On "time-periodic" black-hole solutions to certain spherically symmetric Einstein-Matter systems

This paper explores black hole solutions of various Einstein-wave matter systems admitting a time-orientation preserving isometry of their domain of outer communications taking some point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordström. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [13] from the static to the periodic case. In the third part, which is independent of the first two, it is shown that Dirac fields preserved by an isometry of a spherically symmetric domain of outer communications of the type described above must vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster, Smoller and Yau [10, 8, 9 and also 7]. For equations of evolution, time-periodic or stationary solutions often correspond to the late time behavior of solutions for a large class of initial data. In the general theory of relativity, time-periodic "black hole" solutions, if they exist, seem to provide reasonable candidates for the final state of gravitational collapse. Such solutions can be defined as those invariant with respect to an isometry of the domain of outer communications which takes every point to its future, or more generally, such that points sufficiently close to infinity are mapped to their future. In the case of a continuous family of isometries (i.e. stationary and static solutions), this problem has a long history and goes under the name "no hair" conjecture. See [4] for a survey of classical results and recent important refinements. Current proofs depend on various extra assumptions and truly satisfactory theorems have only been obtained in the vacuum and electrovacuum static case. The aim of this paper is to try to generalize some results from the static case to the spherically symmetric "time-periodic" case. The study of periodic solutions to the Einstein equations was initiated in Papapetrou [ 14, 15]; see also [12], The analyses indicate that vacuum solutions which are periodic near null infinity should in fact be static there, but they are far from complete, and depend very much on analyticity assumptions on the nature of null infinity, assumptions which do not appear to be physically valid. The present paper appears to be the first to address the issue of the existence of periodic solutions in general relativity in a non-analytic setting, in particular, in a setting compatible with the evolutionary hypothesis. After briefly setting some basic assumptions (Sect. 1) regarding spherical symmetry, we shall show in Sect. 2 that for a certain class of matter, non-trivial spherically-symmetric black-hole phenomena cannot be described by solutions invariant with respect to a map taking some point to its future. In Sect. 3, we shall enlarge the class of matter for which the result applies by taking another approach, which in effect reduces the problem to the static case. The method of Sect. 3 is related to the arguments of [11, 16]. In the spherically symmetric context, the above two sections generalize in particular results of [1] and [13], and Sect. 2, where it applies, provides a new and easier approach for the static case. Moreover, no assumption of invariance of the matter with respect to the isometry is necessary, nor is any real understanding of the behavior of the isometry on the event horizon. In fact, the results apply equally well when the "periodic" assumption is weakened to an appropriate notion of "almost periodicity". Key to the results are the monotonicity properties of the area radius or the Hawking mass. In Sect. 4, which is independent of Sects. 2 and 3, we shall show that Dirac fields preserved by an isometry of the form described above must vanish. The method exploits conservation of the Dirac current. There has a been a series of recent work [10, 8, 9] where static spherically symmetric solutions of various coupled Einstein-Dirac-matter systems are considered, and also work [7] where periodic solutions of the Dirac equation on & fixed Reissner-Nordström background are considered. Modulo differences in regularity assumptions, all this previous work follows as a special case of the result of this section, which furthermore excludes non-trivial periodic solutions to a large class of coupled Einstein-Dirac-matter systems. The assumptions in this paper have been laid out in such a way so as to refer to the details of the matter fields as little as possible. In particular situations, however, many of the geometric assumptions of the next section can be retrieved from more "primitive" ones, provided one makes explicit assumptions regarding the relationship of the isometry and the matter. For this, the results of [16] and [17] are essential, and we refer the reader there. A brief discussion of this issue, along with a comparison to [8-10 and 7] for the case of the Dirac equation, is included in the end (Sect. 5).