Analogous of Hartman Effect for Relativistic Particles Through “Transparent” Barrier

The phase time in a nonrelativistic monodimensional tunneling is, for short barrier length, greater than the time it takes for the light in vacuum to travel the same path. Nevertheless, if the length is increased to the limit of “opaque” barrier (κL ≫ 1), where κ is the decay constant of the evanescent wave, the phase time becomes independent from the length (Hartman effect), so that apparent superluminal velocities (independently from unresolved questions about their physical meaning) are possible increasing length in this limit. Here is analytically demonstrated that apparent superluminal behavior can be found also in another situation: for highly relativistic particles, in the limit of “transparent barrier” (κL ≪ 1), is found that the phase time is always shorter than the correspondent time for light in vacuum, and that apparent superluminal velocities (following the Hartman definition) are always possible independently from the barrier length.