Wave stress and multiplier currents

Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected Levi-Civita connection $\mathrm{\nabla }$. Let $\mathcal{U}\subseteq M$ be open, let $u,V,s\in C^{\mathrm{\infty }}(\mathcal{U},\mathbb{R})$, and let $X$ be a smooth vector field on $\mathcal{U}$. Define ${\mathrm{\square }}_{g}u:={\mathrm{\nabla }}^a{\mathrm{\nabla }}_{a}u,\mathrm{\mid }du{\mathrm{\mid }}_g^2:=g^{ab}({\mathrm{\nabla }}_{a}u)({\mathrm{\nabla }}_{b}u)\mathrm{.}$ If ${\mathrm{\square }}_{g}u=Vu+s$, define the symmetric covariant tensor $T_{ab}[u,V]:=({\mathrm{\nabla }}_{a}u)({\mathrm{\nabla }}_{b}u)-\frac{1}{2}{g}_{ab}(\mathrm{\mid }du{\mathrm{\mid }}_g^2+V{u}^2),$ the vector current $J_X^a:=g^{ac}{T}_{cb}{X}^b$, and the scalar deformation term $K_X:=T_{ab}{\mathrm{\nabla }}^{(a}{X}^{b)}$, where ${\mathrm{\nabla }}^{(a}{X}^{b)}=({\mathrm{\nabla }}^a{X}^b+{\mathrm{\nabla }}^b{X}^a)\mathrm{/}2$.

For a finite smooth singular $3$-chain $\mathrm{\Sigma }={\sum }_{i=1}^k{c}_i{\sigma }_i$ with $k\ge 0$ and $c_i\in \mathbb{R}$, take each simplex ${\sigma }_i$ to be a map ${\mathrm{\Delta }}^3\to \mathcal{U}$ that extends smoothly to a map into $M$ on an open neighborhood of ${\mathrm{\Delta }}^3$ in its affine span. Here ${\mathrm{\Delta }}^3:=\{(t_0,t_1,t_2,t_3)\in {\mathbb{R}}^4:t_j\ge 0,{\sum }_{j=0}^3{t}_j=1\},$ with its standard orientation from the ordered vertices $e_0,e_1,e_2,e_3$, where each $e_j$ is the $j$th standard basis vector of ${\mathbb{R}}^4$. Define the local flux by ${\mathrm{\Phi }}_{J_X}(\mathrm{\Sigma }):={\sum }_{i=1}^k{c}_i{\int }_{{\mathrm{\Delta }}^3}{\sigma }_i^{\ast }({\iota }_{J_X}{\mathrm{v}\mathrm{o}\mathrm{l}}_g)\mathrm{.}$

Here ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is the oriented metric $4$-form whose expression in an oriented chart is $\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and ${\iota }_{J_X}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is the $3$-form that sends $(Y_1,Y_2,Y_3)\in (T_{p}M{)}^3$ to ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g,p}(J_X(p),Y_1,Y_2,Y_3)$ at each $p\in \mathcal{U}$. The map ${\sigma }_i^{\ast }$ is pullback of differential forms, and each simplex integral uses the pullback form-integration convention of the metric-volume current-flux definition listed in the following references section. Linearity makes the sum independent of the written chain representation. The integral uses the differential-form convention on null faces and assigns no flux to a chain with points outside $\mathcal{U}$; it never extends $J_X$ outside $\mathcal{U}$. In length coordinates, if $u$ has unit $[u]$, then $V$ has unit ${\mathrm{m}}^{-2}$, $s$ has unit $[u]{\mathrm{m}}^{-2}$, and the components of $T$ have unit $[u{]}^2{\mathrm{m}}^{-2}$; no physical normalization of $u$ or $X$ is imposed.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

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

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

A smooth vector field is a smooth section of the tangent bundle

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

The background assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited from the supplied smooth Lorentzian and tangent-bundle definitions. The definition makes no additional selection. The current is a vector field; its flux is fixed by the cited metric-volume definition, so no unit normal to a null hypersurface is introduced here.