On the origin of the particles in black hole evaporation

Abstract

We present an analytic derivation of Hawking radiation for an arbitrary (spatial) dispersion relation $ω(k)$ as a model for ultra-high energy deviations from general covariance. It turns out that the Hawking temperature is proportional to the product of the group $dω/dk$ and phase $ω/k$ velocities evaluated at the frequency $ω$ of the outgoing radiation far away, which suggests that Hawking radiation is basically a low-energy phenomenon. Nevertheless, a group velocity growing too fast at ultra-short distances would generate Hawking radiation at ultra-high energies (``ultra-violet catastrophe'') and hence should not be a realistic model for the microscopic structure of quantum gravity.