For a globally hyperbolic manifold $(M, g)$ we can always pose a well defined Cauchy problem through a foliation in terms of pairs $(t, \Sigma_t)$ where $t$ acts as a time-coordinate and $\Sigma_t$ is a spacelike hypersurface labeled by the time-coordinate, i.e. a Cauchy surface. When doing (classical/ quantum) field theory in general backgrounds, this construction defines the field dynamics. So, for each constant-time "slice" there is a field configuration that is a solution of the EoM.
Consider a general coordinate transformation where the new time-coordinate becomes $t’$. Although a surface being spacelike doesn’t depend on the choice of coordinates, $\Sigma_{t’}$ and $\Sigma_t$ will in general not be the same surface.
In QFTCS, when one wants to compare the vacuum states for different coordinates, a step is to perform Bogoliubov transformation. This involves taking the Klein-Gordon inner product of different modes found for different coordinates at a spacelike surface. My, naive, question is why are we allowed to do this? As, I thought these modes belong to different Hilbert spaces and defined on different sets of spacelike surfaces.
