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

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 $⟨{\phi }^2⟩$ and stress-energy tensor $⟨T_{ab}⟩$ 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 $\mathit{\text{in}}$ vacuum state is used. Instabilities of the Unruh, Hartle-Hawking, and $\mathit{\text{in}}$ states resulting from the behavior of $⟨{\phi }^2⟩$ 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.