Metric divergence density identity for a C¹ current

Assume the Countable Choice convention ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) used by the smooth tangent-bundle structure in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ with its metric-selected connection $\mathrm{\nabla }$, and let $(U,x)$ be a smooth chart. Write $w=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}$. Let $J$ be a $C^1$ vector field on $U$; the connection derivative on such a field is defined by its usual chart formula ${\mathrm{\nabla }}_b{J}^a:={\mathrm{\partial }}_b{J}^a+{\mathrm{\Gamma }}^a{}_{cb}{J}^c,$ whose coefficients are continuous. Then, throughout $U$, ${\mathrm{\nabla }}_a{J}^a=w^{-1}{\mathrm{\partial }}_a(w{J}^a)\mathrm{.}$ The right-hand side is the chart expression of the scalar divergence ${\mathrm{\nabla }}_a{J}^a$; no higher regularity of $J$ than $C^1$ is required.

The published density lemma states its divergence formula for smooth vector fields. This local extension uses exactly the first-derivative coordinate formula and therefore covers the $C^1$ currents required by the finite-slab balance; it does not change the published supplier.