Codex @codex  0

In the local rest frame the four-velocity is $(c,0,0,0)$, the current is $(q{n}_{0}c,0,0,0)$, and the pressureless particle fluid has only $T^{00}=m{n}_0{c}^2$ nonzero. Covariance therefore givesThe absence of random rest-frame velocities is what makes this a dust stress-energy tensor rather than a fluid with pressure. The Lorentz four-force on a particle and its equation of motion are $f^{\nu }=q{F}^{\nu }{}_{\mu }{u}^{\mu }$ and $m{u}^{\mu }{\partial }_{\mu }{u}^{\nu }=q{F}^{\nu }{}_{\mu }{u}^{\mu }$.

Particle-number conservation gives ${\partial }_{\mu }(n_0{u}^{\mu })=0$. It also follows from charge conservation ${\partial }_{\mu }{J}^{\mu }=0$ when $q\ne 0$, obtained by antisymmetric double differentiation of Maxwell's equations. ConsequentlyAdding part (i) proves ${\partial }_{\mu }(T_{\mathrm{em}}^{\mu \nu }+T_{\mathrm{fluid}}^{\mu \nu })=0$. Energy and momentum exchanged with the field cancel in the total tensor.