Complex Instantons and Charged Rotating Black Hole Pair Creation

We consider the general process of pair-creation of charged rotating black holes. We find that instantons which describe this process are necessarily complex due to regularity requirements. However their associated probabilities are real, and fully consistent with the interpretation that the entropy of a charged rotating black hole is the logarithm of the number of its quantum states. Black hole pair production is by now a well-established process in semi-classical quantum gravity. Pair-production is a tunnelling process in which the mass-energy of the created pair of black holes is balanced by their negative potential energy in some background field. Studies of this phenomenon for non-rotating black holes have repeatedly provided us with evidence that the exponential of the entropy of a black hole does indeed correspond to the number of its quantum states. It is quite robust, with many different background fields – an electromagnetic field [1], a positive cosmological constant [2], a cosmic string [3], or a domain wall [4] – yielding qualitatively similar features for the pair production process. The probability amplitude for the process is approximated by ei , where Ii is the action of the relevant instanton i.e. a Euclidean solution to the field equations which interpolates between the states before and after a pair of black holes is produced. In this paper we consider this problem for the rotating case and show that we must necessarily modify the definition of instanton beyond that which is given by the usual periodicimaginary time formalism [5]. This formalism, originally developed to enable the computation of tunnelling processes and canonical partition functions [6] has (apparently) led to the belief that the relevant instanton action must always be obtained from real, positive-definite (i.e. Euclidean) metrics. For rotating black holes, the real Euclidean metric is then obtained by supplementing the analytic continuation t → it with the transformation J → iJ , where J is the real angular momentum [7]. We find that this prescription cannot consistently define an instanton that will mediate the creation of a pair of charged rotating black holes in a background with a positive cosmological constant, and that complex instantons must be employed to describe the pair-production of rotating black holes. To our knowledge this is the first example in which naive analytic continuation of parameters in a stationary black hole metric will not correctly describe the relevant physical process. The relevance of complex metrics for black hole thermodynamics was discussed in ref. [8], and our results are consistent with this interpretation of the functional integral formalism. Although there is no real Euclidean instanton that describes the pair production of rotating black holes, the action associated with the complex instanton is real, and describes the probability of the pair creation of rotating black holes in a manner fully consistent with the interpretation of the entropy as the logarithm of the number of quantum states of the black hole, as in the non-rotating case. To illustrate this we shall take our background energy to be that provided by a cosmological constant, and consider two oppositely charged black holes undergoing uniform acceleration in this background. This physical situation is described by the following metric [9] ds = 1 (p− q)2 [ 1 + pq P dp + P 1 + p2q2 ( dσ − qdτ )2 − 1 + pq Q dq + Q 1 + p2q2 ( dτ + pdσ )2] , (1) generalizing the C metric which describes the spacetime associated with a pair of uniformly accelerating neutral spinless black holes [10]. The accompanying electromagnetic field is