Stokes phenomenon and Hawking Radiation

We compute the semiclassical decay rate for Kerr black hole by deriving a one-way connection formula, relating the near horizon solution to the outgoing solution at infinity. In particular, we discuss the relevance of the Stokes phenomenon and show how it leads to a Boltzmann-like thermal weight factor by making use of the Stokes diagrams. We also give the exact result for the semiclassical greybody factor $e^{-2S},$ where $S$ is the leading order WKB action. We contrast our results with Maldacena and Strominger's work [Phys. Rev. D 56, 4975 (1997)], where the emission spectrum for a rotating black hole was computed locally via asymptotic matching. We find that the relative error of semiclassical decay rate with respect to asymptotic matching formula diminishes in the limit of large angular momentum, $l$, as expected. In this limit, the action assumes a compact form: $2S \sim (2 l+1) (\text{Log} \left( \frac{16 }{z_0}\right)-1)$, where $z_0$ is the cross ratio formed by the critical points (zeroes) of the scattering potential.