  Recent research developments in the area of spacetime metamaterial structures
and systems have raised new questions as to how the physics of fundamental
phenomena is altered in the presence of spacetime modulation. In this context,
we present a generalized and comparative description of the phenomenon of total
internal reflection (TIR) at different dynamic interfaces. Such interfaces
include, beyond the classical interfaces corresponding to the boundaries of
moving bodies (moving interface -- moving matter systems), interfaces formed by
a traveling-wave step modulation of an electromagnetic parameter (e.g.,
refractive index) (moving interface -- stationary matter systems) and fixed
interfaces between moving-matter media (stationary interface -- moving matter
systems). We first resolve the problem using the evanescence of the transmitted
wave as the criterion for TIR and applying the conventional technique of
relative frame hopping (between the laboratory and rest frames), which results
in closed-form formulas for the relevant critical (incidence, reflection, phase
refraction and power refraction) angles. We then introduce the concept of
catch-up limit between the dynamic interface and the transmitted wave as an
alternative criterion for the critical angle. We use this approach both to
analytically verify the critical angle formulas, further validated by full-wave
(FDTD) analysis, and to explain the related physics, using Fresnel-Fizeau drag
and spacetime frequency transition considerations. These results might find
various applications in ultra-fast optics, gravity analogs and quantum
processing.
