Generalized Clausius-Mossotti mixing formula for 3D spacetime crystals

Here, we characterize the effective response of fully 3D travelling-wave spacetime crystals. To this end, we develop an analytical formalism to homogenize spacetime crystals with spherical inclusions, extending in this manner the renowned Clausius-Mossotti formula to spacetime crystals. Our theory shows that the spacetime crystals behave effectively as bianisotropic materials in the long wavelength limit. Moreover, it reveals the possibility of realizing a purely isotropic Tellegen (axion) response in a system formed by interlaced travelling-wave crystals. We introduce a novel class of media that display invariance under arbitrary Lorentz boosts along a fixed spatial direction. In particular, we prove that the most general class of reciprocal materials invariant under a Lorentz boost is formed by certain biaxial crystals.