Static Black Hole and Vacuum Energy: Thin Shell and Incompressible Fluid

TL;DR

This paper investigates static, black-hole-like solutions to semi-classical Einstein equations considering vacuum energy back reaction, revealing no horizons, violations of Buchdahl's inequality, and potential physicality of the Boulware vacuum.

Contribution

It presents novel solutions with no horizons and explores the effects of vacuum energy back reaction on black hole-like objects, challenging traditional assumptions.

Findings

Solutions have no horizon but a radius minimum.

Buchdahl's inequality can be violated without divergence in pressure.

The Boulware vacuum may be physically relevant for black holes.

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-hole-like 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.