Local And Global Energy — Examples

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.

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 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

$U:=\{p\in M:x^1(p)>\mathrm{\mid }{x}^0(p)\mathrm{\mid }\},\xi :=x^1{\mathrm{\partial }}_0+x^0{\mathrm{\partial }}_1,$

let $X:=\sqrt{(x^1{)}^2-(x^0{)}^2}>0$ on $U$, let $n:=\xi \mathrm{/}X$, and let $T^{ab}:=\rho {\mathrm{\partial }}_0^a{\mathrm{\partial }}_0^b$ with $\rho >0$ constant (the tensor of the dust model $(\rho ,{\mathrm{\partial }}_0)$, The dust matter model: rest density, four-velocity, stress-energy and dust system). Then:

1. $U$ is open, $\xi$ is a Killing field on $U$, and $n$ is a smooth future-directed observer congruence on $U$ with $n^a{\mathrm{\nabla }}_{a}X=0$ and acceleration and expansion

$a=\mathrm{\nabla }\ln X=\frac{x^0}{X^2}{\mathrm{\partial }}_0+\frac{x^1}{X^2}{\mathrm{\partial }}_1,\theta =0.$

2. $(\rho ,{\mathrm{\partial }}_0)$ satisfies the dust system and ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $M$.
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

$\mu =\rho \frac{(x^1{)}^2}{X^2},J=\rho \frac{x^1}{X}{\mathrm{\partial }}_0,\mathrm{\Phi }=-\rho \frac{x^1(x^0{)}^2}{X^3}{\mathrm{\partial }}_0-\rho \frac{(x^1{)}^2{x}^0}{X^3}{\mathrm{\partial }}_1\mathrm{.}$

4. ${\mathrm{\nabla }}_a{J}^a=-{\mathrm{\Phi }}^a{a}_a=\rho x^0{x}^1\mathrm{/}{X}^3$ and $D\mu +\mu \theta +{\mathrm{\nabla }}_a{\mathrm{\Phi }}^a=\rho x^0{x}^1\mathrm{/}{X}^3$: the balance of Divergence of the measured energy-momentum current and the finite-chain balance identity clause (b) holds with a nonvanishing acceleration source, so the measured energy current is not conserved.
5. $XJ=\rho x^1{\mathrm{\partial }}_0$ satisfies ${\mathrm{\nabla }}_a(X{J}^a)=0$ and $T^{ab}{\xi }_b=-X{J}^a$: the redshift-weighted current is conserved, in accordance with Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current clause (d). Consequently, for every finite smooth singular $4$-chain $C\subseteq U$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, the flux ${\mathrm{\Phi }}_{XJ}$ satisfies ${\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_{XJ}(B)$; in particular ${\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_2)={\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_1)$ whenever the velocity vectors of every simplex of $B$ span a hyperplane containing ${\mathrm{\partial }}_0$.

• Contrast with the unweighted current. The measured energy current $J$ is not divergence-free: the boost congruence is accelerated, and clause 4 exhibits the source $-{\mathrm{\Phi }}^a{a}_a$. The redshift weight $X$ converts $J$ into the divergence-free current $-T(\cdot ,\xi )$, which is the content of Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current in this explicit model.

• Interpretation of the weight. On $U$ the observers of the congruence move with rapidity $\mathrm{artanh}(x^0\mathrm{/}{x}^1)$ relative to the static dust; $X=\sqrt{(x^1{)}^2-(x^0{)}^2}$ is the corresponding normalization, and the measured energy density $\mu =\rho (x^1{)}^2\mathrm{/}{X}^2$ is the ${\gamma }^2$-enhanced density of the static dust seen by them.

• No measurement. The displayed values are exact identities of the flat model; the example asserts no measured value and no empirical claim.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$, let $U\subseteq M$ be a nonempty open subset and let $(\epsilon ,p,n)$ be a perfect-fluid model on $U$ with stress-energy tensor $T^{ab}=(\epsilon +p)n^a{n}^b+p{g}^{ab}$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy); suppose ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $U$. Take the observer congruence of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress to be the fluid field $n$ itself, and write $D=n^a{\mathrm{\nabla }}_a$ and $\theta ={\mathrm{\nabla }}_a{n}^a$. Then:

1. For this congruence $\mu =\epsilon$, $J=\epsilon n$, $\mathrm{\Phi }=0$ and $S^{ab}=p{h}^{ab}$ with $h^{ab}=g^{ab}+n^a{n}^b$.
2. $T^{ab}{\mathrm{\nabla }}_a{n}_b=p\theta$, so the local balance of Divergence of the measured energy-momentum current and the finite-chain balance identity clause (b) reads

${\mathrm{\nabla }}_a(\epsilon n^a)=-p\theta ,\text{equivalently}D\epsilon +(\epsilon +p)\theta =0.$

3. This is exactly the energy equation of Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations clause (b) under the hypothesis ${\mathrm{\nabla }}^a{T}_{ab}=0$; the momentum (Euler) equation is the complementary projection of ${\mathrm{\nabla }}^a{T}_{ab}=0$ and is not implied by the energy balance alone.

• Consistency across pages. The example connects the new balance identity of Divergence of the measured energy-momentum current and the finite-chain balance identity with the conservation equivalence already proved in Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations; it does not reprove the latter and uses no equation of state.

• No dynamics. Only the algebraic identities and the hypothesis ${\mathrm{\nabla }}^a{T}_{ab}=0$ are used; the example asserts no existence of a fluid solution and no measured value. A model that adds an equation of state restricts the admissible tuples $(\epsilon ,p,n)$ without changing the identities.

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.