Temporal contribution to gravitational WKB-like calculations

Abstract

Recently, it has been shown that the radiation arising from quantum fields placed in a gravitational background (e.g. Hawking radiation) can be derived using a quasi-classical calculation. Here we show that this method has a previously overlooked temporal contribution to the quasi-classical amplitude. The source of this temporal contribution lies in different character of time in general relativity versus quantum mechanics. Only when one takes into account this temporal contribution does one obtain the canonical temperature for the radiation. Although in this letter the specific example of radiation in de Sitter space-time is used, the temporal contribution is a general contribution to the radiation given off by any gravitational background where the time coordinate changes its signature upon crossing a horizon. Thus, the quasi-classical method for gravitational backgrounds contains subtleties not found in the usual quantum mechanical tunneling problem.

Quantum tunneling from three-dimensional black holes

“…Thus they will not violate unitarity. This is taken care of by the temporal contribution to the imaginary part of the action [22,23].…”

Charged Particle’s Tunneling in a Modified Reissner-Nordstrom Black Hole

“…Further, some resent progress and insights for the tunneling program have been proposed [19][20][21][22][44][45][46][47][48][49][50][51][52][53][54]. Therein, with the aid of density matrix techniques, the tunneling mechanism was reformulated to directly find the radiation spectrum [44,45] and the approach was applied in different gravity to obtain the entropy spectrum of black holes [46].…”

“…It is noteworthy that, by this approach, the black body spectrum was obtained [21,22,44,45]. In addition, some proposals on the calculation of particle's action and the acquisition of black hole information have been proposed [47][48][49][50][51][52][53][54]. But, in the literatures, much less attention was paid to the case of charged particle's tunneling in the quantum corrected spacetime [42].…”

Quantum tunneling from a high dimensional Gödel black hole

“…[51][52][53][54][55], the expression of the tunneling rate Γ ∝ exp −Im P dr , instead of Γ ∝ exp −2Im P dr is invariant under canonical transformations. So the fermion's tunneling rate of the five-dimensional Gödel black hole can be written as…”