Effective Potential of a Black Hole in Thermal Equilibrium with Quantum Fields

TL;DR

This paper investigates how quantum fields in thermal equilibrium influence black hole spacetime, revealing new features like stable photon orbits and repulsive regions through semi-classical Einstein equations.

Contribution

It provides the first detailed analysis of back-reaction effects of various quantum fields on black hole geometry at one-loop order.

Findings

Existence of stable circular photon orbits.

Increase in black hole capture cross sections.

Presence of a gravitationally repulsive region.

Abstract

Expectation values of one-loop renormalized thermal equilibrium stress-energy tensors of free conformal scalars, spin-${1 \over 2}$ fermions and U(1) gauge fields on a Schwarzschild black hole background are used as sources in the semi-classical Einstein equation. The back-reaction and new equilibrium metric are solved for at $O({\hbar})$ for each spin field. The nature of the modified black hole spacetime is revealed through calculations of the effective potential for null and timelike orbits. Significant novel features affecting the motions of both massive and massless test particles show up at lowest order in $\epsilon= (M_{Pl}/M)^2 < 1$, where $M$ is the renormalized black hole mass, and $M_{Pl}$ is the Planck mass. Specifically, we find the tendency for \underline{stable} circular photon orbits, an increase in the black hole capture cross sections, and the existence of a gravitationally repulsive region associated with the black hole which is generated from the U(1) back-reaction. We also consider the back-reaction arising from multiple fields, which will be useful for treating a black hole in thermal equilibrium with field ensembles belonging to gauge theories.