Metric of a tidally perturbed spinning black hole

We explicitly construct the metric of a Kerr black hole that is tidally perturbed by the external universe in the slow-motion approximation. This approximation assumes that the external universe changes slowly relative to the rotation rate of the hole, thus allowing the parameterization of the Newman-Penrose scalar ${\psi }_0$ by time-dependent electric and magnetic tidal tensors. This approximation, however, does not constrain how big the spin of the background hole can be and, in principle, the perturbed metric can model rapidly spinning holes. We first generate a potential by acting with a differential operator on ${\psi }_0$. From this potential we arrive at the metric perturbation by use of the Chrzanowski procedure in the ingoing radiation gauge. We provide explicit analytic formulas for this metric perturbation in Kerr coordinates, where the perturbation is finite at the horizon. This perturbation is parametrized by the mass and Kerr spin parameter of the background hole together with the electric and magnetic tidal tensors that describe the time evolution of the perturbation produced by the external universe. In order to make the metric accurate far away from the hole, these tidal tensors should be determined by asymptotically matching this metric to another one valid far from the hole. The tidally perturbed metric constructed here could be useful in initial data constructions to describe the metric near the horizons of a binary system of spinning holes. This perturbed metric could also be used to construct waveforms and study the absorption of mass and angular momentum by a Kerr black hole when external processes generate gravitational radiation.

Figure 1

Plot of the real part of the NP scalar ${\psi }_0$ along the $x$ axis with the plotting parameters described in Sec. 2.

Figure 2

Plot of the potential ${\Psi }_R$ along the $x$ axis with the plotting parameters described in Sec. 2.

Figure 3

Plot of the $00$ component of the metric perturbation along the $x$ axis (solid line) and along the $y$-$z$ diagonal (dashed line) with the plotting parameters described in Sec. 2.