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Tidal deformation of a slowly rotating black hole

In the first part of this article I determine the geometry of a slowly rotating black hole deformed by generic tidal forces created by a remote distribution of matter. The metric of the deformed black hole is obtained by integrating the Einstein field equations in a vacuum region of spacetime bounded by r < r_max, with r_max a maximum radius taken to be much smaller than the distance to the remote matter. The tidal forces are assumed to be weak and to vary slowly in time, and the metric is expressed in terms of generic tidal quadrupole moments E_{ab} and B_{ab} that characterize the tidal environment. The metric incorporates couplings between the black hole's spin vector and the tidal moments, and captures all effects associated with the dragging of inertial frames. In the second part of the article I determine the tidal moments by immersing the black hole in a larger post-Newtonian system that includes an external distribution of matter; while the black hole's internal gravity is allowed to be strong, the mutual gravity between the black hole and the external matter is assumed to be weak. The post-Newtonian metric that describes the entire system is valid when r > r_min, where r_min is a minimum distance that must be much larger than the black hole's radius. The black-hole and post-Newtonian metrics provide alternative descriptions of the same gravitational field in an overlap r_min < r < r_max, and matching the metrics determine the tidal moments, which are expressed as post-Newtonian expansions carried out through one-and-a-half post-Newtonian order. Explicit expressions are obtained in the specific case in which the black hole is a member of a post-Newtonian two-body system.