  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.
