9 - Uniqueness theorems for nonrotating holes  pp. 140-165

Uniqueness theorems for nonrotating holes

By Markus Heusler

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In this chapter we present the arguments which establish that the Schwarzschild metric describes the only static, asymptotically flat vacuum spacetime with regular (not necessarily connected) event horizon (Israel 1967, Müller zum Hagen et al. 1973, Robinson 1977, Bunting and Masood–ul–Alam 1987). We then discuss the generalization of this result to the situation with electric fields; that is, we demonstrate the uniqueness of the 2–parameter Reissner–Nordström solution amongst all asymptotically flat, static electrovac black hole configurations with nondegenerate horizon (Israel 1968, Müller zum Hagen et al. 1974, Simon 1985, Ruback 1988, Masood–ul–Alam 1992). Taking magnetic fields into account as well, we finally establish the uniqueness of the 3–parameter Reissner–Nordström metric. We conclude this chapter with a brief discussion of the Papapetrou-Majumdar metric, representing a static configuration with M = |Q| and an arbitrary number of extreme black holes (Papapetrou 1945, Majumdar 1947). This metric is not covered by the static uniqueness theorems, since the latter apply exclusively to electrovac solutions which are subject to the inequality M > |Q|.

Throughout this chapter the domain of outer communications is assumed to be static. In the vacuum or the electrovac case staticity is, as we have argued in the previous chapter, a consequence of the symmetry conditions for the matter fields.

Our main objective in this chapter concerns the “modern” approach to the static uniqueness theorem, which is based on conformal transformations and the positive energy theorem. We shall, however, start this chapter with some comments on the traditional line of reasoning, which is due to Israel, Müller zum Hagen, Robinson and others.