  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.
