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According to Wikipedia, greybody factors are corrections to the black hole Hawking radiation spectrum. They say that at the horizon the emission is that of a perfect black body, but the gravitational potential well makes so that the spectrum at infinity does not correspond to a black body anymore. Greybody factors give this correction.

However, standard derivations of Hawking radiation (such as Hawking's original 1975 paper or the one on Wald's book) do not give the particle spectrum at the horizon. Rather, they make a correspondence between modes at past and future null infinities. Thus, the original calculation already seems to take the potential well into account and get a perfect black body spectrum.

My question then is: what is wrong in the previous two paragraphs and how can we reconcile them with each other?

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    $\begingroup$ The standard derivation does have greybody factors, and so does Wald's treatment: it's essentially what he calls $|t|^2$ in equation 14.3.7. When people say Hawking radiation has temperature $T_H$, they mean that if you put in black hole in a giant box containing a radiation bath at that temperature, the system would be in thermal equilibrium, with the black hole radiating particles at the same rate as it emits them. Both the radiation and emission rates, for a given particle energy, spin, and charge, depend on the greybody factor. $\endgroup$
    – knzhou
    Commented Feb 20 at 23:24
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    $\begingroup$ Greybody factors are in Hawking's original paper too, he called them $\Gamma_{j n}$ in equation (2.29). Actually, I'm not sure I've ever seen a work about Hawking radiation that didn't include greybody factors. $\endgroup$
    – knzhou
    Commented Feb 20 at 23:29
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    $\begingroup$ @knzhou I think your comments should be an answer $\endgroup$
    – Níckolas Alves
    Commented Feb 21 at 0:00

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