= Hawking exponential ray map
{c}
{title2=$U_H-U=Ae^{-\kappa u}$}
The late outgoing rays from a nonextremal collapsing <black hole> have an exponential relation between a regular early affine null coordinate $U$ and late retarded time $u$. Logarithmic divergence of the <tortoise coordinate> at the horizon gives $U_H-U=Ae^{-\kappa u}$ up to subleading terms. Backward propagation turns $e^{-i\omega u}$ into $(U_H-U)^{i\omega/\kappa}$; its positive- and negative-frequency Fourier parts give a thermal <Bogoliubov transformation>.
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