= Charged relativistic dust
{title2=$u^\mu\nabla_\mu u^\nu=(q/m)F^{\nu\alpha}u_\alpha$}
For identical particles of <rest mass> $m$ and <electric charge> $q$, the <four-current> is $j^\mu=(q/m)\rho u^\mu$. The matter <stress-energy tensor> obeys $\nabla_\mu T_{\rm m}^{\mu\nu}=F^{\nu\alpha}j_\alpha$, with the opposite force density in the electromagnetic <stress-energy tensor>. Antisymmetry of the <electromagnetic field tensor> makes its contraction with two <four-velocities> vanish, so mass continuity still holds. Removing that term leaves the displayed covariant <Lorentz force> law. The matter tensor alone is not divergence-free when an electromagnetic force acts.
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