# Electromagnetic stress-energy-momentum tensor and stress tensor

The purely spatial components of the energy-momentum-stress tensor for a perfect fluid are clearly related the components of the ordinary stress tensor inside the fluid. My question if this case is general, for example, if we consider a dielectric material subjected to an external electromagnetic field, is there any relation between the internal mechanical stresses of the solid and the components of the electromagnetic energy-impulse tensor? For example, for a a dielectric solid in equilibrium, is it true that:

$$\frac{\partial T^{ab}}{\partial x^a} + \frac{\partial \sigma^{ab}}{\partial x^a} = 0, \qquad a,b\in\{1,2,3\}$$

where $$T^{ab}$$ is the electromagnetic energy-momentum-stress and $$\sigma^{ab}$$ is the mechanical stress tensor.

• If we simply add the mechanical and electromagnetic Lagrangians and apply the homogeneity of space (See sec. 7 "Momentum" of Landau and Lifshitz "Mechanics" book), I am guessing that the equation you have shown holds. Aug 6 at 4:36