  Using a fully relativistic theory we study the quantum Vavilov-Cherenkov
radiation and quantum friction occurring during relative sliding of the two
transparent dielectric plates with the refractive index $n$. These phenomena
occur above the threshold velocity $v_c=2nc/(n^2+1)$. Close to the threshold
velocity they are dominated by the contribution from $s$--polarized
electromagnetic waves, which agrees with the approximate (relativistic) theory
by Pendry (J. Mod. Opt. \textbf{45}, 2389 (1998)). However, in the ultra
relativistic case ($v\rightarrow c$), the contributions from both polarisations
are strongly enhanced in the comparison with the approximate theory, and a new
contribution occurs from the mixing of the electromagnetic waves with the
different polarization. The numerical results are supplemented by an analytical
treatment close to the threshold velocity and the light velocity.
