Casimir-Lifshitz force for 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. The energy has real and imaginary parts. The real part contains a velocity contribution to the force perpendicular to the sheets. The imaginary part is related to a friction force parallel to the graphene sheets. We prove that the friction appears if the relative velocity becomes greater than the velocity of Fermi and if the modulo of reflection coefficient is greater than the one which is significant for virtual photon production. We analyze in detail the real contribution to the Casimir energy 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.