Dynamics of a small black hole in a background universe

A technique involving matched asymptotic expansions is used to investigate the dynamics of a black hole surrounded by an external background universe, according to the Einstein field equations. The approximation scheme is valid provided that the background curvature is small compared to the curvature near the event horizon of the black hole. In this case the background produces only small perturbations of the basic Kerr geometry near the black hole, while the black hole only affects the background metric slightly at distances of the order of the background length scale. These two perturbation expansions are matched in some common region of validity. It is then shown that the black hole moves approximately along a timelike geodesic in the background, and that its spin is approximately parallelly transported along the geodesic. The largest effects of the black hole on the background and the largest distortions of the Kerr geometry caused by the background are analyzed in some detail. The background curvature induces distortions of a quadrupole nature in the black hole; these then slow down the rotation, so that the basic structure of the Kerr black hole changes over long time scales. A similar approach is used to describe the behavior of a small black hole in a background, under the Brans-Dicke field equations. In particular, it is shown that a black hole moves on a geodesic in the Einstein conformal frame; this confirms a conjecture by Hawking