Quantum self-propulsion of an inhomogeneous object out of thermal equilibrium

In an earlier paper, we explored how quantum vacuum torque can arise: a body or nanoparticle that is out of thermal equilibrium with its environment experiences a spontaneous torque. But this requires that the body be composed of nonreciprocal material, which seems to necessitate the presence of an external influence, such as a magnetic field. Then the electric polarizability of the particle has a real part that is nonsymmetric. This effect occurs to first order in the polarizability. To that order, no self-propulsive force can arise. Here, we consider second-order effects, and show that spontaneous forces can arise in vacuum, without requiring exotic electromagnetic properties. Thermal nonequilibrium is still necessary, but the electric susceptibility of the body need only be inhomogeneous. We investigate four examples of such a body: a needle composed of distinct halves; a sphere and a ball, each hemisphere being made of a different substance; and a thin slab, each face of which is different. The results found are consistent with previous numerical investigations. Here, we take into account the skin depth of metal surfaces. We also consider the frictional forces that would cause the body to acquire a terminal velocity, which might be observable. More likely to be important is relaxation to thermal equilibrium, which can still lead to a terminal velocity that might be experimentally verifiable. A general treatment of such forces on a moving body, expressed in momentum space, is provided, which incorporates both propulsive and frictional forces. The source of the propulsive force is the nonsymmetric pattern of radiation from different parts of the body, the higher reflectivity of the metal portion playing a crucial role.