Hawking Temperature from Quasi-normal Modes

A perturbed black hole has characteristic frequencies (quasi-normal modes). Here I apply a quantum measurement analysis of the quasinormal mode frequency in the limit of high damping. It turns out that a measurement of this mode necessarily adds noise to it. For a Schwarzschild black hole, this corresponds exactly to the Hawking temperature. The situation for other black holes is briefly discussed. PACS numbers: 04.70.Dy, 03.65Ta Stationary black holes in four spacetime dimensions are characterized by just three numbers: mass, angular momentum, and electric charge. If a black hole is perturbed, however, it exhibits characteristic oscillations called quasinormal modes (QNMs) because they are damped in time (and thus have a complex frequency). They correspond to perturbations of the geometry (and other fields, if present) outside the horizon and obey the boundary conditions of being ingoing at the horizon and outgoing at infinity, see for example [1] for reviews. I shall restrict myself here to purely gravitational perturbations. Consider in the following a Schwarzschild black hole, which is fully characterized by its mass, M . Two limits for the quasi-normal modes are of particular interest. First, if the angular momentum, l, of the perturbation goes to infinity, one has for the QNM frequency (setting c = 1), ωl ∼ l+ 1 2 3 √ 3GM − i √ 3 ( n + 1 2 )