Local Hawking temperature for dynamical black holes

Abstract

A local Hawking temperature is derived for any future outer trapping horizon in spherical symmetry, using a Hamilton–Jacobi variant of the Parikh–Wilczek tunneling method. It is given by a dynamical surface gravity as defined geometrically. The operational meaning of the temperature is that Kodama observers just outside the horizon measure an invariantly redshifted temperature, diverging at the horizon itself. In static, asymptotically flat cases, the Hawking temperature as usually defined by the Killing vector agrees in standard cases, but generally differs by a relative redshift factor between the horizon and infinity, this being the temperature measured by static observers at infinity. Likewise, the geometrical surface gravity reduces to the Newtonian surface gravity in the Newtonian limit, while the Killing definition instead reflects measurements at infinity. This may resolve a long-standing puzzle concerning the Hawking temperature for the extremal limit of the charged stringy black hole, namely that it is the local temperature which vanishes. In general, this confirms the quasi-stationary picture of black-hole evaporation in early stages. However, the geometrical surface gravity is generally not the surface gravity of a static black hole with the same parameters.

Black Hole Entropy for Two Higher Derivative Theories of Gravity

“…On the other hand, that the horizons we are concerned are of the bifurcate type means we are in Wald's hypothesis in order to compute their entropy. In this sense, equations (31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43) and Figure 1 represent our main results.…”

“…In static, non-asymptotically flat spaces, they are both ambiguous and can differ by normalizations, but nonetheless the ratio "energy/surface gravity" remains fixed [38][39][40]. This means that as far as the Killing temperature associated with the black holes mentioned here is non-vanishing, also their Kodama-Hayward temperature will be so.…”

Applications of the tunneling method to particle decay and radiation from naked singularities

“…We recall that in previous papers [15,16] we considered the quantum instability of dynamical black holes using a variant of the tunneling method introduced by Parikh and Wilczek in the static case to uncover aspects of back-reaction effects [20]. These approaches are based on WKB relativistic method (see, for example [21] and more recently [22]) and only the leading terms of the production rate probability are taken into account, leaving untouched the pre-factor evaluation.…”

“…With regard to the pre-factor issue, we limit ourselves to mention Volovik's arguments according to which the pre-factor is likely to vanish in the case of horizon tunneling in dS space-time [23]. In our generalization of Volovik's calculation, the use of invariant quantities plays a crucial role [15,16]. In order to illustrate them, let us recall that any spherically symmetric metric can locally be expressed in the form…”

“…Note also that, on the horizon, E M S | H = 1 2 R H . Furthermore, one can introduce a dynamic surface gravity [15] associated with this dynamical horizon, given by the normal-space scalar…”