The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy

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 matter domain (An open subset of a smooth manifold has a canonical restricted smooth structure).

(a) Data. A perfect-fluid model on $U$ is a triple $(\epsilon ,p,n)$ consisting of two smooth functions $\epsilon ,p:U\to \mathbb{R}$, the rest energy density and the pressure, and a smooth future-directed unit timelike vector field $n$ on $U$ with $g_p(n_p,n_p)=-1$ and $g_p(n_p,N_p)<0$ at every $p$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, $C^r$ and smooth maps between smooth manifolds). No positivity of $\epsilon$ or of $p$ is required or implied. A model may carry the additional data of an equation of state: a function $P:D_P\to \mathbb{R}$ on a subset $D_P\subseteq \mathbb{R}$ with $\epsilon (U)\subseteq D_P$, together with the relation $p=P(\epsilon )$; the relation is a further assumption and is not part of the basic triple.

(b) Stress-energy tensor and derived fields. The model's stress-energy tensor is

$T^{ab}:=(\epsilon +p)n^a{n}^b+p{g}^{ab},$

a smooth symmetric contravariant $2$-tensor field (Smooth covariant 2-tensor fields on a smooth manifold, The inverse metric, musical isomorphisms and index raising). The comoving derivative, expansion and acceleration of $n$ are

$Df:=n^a{\mathrm{\partial }}_{a}f(f\in C^{\mathrm{\infty }}(U)),\vartheta :={\mathrm{\nabla }}_a{n}^a,a^b:=n^a{\mathrm{\nabla }}_a{n}^b,$

and the rest-space projection is the smooth tensor field

$h^{ab}:=g^{ab}+n^a{n}^b\mathrm{.}$

The physical reading of $\epsilon$ and $p$ as a rest-frame energy density and pressure, and their units, are fixed by the framework's matter postulate, not by this definition.

(c) Fluid equations. The perfect-fluid equations on $U$ are

$D\epsilon +(\epsilon +p)\vartheta =0,(\epsilon +p)a^b+h^{ba}{\mathrm{\partial }}_{a}p=0.$

That the stress-energy tensor is divergence-free if and only if these two equations hold is proved in Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations; the definition asserts neither divergence-freedom nor the equations.

• Projection convention. Since $n$ has unit length, $h^{ab}$ satisfies $h^{ab}{n}_b=0$ and $h^{ab}{g}_{bc}{h}^{cd}=h^{ad}$; it is the $g$-orthogonal projection onto the rest space $n^{\perp }$ raised with $g^{-1}$. The vector $a^b$ is automatically orthogonal to $n$: $n_b{a}^b=\frac{1}{2}{n}^a{\mathrm{\nabla }}_a(n_b{n}^b)=0$.

• Scope. A model with $\epsilon +p=0$ has a stress-energy tensor proportional to $g^{ab}$ on $U$; the equations above then degenerate and the acceleration is not determined by the pressure gradient. Shock waves, dissipation and constitutive transport lie outside this model, as do equations of state with phase transitions, unless the smooth model is restricted to a phase.