Lecture 9

In the last lecture, we considered the modern information paradox. This starts with Page's observation that although unitarity tells us that the final radiation will be pure, we can get a stronger result by adding the assumption that the black hole is a "generic state". In this situation, we expect that the von Neumann entropy of the outgoing radiation will start to decrease after, what is called the "Page time" (after roughly "half" the black hole has evaporated) and continue to decline monotonically to zero (at which point the outgoing radiation is pure). This observation and the strong subadditivity of entropy leads to the "strong subadditivity paradox".

This paradox can be resolved by black hole complementarity: the hypothesis that the exterior and the interior of the black hole are not made up of exactly independent degrees of freedom.

We then considered paradoxes based on the seemingly unusual properties of operators behind the horizon: creation operators behind the horizon have negative energy, but yet the associated number operator has thermal occupancy. We showed how these paradoxes could all be resolved through state-dependence.

Make sure to see the Mathematica file, which shows how the "strong subadditivity" and other paradoxes can all be resolved in a simple and concrete setting.


The Mirror Operators in a Spin Chain (Mathematica File)