The metric volume form, current flux through 3-faces and finite-chain charges

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $M$ carry an orientation in the sense of Oriented smooth manifolds and oriented charts (so that its charts that are oriented in that orientation have orientation-preserving transition maps), and let $\mathrm{\nabla }$ be its metric-selected connection. This item fixes the metric volume form, the flux of a current through oriented $3$-faces and finite-chain charges. The orientation of $M$ is an independent datum: it is not determined by the metric and not by the time orientation.

(a) Metric volume density and volume form. Let $(U,x)$ be an oriented chart, with metric components $g_{ab}=g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a)). Define the metric volume density

$w:=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}>0\text{on }U,$

a strictly positive smooth function because $g_{ab}$ is smooth and $g_p$ is nondegenerate. In the chart define the local $4$-form

${\mathrm{v}\mathrm{o}\mathrm{l}}_{g,U}:=wd{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3,$

with the exterior product of The wedge product of differential forms and the coordinate covector fields of Local coordinate expression for a differential form. That these local forms agree on chart overlaps and therefore define one smooth nowhere-vanishing $4$-form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ on $M$ — the metric volume form — is proved 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 (a), together with the reading statement of clause (b) there: in every oriented chart ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=wd{x}^0\wedge \cdots \wedge d{x}^3$, and on every positively oriented $g_p$-orthonormal basis $(e_0,e_1,e_2,e_3)$ of $T_{p}M$ one has ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g,p}(e_0,e_1,e_2,e_3)=1$.

(b) Current flux through oriented $3$-faces and chains. Let $J$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). The current $3$-form of $J$ is the contraction

${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$

the interior product of the $4$-form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ with $J$ (Interior product of a form by a vector field).

By a smooth singular $k$-simplex of $M$ — the convention of the library definition def-smooth-singular-simplex (differential-geometry track), restated here so this page is self-contained — we mean a map $\sigma :{\mathrm{\Delta }}^k\to M$ whose domain is the standard $k$-simplex ${\mathrm{\Delta }}^k=\{t\in {\mathbb{R}}^{k+1}:t_i\ge 0,{\sum }_i{t}_i=1\},$ for which there exist an open set $O\subseteq A^k=\{t\in {\mathbb{R}}^{k+1}:{\sum }_i{t}_i=1\}$ containing ${\mathrm{\Delta }}^k$ and a smooth map $\bar{\sigma }:O\to M$ with $\bar{\sigma }{\mathrm{\mid }}_{{\mathrm{\Delta }}^k}=\sigma$; the extension is not part of the simplex data, and degenerate parametrizations are allowed. With the integration of forms over such simplices (Integral of a form over a smooth singular simplex), a $3$-face is a smooth singular $3$-simplex $\sigma$, and the flux of $J$ through $\sigma$ is the real number

${\mathrm{\Phi }}_J(\sigma ):={\int }_{\sigma }{\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

A finite smooth singular $k$-chain — the convention of the library definition def-smooth-singular-chain-and-cochain-complexes, restated here — is a finite formal real linear combination $\mathrm{\Sigma }={\sum }_i{c}_i{\sigma }_i$ of smooth singular $k$-simplices. For such a $3$-chain the flux of $J$ through $\mathrm{\Sigma }$ is defined additively,

${\mathrm{\Phi }}_J(\mathrm{\Sigma }):={\sum }_i{c}_i{\mathrm{\Phi }}_J({\sigma }_i),$

a definition independent of the written expression for $\mathrm{\Sigma }$, because a finite formal sum is evaluated by adding the finitely many coefficients and inserting a zero coefficient changes nothing. ${\mathrm{\Phi }}_J(\mathrm{\Sigma })$ is the charge of the current $J$ assigned to $\mathrm{\Sigma }$ when the model reads $J$ as a conserved current; that reading belongs to the physical postulates and is not part of this definition.

(c) Non-null faces and their unit normals. A $3$-dimensional linear subspace $H\subseteq T_{p}M$ is nondegenerate for $g_p$ when the restriction of $g_p$ to $H$ is nondegenerate, equivalently when $H\cap H^{\perp }=\{0\}$ in the orthogonality of eta-orthogonality and orthogonal complements in a Lorentz vector space. A smooth singular $3$-simplex $\sigma$ is non-null at a parameter point $t$ when the span of the three velocity vectors ${\sigma }_{\ast }{\mathrm{\partial }}_1{\mathrm{\mid }}_t,{\sigma }_{\ast }{\mathrm{\partial }}_2{\mathrm{\mid }}_t,{\sigma }_{\ast }{\mathrm{\partial }}_3{\mathrm{\mid }}_t$ has dimension $3$ and is nondegenerate at $\sigma (t)$, and non-null when this holds at every parameter point. For a non-null face the unit normal $\nu$ at that point is the unique vector of $T_{\sigma (t)}M$ with $\nu \in H^{\perp }$, $g(\nu ,\nu )\in \{-1,+1\}$ and $(\nu ,{\sigma }_{\ast }{\mathrm{\partial }}_1,{\sigma }_{\ast }{\mathrm{\partial }}_2,{\sigma }_{\ast }{\mathrm{\partial }}_3)$ a positively oriented basis of $T_{\sigma (t)}M$; its existence, uniqueness and its sign $\epsilon :=g(\nu ,\nu )$ are proved 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), together with the density identity of clause (c) used to read ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ on such a face. Rank-deficient faces and rank-three faces with degenerate induced metric are not assigned a unit normal; their flux is defined by the $3$-form ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ itself, as in clause (b).