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The open question of whether a Kerr black hole can become tidally deformed or not has profound implications for fundamental physics and gravitational-wave astronomy. We consider a Kerr black hole embedded in a weak and slowly varying, but otherwise arbitrary, multipolar tidal environment. By solving the static Teukolsky equation for the gauge-invariant Weyl scalar $psi_0$, and by reconstructing the corresponding metric perturbation in an ingoing radiation gauge, for a general harmonic index $ell$, we compute the linear response of a Kerr black hole to the tidal field. This linear response vanishes identically for a Schwarzschild black hole and for an axisymmetric perturbation of a spinning black hole. For a nonaxisymmetric perturbation of a spinning black hole, however, the linear response does not vanish, and it contributes to the Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an application, we compute explicitly the rotational black hole tidal Love numbers that couple the induced quadrupole moments to the quadrupolar tidal fields, to linear order in the black hole spin, and we introduce the corresponding notion of tidal Love tensor. Finally, we show that those induced quadrupole moments are closely related to the well-known physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment.
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The tidal Love numbers (TLNs) encode the deformability of a self-gravitating object immersed in a tidal environment and depend significantly both on the objects internal structure and on the dynamics of the gravitational field. An intriguing result in classical general relativity is the vanishing of the TLNs of black holes. We extend this result in three ways, aiming at testing the nature of compact objects: (i) we compute the TLNs of exotic compact objects, including different families of boson stars, gravastars, wormholes, and other toy models for quantum corrections at the horizon scale. In the black-hole limit, we find a universal logarithmic dependence of the TLNs on the location of the surface; (ii) we compute the TLNs of black holes beyond vacuum general relativity, including Einstein-Maxwell, Brans-Dicke and Chern-Simons gravity; (iii) We assess the ability of present and future gravitational-wave detectors to measure the TLNs of these objects, including the first analysis of TLNs with LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars, while LISA is able to probe even extremely compact objects. We argue that the TLNs provide a smoking gun of new physics at the horizon scale, and that future gravitational-wave measurements of the TLNs in a binary inspiral provide a novel way to test black holes and general relativity in the strong-field regime.
To perform realistic tests of theories of gravity, we need to be able to look beyond general relativity and evaluate the consistency of alternative theories with observational data from, especially, gravitational wave detections using, for example, an agnostic Bayesian approach. In this paper we further examine properties of one class of such viable, alternative theories, based on metrics arising from ungauged supergravity. In particular, we examine the massless, neutral, minimally coupled scalar wave equation in a general stationary, axisymmetric background metric such as that of a charged rotating black hole, when the scalar field is either time independent or in the low-frequency, near-zone limit, with a view to calculating the Love numbers of tidal perturbations, and of obtaining harmonic coordinates for the background metric. For a four-parameter family of charged asymptotically flat rotating black hole solutions of ungauged supergravity theory known as STU black holes, which includes Kaluza-Klein black holes and the Kerr-Sen black hole as special cases, we find that all time-independent solutions, and hence the harmonic coordinates of the metrics, are identical to those of the Kerr solution. In the low-frequency limit we find the scalar fields exhibit the same $SL(2,R)$ symmetry as holds in the case of the Kerr solution. We point out extensions of our results to a wider class of metrics, which includes solutions of Einstein-Maxwell-Dilaton theory.
We investigate the spherical photon orbits in near-extremal Kerr spacetimes. We show that the spherical photon orbits with impact parameters in a finite range converge on the event horizon. Furthermore, we demonstrate that the Weyl curvature near the horizon does not generate the shear of a congruence of such light rays. Because of this property, a series of images produced by the light orbiting around a near-extremal Kerr black hole several times can be observable.
We show that rotating black holes do not experience any tidal deformation when they are perturbed by a weak and adiabatic gravitational field. The tidal deformability of an object is quantified by the so-called Love numbers, which describe the objects linear response to its external tidal field. In this work, we compute the Love numbers of Kerr black holes and find that they vanish identically. We also compute the dissipative part of the black holes tidal response, which is non-vanishing due to the absorptive nature of the event horizon. Our results hold for arbitrary values of black hole spin, for both the electric-type and magnetic-type perturbations, and to all orders in the multipole expansion of the tidal field. The boundary conditions at the event horizon and at asymptotic infinity are incorporated in our study, as they are crucial for understanding the way in which these tidal effects are mapped onto gravitational-wave observables. In closing, we address the ambiguity issue of Love numbers in General Relativity, which we argue is resolved when those boundary conditions are taken into account. Our findings provide essential inputs for current efforts to probe the nature of compact objects through the gravitational waves emitted by binary systems.
It was shown recently that the static tidal response coefficients, called Love numbers, vanish identically for Kerr black holes in four dimensions. In this work, we confirm this result and extend it to the case of spin-0 and spin-1 perturbations. We compute the static response of Kerr black holes to scalar, electromagnetic, and gravitational fields at all orders in black hole spin. We use the unambiguous and gauge-invariant definition of Love numbers and their spin-0 and spin-1 analogs as Wilson coefficients of the point particle effective field theory. This definition also allows one to clearly distinguish between conservative and dissipative response contributions. We demonstrate that the behavior of Kerr black holes responses to spin-0 and spin-1 fields is very similar to that of the spin-2 perturbations. In particular, static conservative responses vanish identically for spinning black holes. This implies that vanishing Love numbers are a generic property of black holes in four-dimensional general relativity. We also show that the dissipative part of the response does not vanish even for static perturbations due to frame-dragging.
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