Bakry-\'Emery black holes

Abstract

Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-\'Emery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan frame". Just as, in General Relativity, natural energy conditions on the stress-energy tensor become conditions on the Ricci tensor, in scalar-tensor theories expressed in the Jordan frame natural energy conditions become conditions on the Bakry-\'Emery-Ricci tensor. We show that, if the Bakry-\'Emery tensor obeys the null energy condition with an upper bound on the Bakry-\'Emery scalar function, there is a modified notion of apparent horizon which obeys analogues of familiar theorems from General Relativity. The Bakry-\'Emery modified apparent horizon always lies behind an event horizon and the event horizon obeys a modified area theorem. Under more restrictive conditions, the modified apparent horizon obeys an analogue of the Hawking topology theorem in 4 spacetime dimensions. Since topological censorship is known to yield a horizon topology theorem independent of the Hawking theorem, in an appendix we obtain a Bakry-\'Emery version of the topological censorship theorem. We apply our results to the Brans-Dicke theory, and obtain an area theorem for horizons in that theory. Our theorems can be used to understand behaviour observed in numerical simulations by Scheel, Shapiro, and Teukolsky of dust collapse in Brans-Dicke theory.