In this lecture, we proceeded to derive the formula for Hawking radiation in two different ways. First, we examined Hawking's original geometrical optics approach. This is useful because it gives us insight into where the radiation at future infinity originates from. Then, we started a more precise derivation using the properties of correlation functions in any smooth geometry. The geometry of the collapsing shell tells us that field correlators in position space have to have some universal properties. In turn, this implies a specific correlation for the modes that make up the field and yields the formula for Hawking radiation. The advantage of the latter derivation is that it proceeds from well defined assumptions and properties of field-correlators and gives a precise result.
Also included below is the Mathematica file that calculates null geodesics in the Oppenheimer-Snyder geometry.
lecture_7_serc_school.pdf (Lecture Notes)