The normal Casimir-Lifshitz force for laterally moving graphene.

We consider the system of two parallel sheets of graphene which are moving with relative parallel velocity $\vec{v}$ and calculate the Casimir energy by using the scattering approach. We analyze in detail the normal (perpendicular to the planes) Casimir force for two systems -- graphene/graphene and ideal metal/graphene. In the non-relativistic case $v \ll v_F$, the relative correction to the Casimir energy $(\mathcal{E}_v - \mathcal{E}_0)/\mathcal{E}_0$ is proportional to the $(v/c)^2$ (the maximum value is $0.0033$ for the gapeless case and $v=v_F$) for the first system, and it is zero up to the Fermi velocity $v = v_F$ for system ideal metal/graphene.