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

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.