Non-Existence of Black Hole Solutions to Static, Spherically Symmetric Einstein-Dirac Systems – a Critical Discussion

This short note compares different methods to prove that Einstein-Dirac systems have no static, spherically symmetric solutions. There are several approaches to prove that Einstein-Dirac systems do not admit static, spherically symmetric solutions. The first paper in this direction is [2], where it is shown that the Dirac equation has no normalizable, time-periodic solutions in the ReissnerNordström geometry. In [3]–[5] non-existence results were obtained for coupled static systems by choosing polar coordinates and analyzing the nonlinear radial ODEs. Recently, M. Dafermos [1] proposed a different method where he analyzes the Einstein equations in null coordinates. His method has the advantage that it also applies to other Einsteinmatter systems and generalizes to time-periodic solutions. The mathematical and physical assumptions under which the above methods apply are quite different. Therefore, these methods are not equivalent, and it is rather subtle to decide which approach is preferable for a given physical system. The purpose of this short note is to compare the different approaches by collecting and discussing the necessary assumptions and the obtained results. The physical situation of interest is the spherically symmetric collapse to a black hole. Thus thinking of the Cauchy problem for a coupled Einstein-matter system, we consider an initial Cauchy surface which is topologically R, such that an event horizon forms in its future development. It is a reasonable physical assumption that asymptotically for large time, the system should settle down to a static (or more generally time-periodic) system. Under this assumption, ruling out non-trivial static solutions means that the matter (as described by the Dirac field) is no longer present asymptotically as t → ∞, and thus the matter must either have fallen into the black hole or must have escaped to infinity. For this physical interpretation to hold, it is essential that the assumptions, under which the non-existence result for the static Einstein-Dirac system applies, are satisfied in the gravitational collapse. We now discuss the individual papers in chronological order. In the first paper [2] the situation is particularly simple in that the gravitational field is a given ReissnerNordström background field. In the non-extreme case, the problem is analyzed in the maximally extended Kruskal space-time, where the domain of outer communications D is connected to both a black hole and a white hole through the event horizons H+ and H−, respectively (see the conformal diagram in Figure 1; the figures are taken from [1]). In the context of the gravitational collapse described above, the Reissner-Nordström metric should be considered as physical space-time only asymptotically as t → ∞. In particular, the physical metric should involve a black hole, but no white hole. In order to take this