Closed bulk current with nonzero boundary flux

Let $M={\mathbb{R}}^4$ with coordinates $(X^1,X^2,X^3,X^4)=(t,x,y,z)$, orientation $dt\wedge dx\wedge dy\wedge dz$, and flat metric $-d{t}^2+d{x}^2+d{y}^2+d{z}^2$. For one-forms use $\ast dt=-dx\wedge dy\wedge dz,\ast dx=-dt\wedge dy\wedge dz,\ast dy=dt\wedge dx\wedge dz,\ast dz=-dt\wedge dx\wedge dy\mathrm{.}$ Here the massless wave equation means $(-{\mathrm{\partial }}_t^2+{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2+{\mathrm{\partial }}_z^2)u=0$. Let $u_1:=t^2+x^2$ and $u_2:=t$, and define the smooth field family ${\psi }_{a,b}:=a{u}_1+b{u}_2$ on $P={\mathbb{R}}^2$. Both $u_1,u_2$ solve the massless wave equation, so $\psi =0$ at $(a,b)=(0,0)$ is a solution and ${\delta }_1\psi :={\mathrm{\partial }}_a{\psi }_{a,b}=u_1$, ${\delta }_2\psi :={\mathrm{\partial }}_b{\psi }_{a,b}=u_2$ are commuting variations solving that same equation. Use the massless scalar Lagrangian $\mathcal{L}=-\frac{1}{2}d\psi \wedge \ast d\psi$, with selected potential $\theta (\psi ;\delta \psi )=-\delta \psi \ast d\psi$ and current $\omega ({\delta }_1,{\delta }_2)={\delta }_2\psi \ast d{\delta }_1\psi -{\delta }_1\psi \ast d{\delta }_2\psi$. For $0<t_1<t_2$ and $L>0$, put $a_0=(t_1,0,0,0)$ and $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3,{\mathrm{\ell }}_4)=(t_2-t_1,L,1,1)$. Let $T_4:=\{r\in {\mathbb{R}}^4:r_j\ge 0,{\sum }_{j=1}^4{r}_j\le 1\}$ with orientation $d{r}_1\wedge d{r}_2\wedge d{r}_3\wedge d{r}_4$. For each permutation $\rho \in S_4$, define vertices $v_0^{\rho }:=a_0,v_j^{\rho }:=a_0+{\sum }_{k=1}^j{\mathrm{\ell }}_{\rho (k)}{e}_{\rho (k)}(1\le j\le 4),$ where $e_1,e_2,e_3,e_4$ is the standard basis in the ordered coordinates $(t,x,y,z)$. Define the 24 affine simplex maps and their oriented sum by ${\sigma }_{\rho }(r):=v_0^{\rho }+{\sum }_{j=1}^4{r}_j(v_j^{\rho }-v_0^{\rho }),C:={\sum }_{\rho \in S_4}\mathrm{sgn}(\rho ){\sigma }_{\rho }\mathrm{.}$ Each map has domain orientation $d{r}_1\wedge d{r}_2\wedge d{r}_3\wedge d{r}_4$; the coefficient $\mathrm{sgn}(\rho )$ orients it relative to $dt\wedge dx\wedge dy\wedge dz$.

For $t\in \{t_1,t_2\}$, let $b_t=(t,0,0,0)$ and $(m_1,m_2,m_3)=(L,1,1)$. With $T_3:=\{r\in {\mathbb{R}}^3:r_j\ge 0,{\sum }_{j=1}^3{r}_j\le 1\}$ oriented by $d{r}_1\wedge d{r}_2\wedge d{r}_3$, define, for $\rho \in S_3$, $w_0^{t,\rho }:=b_t,w_j^{t,\rho }:=b_t+{\sum }_{k=1}^j{m}_{\rho (k)}{\widehat{e}}_{\rho (k)}(1\le j\le 3),$ where ${\widehat{e}}_1=e_2$, ${\widehat{e}}_2=e_3$, and ${\widehat{e}}_3=e_4$, and put ${\sigma }_{t,\rho }(r):=w_0^{t,\rho }+{\sum }_{j=1}^3{r}_j(w_j^{t,\rho }-w_0^{t,\rho }),{\mathrm{\Sigma }}_t:={\sum }_{\rho \in S_3}\mathrm{sgn}(\rho ){\sigma }_{t,\rho }\mathrm{.}$ Each ${\mathrm{\Sigma }}_t$ is oriented by $dx\wedge dy\wedge dz$. Define the side chain by $B:=\mathrm{\partial }C-{\mathrm{\Sigma }}_{t_2}+{\mathrm{\Sigma }}_{t_1}$, with the signed simplex-face multiplicities inherited from the boundary of $C$. The coordinate-order boundary calculation shows that $B$ is supported on the six side faces of the box, each with outward-normal-first orientation from $dt\wedge dx\wedge dy\wedge dz$; their oriented volume forms are $+dt\wedge dy\wedge dz$ at $x=0$, $-dt\wedge dy\wedge dz$ at $x=L$, $-dt\wedge dx\wedge dz$ at $y=0$, $+dt\wedge dx\wedge dz$ at $y=1$, $+dt\wedge dx\wedge dy$ at $z=0$, and $-dt\wedge dx\wedge dy$ at $z=1$. Then the bulk current is closed on the box, and by definition of $B$, $\mathrm{\partial }C={\mathrm{\Sigma }}_{t_2}-{\mathrm{\Sigma }}_{t_1}+B$, but ${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_2}}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_1}}=-L(t_2^2-t_1^2)\mathrm{\ne }0.$ The oriented side flux is $L(t_2^2-t_1^2)$, so the finite-region balance is ${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_2}}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_{t_1}}=-{\int }_B\omega$. The box is compact; the example uses no asymptotic region or falloff premise.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §§2.2–2.4

The following is the $t$–$x$ projection of the four-dimensional box; the $y,z$ directions are suppressed. It shows the two constant-time cuts and the faces $x=0$ and $x=L$. Boundary orientations are induced from $dt\wedge dx\wedge dy\wedge dz$ by the outward-normal-first convention.