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.

In this Letter we make a critique of, and comparison between, the anomaly method and WKB/tunneling method for obtaining radiation from non-trivial spacetime backgrounds. We focus on Rindler spacetime (the spacetime of an accelerating observer) and the associated Unruh radiation since this is the prototype of the phenomena of radiation from a spacetime, and it is the simplest model for making clear subtle points in the tunneling and anomaly methods. Our analysis leads to the following conclusions: (i) neither the consistent and covariant anomaly methods gives the correct Unruh temperature for Rindler spacetime and in some cases (e.g. de Sitter spacetime) the consistent and covariant methods disagree with one another; (ii) the tunneling method can be applied in all cases, but it has a previously unnoticed temporal contribution which must be accounted for in order to obtain the correct temperature.

Unruh radiation is the thermal flux seen by an accelerated observer moving through Minkowski spacetime. In this article we study Unruh radiation as tunneling through a barrier. We use a WKB-like method to obtain the tunneling rate and the temperature of the Unruh radiation. This derivation brings together many topics into a single problem -classical mechanics, relativity, relativistic field theory, quantum mechanics, thermodynamics and mathematical physics. Moreover, this gravitational WKB method helps to highlight the following subtle points: (i) the tunneling rate strictly should be written as the closed path integral of the canonical momentum; (ii) for the case of the gravitational WKB problem, there is a time-like contribution to the tunneling rate arising from an imaginary change of the time coordinate upon crossing the horizon. This temporal contribution to the tunneling rate has no analog in the ordinary quantum mechanical WKB calculation.

We consider the WKB/tunneling method explicitly in the context of its application to the Rindler and de Sitter space-times. We also present two gravitational anomaly methods (consistent and covariant). The anomaly methods applied to the Rindler space-time do not reproduce the expected Unruh radiation, because the Rindler space-time does not have an anomaly. The consistent and the covariant anomaly methods give different results for the de Sitter space-time. In contrast to them, the WKB/tunneling method is a semiclassical calculation in which the radiation is regarded as a tunneling of quantum fields across the horizon. The tunneling method is applicable in all the indicated cases. But to recover the correct Gibbons-Hawking temperature, a previously overlooked temporal piece that contributes to the total action must be taken into account.

No abstract