Effective potential of a black hole in thermal equilibrium with quantum fields

Expectation values of one-loop renormalized thermal equilibrium stress-energy tensors of free conformal scalars, spin-1/2 fermions, and U(1) gauge fields on a Schwarzschild black hole background are used as sources in the semiclassical Einstein equation. The back reaction and new equilibrium metric have been found at O(ħ) for each spin field in previous work. In this paper, the nature of the modified black hole spacetime is explored 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 ε=(${\mathit{M}}_{\mathrm{Pl}}$/M${)}^2$M is the black hole mass, and ${\mathit{M}}_{\mathrm{Pl}}$ is the Planck mass. Specifically, we find an increase in the black hole capture cross sections, and the existence of a region near the black hole with a repulsive contribution, generated by the U(1) back reaction, to the gravitational force. There is no such effect for other spins. Extrapolating our results suggests a tendency towards the formation of stable circular orbits, but the result cannot be established in O(ħ): the change in the metric becomes large and it changes its signature. We also consider the back reaction arising from multiple fields, which ultimately should be useful for treating a black hole in equilibrium with field ensembles belonging to gauge theories. In certain circumstances, however, reliable results will require calculations beyond O(ħ).