  When an electromagnetic field is confined in a cavity of variable length,
real photons may be generated from vacuum fluctuations due to highly
nonadiabatic boundary conditions. The corresponding effective Hamiltonian is
time-dependent and contains infinite intermode interactions. Considering one of
the cavity mirrors fixed and the other describing uniform motion (zero
acceleration), we show that it is possible to factorize the entire temporal
dependency and write its formal solution, i.e., the Hamiltonian becomes a
product of a time-dependent function and a time-independent operator. With this
factorization, we prove in detail that the photon production is proportional to
the Planck factor involving a velocity-dependent effective temperature. This
temperature significantly limits photon generation even for ultra-relativistic
motion. The time-dependent unitary transformations we introduce to obtain
temporal factorization help establishing connections with the shortcuts to
adiabaticity of quantum thermodynamics and with the quantum Arnold
transformation.
