Quantum field theory in a colliding plane-wave background.

Abstract

Although linear quantum field theory on the background of a single gravitational plane wave is trivial, colliding plane-wave solutions are likely to feature interesting quantum effects; these might yield insight into similar effects in other more general inhomogeneous and dynamical spacetimes. This paper presents the initial results of an investigation into the behavior of quantum fields in colliding plane-wave backgrounds. Throughout the paper, we restrict our attention to the analysis of quantum field theory on the Khan-Penrose spacetime. Since spacetime is flat before the arrival of either plane wave, there exists a unique, well-defined set of in modes and a corresponding in vacuum state. We introduce a physically plausible prescription for constructing a unique canonical set of out-mode solutions, and we evaluate the Bogolubov transformation between the in and out modes explicitly for a massless scalar field propagating on the Khan-Penrose spacetime. We then use these results to approximately compute the spectrum of created particles in the out region. Next, we study the quantity $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉\ensuremath{\equiv}〈0, \mathrm{in}|{T}_{\ensuremath{\mu}\ensuremath{\nu}}|0, \mathrm{in}〉$, the renormalized in-vacuum expectation value of the stress-energy tensor for a massless, conformally coupled ($\ensuremath{\xi}=\frac{1}{6}$) scalar field. In a colliding plane-wave spacetime, $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ vanishes everywhere except in the interaction region. To compute $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ in the interaction region, we make a number of assumptions about its general form; these assumptions are entirely reasonable for the specific geometry of the Khan-Penrose spacetime, but they may not hold for a general colliding plane-wave solution. Combined with the conservation property of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ and the choice of $\ensuremath{\xi}=\frac{1}{6}$, our assumptions reduce the determination of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ throughout the interaction region to the solution of a coupled system of first-order partial differential equations for two functions. These equations cannot be solved exactly; but they can be used to obtain crucial information on the behavior of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ near the singularity of the Khan-Penrose spacetime. Although our method of computing $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ is unlikely to be adequate for other colliding plane-wave solutions we use the information obtained through our calculations to speculate about $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$ in more general colliding gravitational-wave spacetimes. We argue that these speculations have important consequences for cosmology, but they must be verified by further calculations.