Computation and visualization of Casimir forces in arbitrary geometries: nonmonotonic lateral-wall forces and the failure of proximity-force approximations

We present a method of computing Casimir forces for arbitrary geometries, with any desired accuracy, that can directly exploit the efficiency of standard numerical-electromagnetism techniques. Using the simplest possible finite-difference implementation of this approach, we obtain both agreement with past results for cylinder-plate geometries, and also present results for new geometries. In particular, we examine a pistonlike problem involving two dielectric and metallic squares sliding between two metallic walls, in two and three dimensions, respectively, and demonstrate nonadditive and nonmonotonic changes in the force due to these lateral walls.