In this book the questions of an interaction of two strong-field objects in general relativity such as a black hole in a background universe and the motion of two black holes of comparable mass are considered. In Chapter 3 the dynamics of a black hole in a background universe is given. “One would like [...] to understand how a black hole interacts with the surrounding ‘background’ universe, of which the black hole is only a small part.” For this we must allow the ‘mass’ \(M\) of the black hole to become a small parameter, varying in some interval \([0,k)\). Then for the metric \(g_{ab}=g_{ab}(\tau,x,y,z,M)\) of the true universe \({\mathcal M}\) one takes the following asymptotic expansion \[ g_{ab}(\tau,x,y,z,M)= g_{ab}^{(0)}(\tau,x,y,z)+Mg_{ab}^{(1)}(\tau,x,y,z)+ M^2g_{ab}^{(2)}(\tau,x,y,z)+O(M^3) \] where \(g_{ab}^{(0)}\) is the background geometry, \(Mg_{ab}^{(1)}\) is the first perturbation of the background due to the black hole, etc. This asymptotic expansion describes the ‘external region’ in the far field of the black hole, where geometry is dominated by the background. If the background \(g_{ab}^{(0)}\) were exactly flat, then the geometry \(g_{ab}\) would be exactly the Kerr solution with mass \(M\) and spin parameter \(a=\chi M \;(0\leq \chi < 1 )\). The first-order field \(Mg_{ab}^{(1)}\) would be the linearized field Kerr solution, obtained by taking the first-order terms in \(g_{ab}(\tau,x,y,z,M)\). The lowest-order approximation to the path of the small black hole in the background is always a geodesic, regardless of the black hole’s rotation. Its angular momentum vector is approximately parallel transported along the geodesic in the background metric. Another area in which one would have an interaction of two strong-field objects in general relativity concerns the motion of two black holes of comparable mass. In the case that one black hole has much greater mass than the other, the small black hole can be considered as moving in the background metric of the large one, so that the results of Chapter 3 apply. Again, one employs matched asymptotic expansion (Chapter 4). The following examples of black-hole interactions are considered: 1) interaction of two black holes in the slow-motion limit (Chapter 4); 2) gravitational radiation from high-speed black-hole encounters (Chapter 5); 3) axisymmetric black-hole collisions at the speed of light: gravitational radiation (Chapters 6, 7, 8). Many interesting results can be found in these chapters, for example, a new mass-loss formula for the axisymmetric collision of two black holes.
The book is a good textbook for students who wish to study the technique of describing the interaction of two black holes. The researcher finds excellent guidance for further work.