A divergence-free current with nonzero flux through a timelike 3-simplex

Given: The flat Lorentzian four-manifold $({\mathbb{R}}^4,g)$ with identity chart, $g=\mathrm{diag}(-1,1,1,1)$, chart orientation, metric-selected connection $\mathrm{\nabla }$ and metric volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$; the vector field $J={\mathrm{\partial }}_1$; and the affine smooth singular $3$-simplex $\sigma$ of the standard $3$-simplex ${\mathrm{\Delta }}^3=\{(t_1,t_2,t_3):t_i\ge 0,{\sum }_i{t}_i\le 1\}$ into the plane $x^1=0$, $\sigma (t_1,t_2,t_3):=(t_1,0,t_2,t_3)\mathrm{.}$

On the identity chart $\mathrm{\Gamma }\equiv 0$, so $\mathrm{\nabla }$ is the coordinate connection and ${\mathrm{\nabla }}_a{J}^a={\mathrm{\partial }}_a{J}^a$; ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ (The metric volume form, current flux through 3-faces and finite-chain charges, The derivative of the metric density and the chart formula for the divergence of a vector field, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

The flux of $J$ through a smooth singular $3$-simplex $\sigma$ — the affine map of the Facts, a special case of the simplex convention restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) (library definition def-smooth-singular-simplex) — is ${\mathrm{\Phi }}_J(\sigma )={\int }_{\sigma }{\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g={\int }_{{\mathrm{\Delta }}^3}{\sigma }^{\ast }({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g)$, and the integral is the ordinary multiple integral of the pullback coefficient (The metric volume form, current flux through 3-faces and finite-chain charges clause (b), Integral of a form over a smooth singular simplex).

For a non-null face with velocity basis $(v_1,v_2,v_3)$ and adapted unit normal ${\nu }^{\ast }$ (so that $({\nu }^{\ast },v_1,v_2,v_3)$ is positively oriented, $g({\nu }^{\ast },{\nu }^{\ast })=\epsilon$ and ${\nu }^{\ast }\perp \mathrm{span}(v_1,v_2,v_3)$), the pullback of ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ on $(v_1,v_2,v_3)$ equals $\epsilon g(J,{\nu }^{\ast })\sqrt{\mathrm{\mid }\det (g(v_i,v_j))\mathrm{\mid }}$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (c), Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane).

Using the simplex-integral zero-extension convention on $[0,1{]}^3$, the indicator sections are integrable interval indicators; Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections identifies their repeated integrals with the multiple integral. The standard $3$-simplex ${\mathrm{\Delta }}^3$ has volume $1\mathrm{/}6$: the iterated integral ${\int }_0^1\text{\hspace{0.17em}⁣}{\int }_0^{1-t_1}\text{\hspace{0.17em}⁣}{\int }_0^{1-t_1-t_2}d{t}_{3}d{t}_{2}d{t}_1={\int }_0^1\frac{(1-t_1{)}^2}{2}d{t}_1=\frac{1}{6}$ evaluates by the fundamental theorem of calculus; hence the integral of a constant $c$ over ${\mathrm{\Delta }}^3$ is $c\mathrm{/}6$ (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, Integral of a form over a smooth singular simplex).