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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.

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  • $\begingroup$ 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. $\endgroup$
    – HEMMI
    Aug 6 at 4:36

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