Particle number conservation in a relativistic fluid (${\nabla }_a(n{u}^a)=0$)

The number current is $J^a=n{u}^a$, where $n$ is the rest-frame number density and $u^a$ the unit four-velocity. Its covariant continuity equation is ${\nabla }_a{J}^a=0$, equivalently $u^a{\nabla }_{a}n+n{\nabla }_a{u}^a=0$. Integration over a spacetime volume and the divergence theorem equate the particle numbers crossing its boundary. This conservation law is separate from stress-energy conservation.