Vacuum polarization and stress-energy of a quantum field inside of two-dimensional black holes

Abstract

Quantum effects are studied in both Schwarzschild spacetime and a spacetime in which a null shell collapses to form a black hole via the vacuum polarization $\langle \phi^2 \rangle$ and stress-energy tensor $\langle T_{ab} \rangle$ for a massless minimally-coupled scalar field in two dimensions. For Schwarzschild spacetime, the Boulware, Unruh, and Hartle-Hawking states are considered. For the collapsing null shell spacetime, the \textit{in} vacuum state is used. Instabilities of the Unruh, Hartle-Hawking, and \textit{in} states resulting from the behavior of $\langle \phi^2 \rangle$ in the regions inside and outside of the horizon are found. The question of how well the Unruh state for the eternal black hole approximates quantum effects in the interior of a black hole that forms from collapse is addressed.