensor are expressed in terms of separable modes that satisfy ordinary differential equations. However,
for certain applications it is desirable to construct the full metric perturbation in the Lorenz gauge, in which the linearized Einstein field equations take a manifestly hyperbolic form. Here we obtain a set of Lorenz-gauge solutions to the vacuum field equations in terms of homogeneous solutions to the spin-2, spin-1 and spin-0 Teukolsky equations; and completion pieces that represent perturbations to the mass and angular momentum of the spacetime. The solutions are valid in vacuum Petrov type-D spacetimes that admit a conformal Killing-Yano tensor.
We present the gravitational-wave flux balance law in an extreme mass-ratio binary with a spinning secondary. This law relates the flux of energy (angular momentum) radiated to null infinity and through the event horizon to the local change in the se
condarys orbital energy (angular momentum) for generic (non-resonant) bound orbits in Kerr spacetime. As an explicit example we compute these quantities for a spin-aligned body moving on a circular orbit around a Schwarzschild black hole. We perform this calculation both analytically, via a high-order post-Newtonian expansion, and numerically in two different gauges. Using these results we demonstrate explicitly that our new balance law holds.
We study the time-independent scattering of a planar gravitational wave propagating in the curved spacetime of a compact body with a polytropic equation of state. We begin by considering the geometric-optics limit, in which the gravitational wave pro
pagates along null geodesics of the spacetime; we show that a wavefront passing through a neutron star of tenuity $R/M = 6$ will be focussed at a cusp caustic near the stars surface. Next, using the linearized Einstein Field Equations on a spherically-symmetric spacetime, we construct the metric perturbations in the odd and even parity sectors; and, with partial-wave methods, we numerically compute the gravitational scattering cross section from helicity-conserving and helicity-reversing amplitudes. At long wavelengths, the cross section is insensitive to stellar structure and, in the limit $M omega rightarrow 0$, it reduces to the known low-frequency approximation of the black hole case. At higher frequencies $M omega gtrsim 1$, the gravitational wave probes the internal structure of the body. In essence, we find that the gravitational wave cross section is similar to that for a massless scalar field, although with subtle effects arising from the non-zero helicity-reversing amplitude, and the coupling in the even-parity sector between the gravitational wave and the fluid of the body. The cross section exhibits emph{rainbow scattering} with an Airy-type oscillation superposed on a Rutherford cross section. We show that the rainbow angle, which arises from a stationary point in the geodesic deflection function, depends on the polytropic index. In principle, rainbow scattering provides a diagnostic of the equation of state of the compact body; but, in practice, this requires a high-frequency astrophysical source of gravitational waves.
A key goal of the Event Horizon Telescope is to observe the shadow cast by a black hole. Recent simulations have shown that binary black holes, the progenitors of gravitational waves, present shadows with fractal structure. Here we study the binary s
hadow structure using techniques from nonlinear dynamics, recognising shadows as exit basins of open Hamiltonian dynamical systems. We apply a recently developed numerical algorithm to demonstrate that parts of the Majumdar-Papapetrou binary black hole shadow exhibit the Wada property: any point of the boundary of one basin is also on the boundary of at least two additional basins. We show that the algorithm successfully distinguishes between the fractal and regular (i.e., non-fractal) parts of the binary shadow.
Spinning horizonless compact objects may be unstable against an ergoregion instability. We investigate this mechanism for electromagnetic perturbations of ultracompact Kerr-like objects with a reflecting surface, extending previous (numerical and ana
lytical) work limited to the scalar case. We derive an analytical result for the frequency and the instability time scale of unstable modes which is valid at small frequencies. We argue that our analysis can be directly extended to gravitational perturbations of exotic compact objects in the black-hole limit. The instability for electromagnetic and gravitational perturbations is generically stronger than in the scalar case and it requires larger absorption to be quenched. We argue that exotic compact objects with spin $chilesssim 0.7$ ($chilesssim 0.9$) should have an absorption coefficient of at least $0.3%$ ($6%$) to remain linearly stable, and that an absorption coefficient of at least $approx60%$ would quench the instability for any spin. We also show that - in the static limit - the scalar, electromagnetic, and gravitatonal perturbations of the Kerr metric are related to one another through Darboux transformations.
A massive vector boson field in the vicinity of a rotating black hole is known to suffer an instability, due to the exponential amplification of (co-rotating, low-frequency) bound states by black hole superradiance. Here we calculate the bound state
spectrum by exploiting the separation of variables recently achieved by Frolov, Krtous, Kubiznak and Santos (FKKS) for the Proca field on Kerr-(A)dS-NUT spacetimes of arbitrary dimension. Restricting to the 4D Kerr case, we first establish the relationship between the FKKS and Teukolsky variables in the massless case; obtain exact results for the angular eigenvalues in the marginally-bound case; and present a spectral method for solving the angular equation in the general case. We demonstrate that all three physical polarizations can be recovered from the FKKS ansatz, resolving an open question. We present numerical results for the instability growth rate for a selection of modes of all three polarizations, and discuss physical implications.
In Phys.Rev.D89, 104053 (2014) we studied the absorption cross section of a scalar field of mass $m$ impinging on a static black hole of mass $M$ and charge $Q$. We presented numerical results using the partial-wave method, and analytical results in
the high- and low-frequency limit. Our low-frequency approximation was only valid if the (dimensionless) field velocity $v$ exceeds $v_c = 2 pi M m$. In this Addendum we give the complementary result for $v lesssim v_c$, and we consider the possible physical relevance of this regime.
We present new regular solutions of Einstein-charged scalar field theory in a cavity. The system is enclosed inside a reflecting mirror-like boundary, on which the scalar field vanishes. The mirror is placed at the zero of the scalar field closest to
the origin, and inside this boundary our solutions are regular. We study the stability of these solitons under linear, spherically symmetric perturbations of the metric, scalar and electromagnetic fields. If the radius of the mirror is sufficiently large, we present numerical evidence for the stability of the solitons. For small mirror radius, some of the solitons are unstable. We discuss the physical interpretation of this instability.
We compute the scattering cross section of Reissner-Nordstrom black holes for the case of an incident electromagnetic wave. We describe how scattering is affected by both the conversion of electromagnetic to gravitational radiation, and the parity-de
pendence of phase shifts induced by the black hole charge. The latter effect creates a helicity-reversed scattering amplitude that is non-zero in the backward direction. We show that from the character of the electromagnetic wave scattered in the backward direction it is possible, in principle, to infer if a static black hole is charged.
We compute the quasinormal mode frequencies and Regge poles of the canonical acoustic hole (a black hole analogue), using three methods. First, we show how damped oscillations arise by evolving generic perturbations in the time domain using a simple
finite-difference scheme. We use our results to estimate the fundamental QN frequencies of the low multipolar modes $l=1, 2, ldots$. Next, we apply an asymptotic method to obtain an expansion for the frequency in inverse powers of $l+1/2$ for low overtones. We test the expansion by comparing against our time-domain results, and (existing) WKB results. The expansion method is then extended to locate the Regge poles. Finally, to check the expansion of Regge poles we compute the spectrum numerically by direct integration in the frequency domain. We give a geometric interpretation of our results and comment on experimental verification.
