Electromagnetic Boundary Conditions for Space--time Interfaces

We give a general family of electromagnetic boundary conditions applicable to arbitrary space–time interfaces between electromagnetic media, which include the known space–only and time–only boundary conditions as special cases. These boundary conditions describe a broad class of electromagnetic interfaces, including surfaces in arbitrary motion, ultra-thin (metasurface) media, and cases where the media on one or both sides of the boundary can be both spatially and temporally dispersive. Our approach utilizes 4-dimensional spacetime and addresses the question of how, and in what ways, an electromagnetic field may be connected across a 3-dimensional hypersurface. We show that our proposed boundary conditions are the most general conditions consistent with causality and linearity.