The Energy-momentum Tensor in Fulling-rindler Vacuum

The energy density in Fulling-Rindler vacuum, which is known to be negative " everywhere " is shown to be positive and singular on the horizons in such a fashion as to guarantee the positivity of the total energy. The mechanism of compensation is displayed in detail. 1 1. Introduction In connection with the problem of the radiation emitted and absorbed by an accelerating observer [1, 2, 3, 10, 14], an interesting conceptual problem has arisen. What is the behavior of the energy density in Fulling-Rindler (FR) vacuum? We remind the reader that Minkovski vacuum presents itself as a thermal distribution of FR quanta within the quadrant containing the accelerating trajectory [1, 4, 5]. One often describes excitation of the detector in terms of absorbtion of these FR quanta so the above question naturally arises (i.e. what would the state look like when all FR quanta are absorbed?). Existent calculations [6, 7] in the literature give the result that the energy density in FR vacuum is negative. This has been interpreted as the absence of the thermal FR quanta which characterize Minkovski vacuum. When this density is integrated over all space, the total energy in FR vacuum would be therefore less than that of Minkovski vacuum. But this later is the ground state. In this paper, this dilemma is solved. We show that the energy density is singular and positive in such a fashion that the total (integrated) energy is positive. Our method proceeds through the introduction of an ultra violet regulator (ε) which permits the evaluation of the energy density close and on the horizon. As ε → 0, the energy density tends to a distribution. For all points off the horizon, this density is the negative value previously obtained. Yet its integral is positive. We note in passing that the same behaviour exists in Boulware vacuum near the Schwarzschild horizon [6, 7], as well as near the horizons of the static patch in de Sitter space, in the state of vacuum defined by the static system.