Macroscopic approach to the Casimir friction force

The general formula is derived for the vacuum friction force between two parallel perfectly flat planes bounding two material media separated by a vacuum gap and moving relative to each other with a constant velocity $\mathbf{v}$. The material media are described in the framework of macroscopic electrodynamics whereas the nonzero temperature and dissipation are taken into account by making use of the Kubo formulae from non-equilibrium statistical thermodynamics. The formula obtained provides a rigorous basis for calculation of the vacuum friction force within the quantum field theory methods in the condensed matter physics. The revealed $v$-dependence of the vacuum friction force proves to be the following: for zero temperature ($T=0$) it is proportional to $(v/c)^3$ and for $T>0$ this force is linear in $v/c$.