Lecture 16: A Local Derivation of Hawking Radiation
The ray-tracing derivation of Hawking radiation suffers from a trans-Planckian problem. The frequencies that we see at late times originate from very blueshifted frequencies in the past. To sidestep this issue, we derived the formula for Hawking radiation using only local assumptions.
Our main assumption was that the future horizon remains smooth. This is very reasonable --- it is simply the statement that quantum corrections are small and do not affect the geometry at leading order. Then, by examining the two point function of a scalar field and demanding that it remain regular as we approach the horizon, we were able to show that the final state of the scalar field must be thermal.
This derivation also gave us an understanding of greybody factors, which were not visible in the ray-tracing derivation.
