Global structure of exact scalar hairy dynamical black holes

Abstract

We study the global structure of some exact scalar hairy dynamical black holes which were constructed in Einstein gravity either minimally or non-minimally coupled to a scalar field. We find that both the apparent horizon and the local event horizon (measured in luminosity coordinate) monotonically increase with the advanced time as well as the Vaidya mass. At late advanced times, the apparent horizon approaches the event horizon and gradually becomes future outer. Correspondingly, the space-time arrives at stationary black hole states with the relaxation time inversely proportional to the 1/(n−1) power of the final black hole mass, where n is the space-time dimension. These results strongly support the solutions describing the formation of black holes with scalar hair. We also obtain new charged dynamical solutions in the non-minimal theory by introducing an Maxwell field which is non-minimally coupled to the scalar. The presence of the electric charge strongly modifies the dynamical evolution of the space-time.

Static and dynamic charged black holes

“…Since the event horizon is of great difficult to solve in dynamical spacetimes, it is useful to use the apparent horizon ( or trapping horizon ) to characterize the evolution of dynamic black holes [25,28]. The apparent horizon is defined by…”

“…where c is an integration constant. The evolution equation of this type was first examined in [25], where dynamical charged black holes are constructed in nonminimally coupled Einstein-Scalar gravity.…”

“…Some of these solutions can be promoted to become dynamical ones where the static black holes are the end points of the time evolution [19,20]. Further such exact dynamical solutions were later constructed in [22][23][24][25][26][27].…”

Dynamical black holes in low-energy string theory

“…They can be regarded as a generalization of the well-known Roberts' selfsimilar solution for the Einstein-massless scalar system [26,27] to include a Maxwell field. More recently, this solution was embedded in N = 4 gauged supergravity [28] and similar time-dependent solutions with supergravity-inspired potentials for the scalar field have been discussed in [29][30][31].…”

Holographic complexity under a global quantum quench

“…However, the construction there cannot be directly generalized to general dynamical black holes since the matter sources needed may not be null any longer. Interestingly, in recent years there are a list of exact dynamical hairy black holes having been successfully constructed in Einstein-Scalar gravity [41,42,43,44,45,46,47,48]. The Lagrangian density is given by…”