Gravitational shock waves and vacuum fluctuations

Abstract

We show that the vacuum expectation value of the stress–energy tensor of a scalar particle on the background of a spherical gravitational shock wave does not give a finite expression in second-order perturbation theory, in contrast to the case seen for the impulsive wave. No infrared divergences appear at this order. This result shows that there is a qualitative difference between the shock and impulsive wave solutions which is not exhibited in first order. PACS numbers: 0420J, 0462, 9880 Both from physical and mathematical points of view, the cosmic string solutions [1] of Einstein's field equations are interesting [2]. An immediate question is whether these strings decay. Exact solutions describing such decays are given for impulsive waves by Nutku and Penrose [3] and Gleiser and Pullin [4], and for shock waves by Nutku [5]. Once these solutions are found one may question whether they give rise to vacuum fluctuations. We have investigated these fluctuations in several papers. In first-order perturbation theory, we have found that we could not isolate a finite part for the vacuum expectation value (VEV) of the stress–energy tensor both for the impulsive [6] and shock-wave solutions [7]. When the calculation is carried out to second order for the impulsive wave case, a finite result is found [8] if a detour is taken to de Sitter space. The essential point in this calculation is the generation of an infrared divergence in second-order perturbation theory which is regulated by an infrared mass. We go to de Sitter space and there cancel this mass by the cosmological constant. At the end we let both the infrared mass and the cosmological constant go to zero and obtain a finite result. In this addendum we carry out the calculation in the background of the shock-wave metric to second order and investigate whether the same trick gives us a finite expression for the VEV of the stress–energy tensor for this case. If the calculation does not generate an infrared divergence, going to de Sitter space gives us a finite result only in this space which vanishes when we go back to Minkowski space [9]. In first-order perturbation theory both the impulsive and the shock-wave cases showed similar behaviour. The two solutions are essentially different, though. The shock-wave solution has a dimensional constant which is lacking in the impulsive wave solution. Since in quantum field theory models with dimensional and dimensionless constants belong to different classes, we thought a similar distinction may exist between these two models. Relying on these motivations we planned to check whether there is a qualitative difference between the two solutions exhibited by their behaviour at higher orders.