Finite spacelike slab and normal flux charge

Let $m\ge 1$ be an integer. For a canonical coordinate vector $x:\{0,\dots ,m-1\}\to \mathbb{R}$, write $x_j:=x(j-1)$ for $1\le j\le m$. Accordingly, ${\prod }_{j=1}^m{I}_j$ denotes $\{x\in {\mathbb{R}}^m:x(j-1)\in I_j\text{ for }1\le j\le m\}$; this is only a relabelling of the coordinates in the cited suppliers. Let $Q={\prod }_{j=1}^m[a_j,b_j]\subseteq {\mathbb{R}}^m,a_j<b_j\mathrm{.}$ Then $Q$ is compact and Jordan measurable, and its boundary has Jordan content zero. For any product $R={\prod }_{j=1}^m{I}_j$ where each $I_j$ is one of $[a_j,b_j]$, $(a_j,b_j)$, $[a_j,b_j)$ or $(a_j,b_j]$, the set $R$ is a bounded Jordan set with closure contained in $Q$. Every continuous $f:Q\to \mathbb{R}$ is Riemann integrable over $R$. When $m\ge 2$, each closed coordinate face of $Q$ is compact and Jordan measurable in its own $(m-1)$-dimensional coordinate space; the same endpoint-variant and integrability conclusions hold for products in that face.

This lemma proves only finite-dimensional box and face regularity. It asserts no integrability or convergence result for unbounded sets or exhaustions.

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.

Use a time-oriented smooth Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$ and an independent orientation, with the exact meanings in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Oriented smooth manifolds and oriented charts. The tangent-bundle smooth structure follows the stated Countable Choice convention The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $(U,x)$ be a positively oriented chart with $x=(x^0,x^1,x^2,x^3)$ and $x(U)=V\subseteq {\mathbb{R}}^4$. Coordinates here are mathematical real coordinates; no physical units are assigned.

Choose real numbers $t_0<t_1$ and $a_i<b_i$ for $i=1,2,3$, and let $\mathrm{\Omega }={\prod }_{i=1}^3(a_i,b_i),\bar{\mathrm{\Omega }}={\prod }_{i=1}^3[a_i,b_i],K=[t_0,t_1]\times \mathrm{\Omega },\bar{K}=[t_0,t_1]\times \bar{\mathrm{\Omega }}\subset V\mathrm{.}$ The finite integration slab $K$ is a bounded Jordan set and its closure $\bar{K}$ is a compact coordinate box. Assume $\bar{K}$ has an open neighborhood in $V$ on which the spatial metric block $h_{ij}:=g_{ij}$ is positive definite and the inverse-metric component $g^{00}$ is negative. Thus each slice ${\mathrm{\Sigma }}_t=\{x^{-1}(t,x^1,x^2,x^3):(x^1,x^2,x^3)\in \mathrm{\Omega }\}$ is spacelike. Assume also that the future unit normal defined below is future-directed in the given time orientation. The closure $\bar{K}$, its closed coordinate faces, the open spatial rectangle $\mathrm{\Omega }$, the mixed-endpoint slab $K$, and the corresponding mixed-endpoint side-face rectangles are Jordan sets by Finite coordinate boxes and their faces are compact Jordan sets.

Write $g^{ab}$ for the inverse matrix to $g_{ab}$ and put $w:=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }},\alpha :=(-g^{00}{)}^{-1\mathrm{/}2},n^a:=-\alpha g^{a0},n_a:=-\alpha {\delta }_a^0\mathrm{.}$ Then $n$ is the future unit normal to every ${\mathrm{\Sigma }}_t$; its unit and normal properties follow from $g^{ab}{n}_a{n}_b=-1$ and $g_{ab}{n}^b=n_a$. Let $d{V}_h:=\sqrt{\det (h_{ij})}d{x}^{1}d{x}^{2}d{x}^3$ be the Riemannian volume element of the induced positive-definite slice metric. In this positively oriented chart the metric volume form is ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=wd{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ by The metric volume form, current flux through 3-faces and finite-chain charges clause (a).

A current here is a $C^1$ vector field $J=J^a{\mathrm{\partial }}_a$ on an open neighborhood of $x^{-1}(\bar{K})$; equivalently, its coordinate components $J^a$ are $C^1$. Its current three-form is the pointwise contraction ${\omega }_J:={\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, where $({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g)(X_1,X_2,X_3):={\mathrm{v}\mathrm{o}\mathrm{l}}_g(J,X_1,X_2,X_3)$. In coordinates, ${\omega }_J=w(J^{0}d{x}^1\wedge d{x}^2\wedge d{x}^3-J^{1}d{x}^0\wedge d{x}^2\wedge d{x}^3+J^{2}d{x}^0\wedge d{x}^1\wedge d{x}^3-J^{3}d{x}^0\wedge d{x}^1\wedge d{x}^2)\mathrm{.}$ The slice orientation is chosen so that the future unit normal precedes a positive slice basis. Since $n^0=1\mathrm{/}\alpha >0$ and the spacetime chart is positive, this slice orientation is $d{x}^1\wedge d{x}^2\wedge d{x}^3$; the pullback of ${\omega }_J$ to ${\mathrm{\Sigma }}_t$ is $w{J}^{0}d{x}^1\wedge d{x}^2\wedge d{x}^3$. Define its signed slice charge (the normal flux through the spacelike slice) by $Q_J(t):={\int }_{\mathrm{\Omega }}-g(J,n)d{V}_h\mathrm{.}$ Every displayed integral is the Riemann integral over the indicated bounded Jordan set, as defined in The Riemann integral of a bounded function over a bounded Jordan measurable set. The current and metric extend continuously to the compact coordinate closure, and Finite coordinate boxes and their faces are compact Jordan sets proves that the open spatial and mixed-endpoint rectangles are Jordan measurable and that these continuous densities are Riemann integrable there.

Let $B$ denote the union of the six side faces of $K$, with outward boundary orientation; each face is integrated over its relative interior, a bounded Jordan rectangle whose closure is compact. The upper $x^i$ face has induced orientation $(-1{)}^{i}d{x}^0\wedge \cdots \widehat{d{x}^i}\cdots \wedge d{x}^3$; the lower face has its opposite. Thus the $i$th current-three-form pullback integrates to the upper-minus-lower $w{J}^i$ coordinate expression below. Define the net outward side flux ${\mathrm{\Phi }}_J(B)$ as the sum over the three spatial coordinate directions of upper-face flux minus lower-face flux. More explicitly, for $i\in \{1,2,3\}$ let ${\mathrm{\Omega }}_i$ be the product Jordan rectangle in the three coordinates other than $x^i$, with endpoint convention $x^0\in [t_0,t_1]$ and $x^j\in (a_j,b_j)$ for $j\mathrm{\ne }i$; then ${\mathrm{\Phi }}_J(B):={\sum }_{i=1}^3{\int }_{{\mathrm{\Omega }}_i}{w{J}^i\mid }_{x^i=b_i}-{w{J}^i\mid }_{x^i=a_i}{d}^3{x}_{\widehat{i}}\mathrm{.}$ The integrals use the standard increasing-coordinate order on each face. For a $C^1$ current the three-form coefficients are continuous, so the displayed rectangular integrals exist. This extends the smooth-current flux convention of The metric volume form, current flux through 3-faces and finite-chain charges clause (b) to the present finite coordinate faces. No unit normal to a side face is part of this definition; in particular the flux form remains defined if a side face is null. The term “charge” names this mathematical flux integral only; it does not assert a physical conservation law or an entropy interpretation.

• The definition is local to the supplied chart and finite product slab with compact closure. It makes no claim about arbitrary regions, rough boundaries, noncompact exhaustions, or limits at infinity.
• If a later physical model assigns units and identifies $J$ with an entropy current, those units and that interpretation are additional premises. This mathematical definition has no unit convention.

Assume the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention and the spacelike slab data of Finite spacelike slab and normal flux charge. For every slice ${\mathrm{\Sigma }}_t$ and every $C^1$ current $J$ in that definition, its normal flux charge is $Q_J(t)={\int }_{\mathrm{\Omega }}w{J}^0{d}^{3}x,$ where $w=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}$ is the metric density in the supplied chart.

The calculation is restricted to the supplied spacelike coordinate slices. It makes no assertion that the coordinate component $J^0$ alone is invariant or that charges from different slicings agree.

Assume the ${\mathrm{A}\mathrm{C}}_{\omega }$ background of Finite spacelike slab and normal flux charge. Let $K$, ${\mathrm{\Sigma }}_t$, the $C^1$ current $J$, its slice charge $Q_J(t)$ and its net outward side flux ${\mathrm{\Phi }}_J(B)$ be as in Finite spacelike slab and normal flux charge. Use the oriented metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges clause (a). Put $\mathrm{\Pi }:={\mathrm{\nabla }}_a{J}^a$, using the $C^1$ divergence of Metric divergence density identity for a C¹ current. Then the exact finite-slab balance is $Q_J(t_1)-Q_J(t_0)+{\mathrm{\Phi }}_J(B)={\int }_K\mathrm{\Pi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ Here the four-volume integral is the Riemann integral on the bounded Jordan slab $K$, whose compact closure is the supplied coordinate box. If $\mathrm{\Pi }\ge 0$ everywhere on $K$, then $Q_J(t_1)+{\mathrm{\Phi }}_J(B)\ge Q_J(t_0)$. In particular, nonnegative local production implies $Q_J(t_1)\ge Q_J(t_0)$ when the net outward side flux is zero. If $\mathrm{\Pi }=0$, the identity instead reads $Q_J(t_1)-Q_J(t_0)=-{\mathrm{\Phi }}_J(B)$, so zero production alone does not imply constant slice charge.

• This is an equality on the stated finite rectangular coordinate slab. It asserts no arbitrary-domain Stokes theorem, infinite exhaustion, boundary at infinity, or convergence result.
• For nonnegative production, zero net outward side flux is sufficient for endpoint charge monotonicity. The exact criterion is that integrated production is at least the net outward side flux, as illustrated in proof steps 1.4, 2.2 and 4.1. The theorem places no causal-character restriction on the side faces; their flux is defined by the three-form/coordinate face integral, so null side faces need no unit normal.
• This is a pure mathematical identity. A physical entropy interpretation, units, and a local second-law premise must be supplied separately.