Bakry-Émery black holes. (English) Zbl 1302.83023

Summary: 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-Émery-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-Émery-Ricci tensor. We show that, if the Bakry-Émery tensor obeys the null energy condition with an upper bound on the Bakry-Émery scalar function, there is a modified notion of apparent horizon which obeys analogues of familiar theorems from General Relativity. The Bakry-Émery 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 four 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-Émery 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 M. A. Scheel et al. [Phys. Rev. D (3) 51, No. 8, 4236–4249 (1995; doi:10.1103/PhysRevD.51.4236)] of dust collapse in Brans-Dicke theory.


83C57 Black holes
83C75 Space-time singularities, cosmic censorship, etc.
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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