A translating dust in flat space: measured energy density, flux and the local balance

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$ and the time orientation $N={\mathrm{\partial }}_0$, with the standard coordinate orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and the 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). Let $\beta \in (-1,1)$, put $\gamma :=(1-{\beta }^2{)}^{-1\mathrm{/}2}$, let $\phi :\mathbb{R}\to (0,\mathrm{\infty })$ be smooth, and define

$\rho (x):=\phi (x^1-\beta x^0),u:=\gamma ({\mathrm{\partial }}_0+\beta {\mathrm{\partial }}_1)\mathrm{.}$

Further let $n:={\mathrm{\partial }}_0$, and let $T^{ab}:=\rho u^a{u}^b$ (the tensor of the dust model $(\rho ,u)$, The dust matter model: rest density, four-velocity, stress-energy and dust system). Then:

1. $u$ is a smooth future-directed unit timelike vector field and $(\rho ,u)$ is a dust model on $M$; the dust system holds, so ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $M$.
2. $n$ is a smooth observer congruence with $a={\mathrm{\nabla }}_{n}n=0$ and $\theta ={\mathrm{\nabla }}_a{n}^a=0$.
3. The measured fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress for the congruence $n$ are

$\mu =\rho {\gamma }^2,J=\rho {\gamma }^2({\mathrm{\partial }}_0+\beta {\mathrm{\partial }}_1),\mathrm{\Phi }=\rho {\gamma }^2\beta {\mathrm{\partial }}_1,S=\rho {\gamma }^2{\beta }^2{\mathrm{\partial }}_1\otimes {\mathrm{\partial }}_1,$

and they satisfy the decomposition $T^{ab}=\mu n^a{n}^b+n^a{\mathrm{\Phi }}^b+n^b{\mathrm{\Phi }}^a+S^{ab}$ of The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness clause (b). 4. ${\mathrm{\nabla }}_a{J}^a=0$ on $M$, and the local balance of Divergence of the measured energy-momentum current and the finite-chain balance identity clause (b) holds:

$D\mu +\mu \theta +{\mathrm{\nabla }}_a{\mathrm{\Phi }}^a=0=-T^{ab}{\mathrm{\nabla }}_a{n}_b\mathrm{.}$

In this example the balance is the continuity equation for the translating density current $\rho \gamma ({\mathrm{\partial }}_0+\beta {\mathrm{\partial }}_1)$.

• Unit reading. In the SI stress-energy reading, $\rho =\epsilon$ is rest energy density in $\mathrm{J}{\mathrm{m}}^{-3}$, and ${\rho }_m=\rho \mathrm{/}{c}^2$ is rest-mass density. The values of $\mu$, $\mathrm{\Phi }$ and $S$ displayed above are the normalized component values of the tensor split. In particular $\mathrm{\Phi }$ is the spatial tensor current with energy-density component units, and its SI measured energy flux is $q=c\mathrm{\Phi }$ under the framework convention; the mathematical Example asserts the algebraic identities only and no measured value. The factor ${\gamma }^2$ is the familiar enhancement of the measured energy density of a moving medium relative to the congruence at rest in the chart.

• Nonzero flux with conserved charge. The current $J$ is divergence-free although its spatial part $\mathrm{\Phi }$ is nonzero: the translating profile carries energy across each constant-$x^0$ slice, and the balance records that the loss of energy in a region is compensated by the flux through its boundary, exactly as in Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.