Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with an orientation, metric-selected connection $\mathrm{\nabla }$, metric volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ and current flux ${\mathrm{\Phi }}_J$ as in The metric volume form, current flux through 3-faces and finite-chain charges, and let $J$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). Write ${\mathrm{\nabla }}_a{J}^a$ for the divergence of $J$.

(a) Exterior derivative of the current form. As smooth $4$-forms on $M$,

$d\left({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g\right)=\left({\mathrm{\nabla }}_a{J}^a\right){\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

(b) Flux balance (Stokes). For every finite smooth singular $4$-chain $C$ of $M$ — a finite formal real sum of smooth singular $4$-simplices, the library convention def-smooth-singular-chain-and-cochain-complexes restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) — and the flux ${\mathrm{\Phi }}_J$ of the induced boundary $3$-chain (Integral of a form over a smooth singular simplex, Stokes theorem for smooth singular chains),

${\mathrm{\Phi }}_J(\mathrm{\partial }C)={\int }_C\left({\mathrm{\nabla }}_a{J}^a\right){\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

Consequently, if $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$ for smooth singular $3$-chains ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2,B$, then

${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_J(B)+{\int }_C\left({\mathrm{\nabla }}_a{J}^a\right){\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

In particular, if ${\mathrm{\nabla }}_a{J}^a=0$ on $C$ then ${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_J(B)$, and if moreover ${\mathrm{\Phi }}_J(B)=0$ then ${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)={\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)$: equal charges need a vanishing side flux, not merely a divergence-free current.

(c) Non-null face formula. Let $\sigma$ be a smooth singular $3$-simplex — a map $\sigma :{\mathrm{\Delta }}^3\to M$ admitting a smooth extension to a neighbourhood of the standard $3$-simplex inside its affine plane, the convention restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) — that is non-null at a parameter point $t$ (The metric volume form, current flux through 3-faces and finite-chain charges clause (c)), with $v_i:={\sigma }_{\ast }{\mathrm{\partial }}_i{\mathrm{\mid }}_t$ spanning the hyperplane $H$ and with the unit normal ${\nu }^{\ast }$ adapted to the ordered basis $(v_1,v_2,v_3)$ as in Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane clause (c); write $\epsilon :=g_{\sigma (t)}({\nu }^{\ast },{\nu }^{\ast })\in \{-1,1\}$. Then

${\sigma }^{\ast }\left({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g\right)({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3){\mid }_t=\epsilon g_{\sigma (t)}(J,{\nu }^{\ast })\sqrt{\mid \det (g_{\sigma (t)}(v_i,v_j))\mid }\mathrm{.}$

Equivalently, the tangent vector $J$ is decomposed as $J=\epsilon g(J,{\nu }^{\ast }){\nu }^{\ast }+w^{\mathrm{\prime }}$ with $w^{\mathrm{\prime }}\in H$, and only the normal component contributes to the flux through the face.

• The side flux is a genuine hypothesis. The balance in (b) is an identity, not an inequality: a divergence-free current can have nonzero flux through a bounded region's side wall, and the charges of the two slices differ by exactly that side flux. The compact-support case (B absent or of zero flux) is the setting used for conserved charges.

• Regularity. All chains are finite smooth singular chains; no arbitrary rough or infinite region is integrated, no boundary at infinity is treated and no convergence of an exhaustion is asserted. Finiteness of the chain makes every sum involved finite.