Hamilton–Jacobi tunneling method for dynamical horizons in different coordinate gauges

Abstract

Previous work on dynamical black hole instability is further elucidated within the Hamilton–Jacobi method for horizon tunneling and the reconstruction of the classical action by means of the null expansion method. Everything is based on two natural requirements, namely that the tunneling rate is an observable and therefore it must be based on invariantly defined quantities, and that coordinate systems which do not cover the horizon should not be admitted. These simple observations can help to clarify some ambiguities, like the doubling of the temperature occurring in the static case when using singular coordinates and the role, if any, of the temporal contribution of the action to the emission rate. The formalism is also applied to FRW cosmological models, where it is observed that it predicts the positivity of the temperature naturally, without further assumptions on the sign of energy.

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

“…As showed in [16], this result is invariant since the quantities appearing in the imaginary part are manifestly invariant. As a consequence, we may interpret T = −κ H /2π as the dynamical temperature associated with FRW space-times.…”

“…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…”

“…As an application of the last formula we may derive, following [16], the cosmic horizon tunneling rate. To this aim, as we have anticipated, the energy scale is such that near the horizon, we may neglect the particle's mass, and note that radially moving massless particles follow a null direction.…”

“…In order to deal with the simple pole in the integrand, we implement Feynman's iǫ -prescription. In the final result, beside a real (irrelevant) contribution, we obtain the following imaginary part [16] …”