Pure states and black hole complementarity

The future apparent horizon of a black hole develops large stress energy due to quantum effects, unless the outgoing modes are in a thermal density matrix at the local Hawking temperature. It is shown for generic pure states that the deviation from thermality is so small that infalling observers will see no drama on their way to the stretched horizon, providing a derivation of black hole complementarity after the Page time. Atypical pure states, and atypical observers, may of course see surprises, but that is not surprising.

Figure 1

Scalar mode fluctuation near the horizon. Here we set $M=1$ and $\omega =1$.

Figure 2

The expectation value of an ingoing fluctuation contracted with the velocity of an infalling timelike geodesic near the future horizon. Here we set $M=1$ and $\omega =1$.

Figure 3

The Penrose diagram for the outgoing Vaidya metric. The lower dashed line shows the apparent horizon below the global horizon. The upper dot-dashed line indicates where the metric is geodesically incomplete and may be patched onto a variety of interior solutions. The dotted timelike line shows the position of the stretched horizon.