The Cerenkov radiation and the Smith-Purcell effect state that free electron
emission occurs exclusively in dielectrics when the velocity of the particles
exceeds the speed of light in the medium or in the vicinity of periodic
gratings close to each other within a vacuum. We demonstrate that free
electrons in a vacuum can also emit highly directional monochromatic waves when
they are in close proximity to a medium that is periodically modulated
temporally, suggesting the existence of temporal Smith-Purcell effect. The
momentum band gaps of time-varying media, such as photonic time crystals
(PTCs), create new pathways for the injection of external energy, allowing the
frequency, intensity, and spatial distribution of the electromagnetic fields to
be controlled. Moreover, the PTC substrate enables the conversion of localized
evanescent fields into amplified, highly directional propagating plane waves
that are only sensitive to the velocity of particles and the modulation
frequency, which allows us to observe and utilize Cerenkov-like radiation in
free space. Our work exhibits significant opportunities for the utilization of
time-varying structures in various fields, including particle identification,
ultraweak signal detection, and improved radiation source design