Electromagnetic Energy Momentum Tensor in a Spatially Dispersive Medium

As interest in photonic crystals and metamaterials expands, it is becoming increasingly clear that these materials often exhibit strong spatial dispersion. Understanding spatial dispersion and its consequences is essential to understanding these materials. It has long been known that temporal dispersion alters the expression for the electromagnetic energy density of a material. Similarly, it is known that spatial dispersion alters the expression for the Poynting energy flux. A simple generalization of these arguments shows that temporal and spatial dispersion affect the Energy Momentum Tensor (EM Tensor) as a whole. Here we derive the entire EM Tensor for a monochromatic wave in a temporally and spatially dispersive medium with no loss. We then show that for a freely propagating wave the various components of the EM Tensor are related by the group and phase velocities. From the EM Tensor, we then derive more familiar expressions for the energy density, energy flux, momentum density, and momentum flux. Finally, we introduce a small amount of loss to see how this effects energy-momentum conservation.