= Relativistic dust
{title2=$T^{\mu\nu}=\rho u^\mu u^\nu$}
= Pressureless relativistic fluid
{synonym}
<Relativistic dust> is a continuum of noninteracting massive particles with one local <four-velocity> $u^\mu$ and negligible <pressure>. With signature $(+---)$, $u^\mu u_\mu=c^2$ and rest <mass density> $\rho$, its <stress-energy tensor> has the displayed form. Contracting <stress-energy conservation> with $u_\nu$ and using <metric compatibility> gives $c^2\nabla_\mu(\rho u^\mu)=0$. Substituting back yields $u^\mu\nabla_\mu u^\nu=0$ wherever $\rho>0$: each particle follows a <geodesic>. The cosmological <pressureless matter> model is a homogeneous application.
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