The dust matter model: rest density, four-velocity, stress-energy and dust system

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $U\subseteq M$ be a nonempty open subset, the matter domain, carrying its restricted smooth structure (An open subset of a smooth manifold has a canonical restricted smooth structure).

(a) Data. A dust model on $U$ is a pair $(\rho ,u)$ consisting of

• a smooth function $\rho :U\to \mathbb{R}$ with $\rho (p)>0$ for every $p\in U$, the rest density; and
• a smooth future-directed unit timelike vector field $u$ on $U$ ($C^r$ and smooth maps between smooth manifolds with target $TM$ described through the tangent bundle; concretely, $u$ is a smooth vector field on $U$ with $g_p(u_p,u_p)=-1$ and $g_p(u_p,N_p)<0$ at every $p$, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

The parameter $\rho$ is the coefficient in the unit-normalized tensor of clause (b); this mathematical definition assigns it no physical unit. In the framework's SI stress-energy reading it is the rest energy density $\epsilon :=\rho$, in $\mathrm{J}{\mathrm{m}}^{-3}$. The rest-mass density is instead ${\rho }_m:=\epsilon \mathrm{/}{c}^2$, in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, and the physical four-velocity is $U:=cu$, with $g(U,U)=-c^2$. Thus the same tensor is $T^{ab}=\epsilon u^a{u}^b={\rho }_m{U}^a{U}^b$. A purely mathematical or $c=1$ convention may rescale the density coefficient; interpreting $\rho$ as an SI mass density while keeping the unit tangent $u$ would require the explicit tensor $c^2\rho u\otimes u$. Physical realizations and readings are adopted by the matter postulate, not deduced here.

(b) Stress-energy tensor. The model's stress-energy tensor has the smooth symmetric contravariant components and their covariant lowering

$T^{ab}:=\rho u^a{u}^b\text{equivalently}{T}_{ab}=\rho u_a{u}_b,$

with $u_a=g_{ac}{u}^c$ (The inverse metric, musical isomorphisms and index raising, Smooth covariant 2-tensor fields on a smooth manifold); the two expressions correspond under raising both indices, by $g^{ac}{g}^{bd}{T}_{cd}=T^{ab}$.

(c) Dust system. The dust system on $U$ is the pair of equations

${\mathrm{\nabla }}_a(\rho u^a)=0,u^a{\mathrm{\nabla }}_a{u}^b=0.$

The first is a continuity equation for the density current $\rho u$; the second says that the integral curves of $u$ are geodesics in the parameter that makes $u$ their unit tangent, i.e. that the dust particles are in free fall (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

• What is and is not asserted. The definition introduces the data, the tensor and the system; it asserts neither that the dust system holds, nor that $T$ is divergence-free, nor that $T$ is the Hilbert tensor of any action. The equivalence of the dust system with ${\mathrm{\nabla }}^a{T}_{ab}=0$ (where $\rho >0$) is proved in Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations.

• Smooth-branch restriction. Caustics or shocks, where the congruence $u$ fails to be a smooth unit field or $\rho$ ceases to be smooth, lie outside this model; a pressureless fluid with vanishing density in some region does not determine $u$ there, which is why $\rho >0$ on the whole matter domain is part of the data.