  Lorentzian-type media support optical phonons that oscillate with
longitudinal polarization parallel to the wave direction, at a wave
vector-independent frequency at which the permittivity becomes zero. Here, we
study the interactions between the longitudinal optical phonons and Lorentzian
medium-based dispersive photonic time crystals (PTCs). We demonstrate that a
stationary charge embedded in the PTCs can excite these longitudinal modes
through the conversion of the static polarization field induced by the charge.
Furthermore, the PTCs can develop a momentum bandgap across the entire wave
vector space to amplify the longitudinal modes. Remarkably, this infinite
momentum bandgap can be established with minimal temporal modulation of the
refractive index when creating the PTCs. Our approach expands the range of
waves that can be manipulated in PTCs and shows potential for observing
momentum bandgap phenomenon in realistic optical experiments, where the
modulation depth of the refractive index is severely constrained.
