In this final section we draw attention to some of the current problems and developments, without claiming completeness.

(1) Given black hole data for n holes of fixed masses and mutual separations (whatever definitions one uses here), one would like to minimize these data on the amount of outgoing radiation energy. Any excess over the minimal amount can be said to be ‘already contained’ initially. But so far no local (in time) criterion is known which quantifies the amount of gravitational radiation in an initial data set. First hints at the possibility that some (Newman-Penrose) conserved quantities could be useful.

(2) Restricting ourselves to spatially conformally flat metrics seems to be too narrow. It has been shown that there are no conformally flat spatial slices in Kerr spacetime which are axisymmetric and reduce to slices of constant Schwarzschild time in the limit of vanishing angular momentum. Accordingly, Bowen-York data, even for a single black hole, contain excess gravitational radiation due to the relaxation of the individual holes to Kerr for an informal discussion of this and related problems. An alternative to the Bowen–York data, which describe two spinning black holes and which reduce to Kerr data if the mass of one hole goes to zero.

(3) Even for the simplest two-hole data (Schwarzschild or Einstein-Rosen) it is not known whether the evolving spacetime will have a suitably smooth asymptotic structure at future-lightlike infinity (i.e. ‘scri-plus’). As a consequence, we still do not know whether we can give a rigorous mathematical meaning to the notion of ‘energy loss by gravitational radiation’ in this case of the simplest head-on collision of two black holes! The difficult analytical problems involved are studied in the framework of the so called ‘conformal field equations’, for a summary and references.

(4) We usually like to ask ‘Newtonian’ questions, like: given two black holes of individual masses m_1,2 and mutual separation ℓ, what is their binding energy? For such a question to make sense, we need good concepts of quasi local mass and distance. But these are ambiguous concepts in GR. Different definitions of ‘quasi-local mass’ and ‘distance’ amount to differences in the calculated binding energies which can be a few 10⁻³ times total energy at closest. This is of the same order of magnitude as the total energy lost into gravitational radiation for the head-on (i.e. zero angular momentum) collision of two black holes modelled with Misner data.