Evolution of confined quantum scalar fields in curved spacetime. Part I

We develop a method for computing the Bogoliubov transformation experienced by a confined quantum scalar field in a globally hyperbolic spacetime, due to the changes in the geometry and/or the confining boundaries. The method constructs a basis of modes of the field associated to each Cauchy hypersurface, by means of an eigenvalue problem posed in the hypersurface. The Bogoliubov transformation between bases associated to different times can be computed through a differential equation, which coefficients have simple expressions in terms of the solutions to the eigenvalue problem. This transformation can be interpreted physically when it connects two regions of the spacetime where the metric is static. Conceptually, the method is a generalisation of Parker's early work on cosmological particle creation. It proves especially useful in the regime of small perturbations, where it allows one to easily make quantitative predictions on the amplitude of the resonances of the field, providing an important tool in the growing research area of confined quantum fields in table-top experiments. We give examples within the perturbative regime (gravitational waves) and the non-perturbative regime (cosmological particle creation). This is the first of two articles introducing the method, dedicated to spacetimes without boundaries or which boundaries remain static in some synchronous gauge.