Static black hole and vacuum energy: thin shell and incompressible fluid

Abstract

With the back reaction of the vacuum energy-momentum tensor consistently taken into account, we study static spherically symmetric black-hole-like solutions to the semi-classical Einstein equation. The vacuum energy is assumed to be given by that of 2-dimensional massless scalar fields, as a widely used model in the literature for black holes. The solutions have no horizon. Instead, there is a local minimum in the radius. We consider thin shells as well as incompressible fluid as the matter content of the black-holelike geometry. The geometry has several interesting features due to the back reaction of vacuum energy. In particular, Buchdahl's inequality can be violated without divergence in pressure, even if the surface is below the Schwarzschild radius. At the same time, the surface of the star can not be far below the Schwarzschild radius for a density not much higher than the Planck scale, and the proper distance from its surface to the origin can be very short even for very large Schwarzschild radius. The results also imply that, contrary to the folklore, in principle the Boulware vacuum can be physical for black holes.

On the near-horizon geometry of an evaporating black hole

“…Recently, in refs. [2,3], this idea was realized in a static solution to the semi-classical Einstein equation, which adopted the vacuum energymomentum tensor derived from a 2D quantum field theory [12,13]. A similar horizonless wormhole-like solution was later also found for the vacuum energy-momentum tensor of a 4D quantum field theory in ref.…”

“…[12,13], so the dynamical solutions can be compared with the static solutions found in refs. [2,3]. In fact, the black holes defined by the same equations have already been considered in refs.…”

“…We define in section 2 the semi-classical Einstein equation that is the basis of our analysis, and review in section 3 the results of refs. [2,3] on the static solutions with the wormhole-like structure. In section 4, we use the derivative expansion to define a perturbation theory to deal with the extremely weak time dependence due to the Hawking radiation, in particular, to determine the timedependence of the wormhole-like structure.…”

“…For this model, it is shown that static solutions have no horizon in the absence of outgoing or ingoing energy fluxes (f u (u) = f v (v) = 0) [2]. The static configuration of a star composed of the perfect fluid was also studied as a solution to the semi-classical Einstein equation, by introducing the classical energy-momentum tensor of the fluid in addition to the vacuum energy-momentum tensor above [3]. It was found that the vacuum energy-momentum tensor of this toy model automatically regularizes the geometry such that C(u, v) never goes to 0 and the pressure T θθ never diverges.…”

“…With the vacuum energy-momentum tensor taken into account, there can be regular static solutions for which the Buchdahl theorem is inapplicable [2,3]. Due to the back reaction of the vacuum energy-momentum tensor, the geometry around the Schwarzschild radius is wormhole-like [2][3][4][5][6], i.e., it has a local minimum in the areal radius r, but the static solution is horizonless. 1…”