Nonrotating black hole in a post-Newtonian tidal environment

Abstract

In the first part of the paper we construct the metric of a tidally deformed, nonrotating black hole. The metric is presented as an expansion in powers of $r/b\ensuremath{\ll}1$, in which $r$ is the distance to the black hole and $b$ the characteristic length scale of the tidal field---the typical distance to the remote bodies responsible for the tidal environment. The metric is expanded through order $(r/b{)}^{4}$ and written in terms of a number of tidal multipole moments, the gravitoelectric moments ${\mathcal{E}}_{ab}$, ${\mathcal{E}}_{abc}$, ${\mathcal{E}}_{abcd}$, and the gravitomagnetic moments ${\mathcal{B}}_{ab}$, ${\mathcal{B}}_{abc}$, ${\mathcal{B}}_{abcd}$. It differs from the similar construction of Poisson and Vlasov in that the tidal perturbation is presented in Regge-Wheeler gauge instead of the light-cone gauge employed previously. In the second part of the paper we determine the tidal moments by matching the black-hole metric to a post-Newtonian metric that describes a system of bodies with weak mutual gravity. This extends the previous work of Taylor and Poisson (Paper I in this sequence), which computed only the leading-order tidal moments, ${\mathcal{E}}_{ab}$ and ${\mathcal{B}}_{ab}$. The matching is greatly facilitated by the Regge-Wheeler form of the black-hole metric, and this motivates the work carried out in the first part of the paper. The tidal moments are calculated accurately through the first post-Newtonian approximation, and at this order they are independent of the precise nature of the compact body. The moments therefore apply equally well to a rotating black hole, or to a (rotating or nonrotating) neutron star. As an application of this formalism, we examine the intrinsic geometry of a tidally deformed event horizon and describe it in terms of a deformation function that represents a quadrupolar and octupolar tidal bulge.