Particle creation by a nonstationary spacetime ($⟨N_i^{\mathrm{out}}{⟩}_{\mathrm{in}}={\sum }_j|{\beta }_{ij}{|}^2$)

Positive-frequency modes in stationary asymptotic past and future regions can differ by a Bogoliubov transformation mixing positive and negative frequency solutions. The conserved Klein-Gordon inner product gives its canonical normalization. The out annihilation operator then contains in creation operators, yielding the displayed number expectation in the initial vacuum. A nonzero mixing coefficient changes the particle content relative to final observers without violating deterministic field evolution.