Exact black hole solution with a minimally coupled scalar field

Abstract

An exact four-dimensional black hole solution of gravity with a minimally coupled self-interacting scalar field is reported. The event horizon is a surface of negative constant curvature enclosing the curvature singularity at the origin, and the scalar field is regular everywhere outside the origin. This solution is an asymptotically locally AdS spacetime. The strong energy condition is satisfied on and outside the event horizon. The thermodynamical analysis shows the existence of a critical temperature, below which a black hole in vacuum undergoes a spontaneous dressing up with a nontrivial scalar field in a process reminiscent of

Non-supersymmetric membrane flows from fake supergravity and multi-trace deformations

“…Interestingly, the same scalar potential (4.24), corresponding to k = 6, led to the MTZ black hole in four dimensions [19]. Since we know how to uplift solutions of the four-dimensional scalar-gravity system with this potential to eleven dimensions, we find it tempting to present the eleven-dimensional black hole metric explicitly, which we do in appendix B.…”

“…In 18 We assume that the scalar fields all have the same mass squared, but the argument generalizes in an obvious way to unequal masses. 19 This fact, in combination with the fact that the dimension of the dual operators is unambiguously determined to be ∆+ = 4, implies that all domain walls for the SL(5, R)/ SO(5) scalars in seven dimensions necessarily describe VEVs of the dual theory.…”

“…, −∆ + /2l). 19 Hence, there are simply no solutions with non-zero φ − in this case and so W o (φ) can be safely used as the counterterm, resulting in the expected supersymmetric renormalization scheme. Second, focusing on the case d = 3 which we are interested in, we have seen that there is a continuous family of fake superpotentials, of the generic form…”

“…In section 4 we systematically discuss how to obtain these non-supersymmetric solutions in four dimensions in closed form by consistently reducing the number of scalar fields, and we solve (1.8) exactly, obtaining a family of exact fake superpotentials, for a special case involving a single scalar field. We then uplift this solution to eleven dimensions in section 5 using the ansatz discussed in section 2 and, noting that the MTZ black hole [19] in four dimensions is interestingly a solution of exactly the same action as our exact domain wall, we also give the eleven-dimensional black hole solution (given explicitly in appendix B). The holographic one-and two-point functions for the non-supersymmetric domain walls are then computed respectively in sections 6 and 7.…”

“…This can be easily done by using the equations (7.11). One expands the radial derivative as 18) as well as the functions E and Ω 21 19) and inserts these expansions in equations (7.11). Collecting terms of the same dilatation weight then determines all terms in the expansions (7.19), except for the coefficients E (d) and Ω (2∆−d) .…”