The Killing energy of a static dust in flat space and its conservation across slices

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$, the time orientation $N={\mathrm{\partial }}_0$ and the chart orientation, 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). Let $\xi :={\mathrm{\partial }}_0$, let $\rho >0$ be constant with the framework reading $\rho =\epsilon$ in $\mathrm{J}{\mathrm{m}}^{-3}$ (rest-mass density ${\rho }_m=\rho \mathrm{/}{c}^2$), let $u:={\mathrm{\partial }}_0$ 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. $\xi$ is a Killing field on $M$ that is everywhere timelike and future-directed, with $X=\sqrt{-g(\xi ,\xi )}=1$ and congruence $n=\xi$; and the dust system holds, so ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $M$.
2. The Killing current of The timelike Killing current and the charge of a three-chain is $J_{\xi }{}^a=T^{ab}{\xi }_b=-\rho {\mathrm{\partial }}_0^a$, and its current $3$-form is ${\iota }_{J_{\xi }}{\mathrm{v}\mathrm{o}\mathrm{l}}_g=-\rho d{x}^1\wedge d{x}^2\wedge d{x}^3$.
3. For every finite smooth singular $3$-chain $\mathrm{\Sigma }$ of $M$ whose simplices have chart velocity vectors $({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)$ at every parameter point — in particular for simplices parametrized by the spatial chart coordinates on a constant-$x^0$ slice — the Killing energy of $\mathrm{\Sigma }$ is

$Q_{\xi }(\mathrm{\Sigma })=\rho {\int }_{\mathrm{\Sigma }}d{x}^1\wedge d{x}^2\wedge d{x}^3,$

the integral of the model's rest-energy-density form over the chain in the framework's unit reading. 4. For every finite smooth singular $4$-chain $C$ of $M$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$,

$Q_{\xi }({\mathrm{\Sigma }}_2)-Q_{\xi }({\mathrm{\Sigma }}_1)=-Q_{\xi }(B);$

in particular $Q_{\xi }({\mathrm{\Sigma }}_2)=Q_{\xi }({\mathrm{\Sigma }}_1)$ whenever the velocity vectors of every simplex of $B$ span a hyperplane containing ${\mathrm{\partial }}_0$, since then the side flux vanishes. 5. Consequently this prescribed flat-source calculation realizes a conserved Killing charge: it is $\rho$ times the coordinate-volume reading of the slice region, read as the model's rest energy. The prescribed nonzero dust source is not asserted to solve the gravitational field law on this flat metric. No measured value is asserted.

• Reading of the value. In the framework's stress-energy reading, $T$ has components in $\mathrm{J}{\mathrm{m}}^{-3}$. Here $\xi ={\mathrm{\partial }}_0$ is length-normalized because $x^0=ct$ is a length coordinate, and $X=1$; hence this $Q_{\xi }(\mathrm{\Sigma })$ is read in joules as the rest energy assigned to the chain. The parameter $\rho$ is the dust model's rest-energy-density parameter in the SI reading; ${\rho }_m=\rho \mathrm{/}{c}^2$ is the corresponding mass density. The identification of $\rho {\int }_{\mathrm{\Sigma }}d{x}^1\wedge d{x}^2\wedge d{x}^3$ with a physical mass or energy requires separate empirical premises and is not asserted.

• Why conservation needs the side flux. The current $J_{\xi }$ has no spatial components, so a side face swept out along ${\mathrm{\partial }}_0$ carries no flux; a side face tilted so that its velocity vectors span a hyperplane containing ${\mathrm{\partial }}_0$ also carries none, by the linear-dependence argument of step 4.1. For a general side chain the flux need not vanish and the equality of clause 4 is a balance identity of bookkeeping, exactly as in Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.