Vacuum for a massless quantum scalar field outside a collapsing shell in anti-de Sitter space-time

TL;DR

This paper analyzes the behavior of a massless quantum scalar field outside a collapsing shell forming a Schwarzschild-anti-de Sitter black hole, showing the transition from vacuum to thermal equilibrium at late times.

Contribution

It models the quantum field dynamics during gravitational collapse in AdS space and demonstrates the emergence of a thermal state analogous to the Hartle-Hawking vacuum.

Findings

Quantum field starts in Boulware vacuum before collapse

Field reaches thermal equilibrium at Hawking temperature after collapse

Reflecting boundary conditions influence the field's late-time state

Abstract

We consider a massless quantum scalar field on a two-dimensional space-time describing a thin shell of matter collapsing to form a Schwarzschild-anti-de Sitter black hole. At early times, before the shell starts to collapse, the quantum field is in the vacuum state, corresponding to the Boulware vacuum on an eternal black hole space-time. The scalar field satisfies reflecting boundary conditions on the anti-de Sitter boundary. Using the Davies-Fulling-Unruh prescription for computing the renormalized expectation value of the stress-energy tensor, we find that at late times the black hole is in thermal equilibrium with a heat bath at the Hawking temperature, so the quantum field is in a state analogous to the Hartle-Hawking vacuum on an eternal black hole space-time.