Observer energy conditions for selected matter plus EM

Use the smooth electromagnetic stress, observer contractions and selected matter tensors of Smooth matter quantities, current and observer observables, with ${\mu }_0>0$ and a future unit observer $n$. The electromagnetic tensor is symmetric and trace-free, and it satisfies WEC, NEC, SEC and DEC. In a positively oriented orthonormal rest tetrad, with the cross product taken in the oriented Euclidean rest space, its energy density and spatial energy-current components are

$u=\frac{{ϵ}_0\mathrm{\mid }{E}_n{\mathrm{\mid }}^2+\mathrm{\mid }{B}_n{\mathrm{\mid }}^2\mathrm{/}{\mu }_0}{2},{\mathcal{S}}_n=\frac{E_n\times B_n}{{\mu }_{0}c}\mathrm{.}$

For charged dust with $\rho \ge 0$, the matter tensor and its sum with the electromagnetic tensor satisfy all four conditions. For a selected perfect-fluid tensor with rest energy density $ϵ$ and pressure $p$, the exact pointwise conditions on the matter tensor are

$\mathrm{W}\mathrm{E}\mathrm{C}⟺ϵ\ge 0\text{ and }ϵ+p\ge 0,\mathrm{N}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0,$

$\mathrm{S}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0\text{ and }ϵ+3p\ge 0,\mathrm{D}\mathrm{E}\mathrm{C}⟺ϵ\ge \mathrm{\mid }p\mathrm{\mid }\mathrm{.}$

If the perfect-fluid tensor satisfies any one of these conditions, adding the electromagnetic tensor preserves that condition. The inequalities are not assumed universally, and adding the electromagnetic tensor to arbitrary matter need not repair a violated condition.