Uniqueness and Non-Uniqueness of Static Vacuum Black Holes in Higher Dimensions

With the development of string theory, black holes in higher-dimensional spacetimes have come to play a fundamental role in physics 1). Furthermore, the possibility of black hole production in high energy experiments has recently been suggested in the context of the so-called TeV gravity 2). A TeV-size black hole in TeV gravity is small enough to be well approximated by an asymptotically flat black hole in higher dimensions. To predict phenomenological results, we need reliable knowledge about higher-dimensional black holes. However, some essential features of black hole theory have not so far been fully explored. Among these, the equilibrium problem for black holes is one of most important issues. The final equilibrium state of the black hole is known to drastically simplify in the case of four space-time dimensions, because of the uniqueness properties of static or stationary black hole solutions. The uniqueness theorem for the vacuum black hole is well established in four-dimensional spacetimes. 3) 5) (See also Ref. 6) for comprehensive review) Although this no hair property is fundamental to the nature of black holes, it is at the same time a quite non-trivial result derived from the Einstein equations. So far, there is no evidence that in higher dimensions the final state of the black hole is unique. Remarkably, five-dimensional stationary vacuum black holes are not unique; there is a Myers-Perry solution 1), which is a generalization of the Kerr solution to arbitrary dimensions, while Emparan and Reall 7) have recently found five-dimensional rotating black ring solutions with the same angular momenta and mass but with an event horizon homeomorphic to S2 × S1. In the static case, such a counter-example has not yet been presented. The only known asymptotically flat static vacuum black hole is the n-dimensional hyperspherically symmetric Schwarzschild-Tangherlini solution 8). We shall show in what follows that there are no others∗). Our proof can be extended to charged ∗) After submitting this manuscript to gr-qc, we were informed by M. Anderson that a similar proof has been given by Hwang .