Invariance of Positive-Frequency Kernels in Generalized FRW Spacetimes

TL;DR

This paper investigates the invariance properties of positive-frequency kernels in generalized FRW spacetimes, showing that under broad conditions, all such kernels are related by unitary transformations that preserve spacetime symmetries.

Contribution

It demonstrates that, except for special scale factor evolutions, all positive-frequency kernels in these spacetimes are connected by symmetry-preserving unitary transformations.

Findings

All admissible kernels are related by unitary transformations.

Kernels are invariant under spacetime isometries.

Invariance holds generally, with exceptions for special scale factors.

Abstract

We consider the Klein-Gordon equation in FRW-like spacetimes, with compact space sections (not necessarily isotropic neither homogeneous). The bi-scalar kernel allowing to select the positive-frequency part of any solution is developed on mode solutions, using the eigenfunctions of the three-dimensional Laplacian. Of course this kernel is not unique but, except (perhaps) when the scale factor undergoes a special law of evolution, the metric has no more symmetries (connected with the identity) than those inherited from the space sections. As a result, all admissible definitions of the positive-frequency kernel are related one to another by a unitary transformation which commutes with the connected isometries of spacetime; any such kernel is invariant under these isometries isometries. A physical interpretation is tentatively suggested.