Vacuum Polarization and Persistence on the Black Hole Horizon

TL;DR

This paper derives an exact one-loop effective action for a Schwarzschild black hole, linking vacuum polarization and persistence to Hawking radiation, revealing thermal and flux properties of quantum fields near the horizon.

Contribution

It provides a novel exact calculation of the effective action in black hole spacetime using the proper-time integral and Schwinger variational principle, connecting vacuum effects to Hawking radiation.

Findings

Vacuum persistence equals the total Hawking radiation flux.

Vacuum polarization exhibits a thermal distribution.

Effective action parallels the Heisenberg-Euler form in electric fields.

Abstract

We find the exact one-loop effective action in a Schwarzschild black hole in the proper-time integral from the Schwinger variational principle, which has the same form up to number of states as the Heisenberg-Euler and Schwinger QED effective action in a constant electric field. The leading term for Hawking radiation comes from the first simple pole of the vacuum polarization and the sum of residues from all simple poles exactly leads to the vacuum persistence. We show that the vacuum persistence is the total flux of Hawking radiation for bosons and fermions and the vacuum polarization has an expression of thermal distribution and the propagator.