Quantum Vavilov-Cherenkov radiation from shearing two transparent dielectric plates

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 $v_c=2nc\mathrm{/}(n^2+1)$. 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. $\mathbf{\text{45}}$, 2389 (1998)). However, in the ultra relativistic case ($v\to c$), 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.