Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum

Retain the fields $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of The electric and magnetic fields as fixed-time spatial vector fields, the energy density $u$ and Poynting vector $S$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector), the laboratory region, its frame basis $(e_1,e_2,e_3)$ and its spatial Euclidean structure of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, the constants and units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and the partial derivatives ${\mathrm{\partial }}_j$ of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. Write $x=(x^1,x^2,x^3)$ for the position in the spatial chart, ${\delta }_{ij}$ for the Kronecker symbol (${\delta }_{ij}=1$ if $i=j$ and $0$ otherwise), $\mid E{\mid }^2=⟨E,E⟩$ with the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and $\times$ for the cross product of The cross product in ${\mathbb{R}}^3$ in the frame basis. Let ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and ${\mu }_0>0$ be the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Momentum density] The electromagnetic momentum density is the map $g:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ defined by

$g:=\frac{S}{c^2}={\epsilon }_{0}E\times B,$

in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$; the second equality uses ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$.

(b) [Maxwell stress tensor] The Maxwell stress tensor is the spatial tensor field $\sigma$ on $\mathrm{\Omega }\times I$ with components

${\sigma }_{ij}:={\epsilon }_0(E_i{E}_j-\frac{1}{2}{\delta }_{ij}\mid E{\mid }^2)+\frac{1}{{\mu }_0}(B_i{B}_j-\frac{1}{2}{\delta }_{ij}\mid B{\mid }^2),$

in $\mathrm{N}{\mathrm{m}}^{-2}=\mathrm{J}{\mathrm{m}}^{-3}$; the indices range over $\{1,2,3\}$ and repeated symbols are not summed here, every sum being written explicitly. The tensor is symmetric, ${\sigma }_{ij}={\sigma }_{ji}$ for all $i,j$, by inspection of the displayed expression. Its divergence is the vector field with components $(\mathrm{div}\sigma {)}_i:={\sum }_{j=1}^3{\mathrm{\partial }}_j{\sigma }_{ij}$, the partial derivatives being taken at fixed $t$.

(c) [Momentum flux] The momentum flux tensor is $\mathrm{\Pi }:=-\sigma$, with $(\mathrm{div}\mathrm{\Pi }{)}_i:={\sum }_{j=1}^3{\mathrm{\partial }}_j{\mathrm{\Pi }}_{ij}$; then $\mathrm{div}\mathrm{\Pi }=-\mathrm{div}\sigma$ componentwise.

(d) [Orbital angular momentum density and its flux] Relative to the origin of the spatial chart, the orbital angular momentum density is $\mathrm{\ell }:=x\times g$, with components ${\mathrm{\ell }}_i={\sum }_{j,k=1}^3{ϵ}_{ijk}{x}_j{g}_k$, and its flux tensor is the spatial tensor field $x\times \mathrm{\Pi }$ with components

$(x\times \mathrm{\Pi }{)}_{i\mathrm{\ell }}={\sum }_{j,k=1}^3{ϵ}_{ijk}{x}_j{\mathrm{\Pi }}_{k\mathrm{\ell }}\mathrm{.}$

Here ${ϵ}_{ijk}$ is the Levi-Civita symbol, defined by ${ϵ}_{ijk}=+1$ if $(i,j,k)$ is an even permutation of $(1,2,3)$, ${ϵ}_{ijk}=-1$ if it is an odd permutation, and ${ϵ}_{ijk}=0$ if two indices coincide, in the positively oriented basis $(e_1,e_2,e_3)$. The object $\mathrm{\ell }$ carries $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ and the components of $x\times \mathrm{\Pi }$ carry $\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}=\mathrm{N}{\mathrm{m}}^{-1}$.

(e) [Origin] Both $\mathrm{\ell }$ and $x\times \mathrm{\Pi }$ depend on the origin of the spatial chart: under a shift $x\mapsto x+a$ of the origin by a constant $a\in {\mathbb{R}}^3$ their values become $\mathrm{\ell }+a\times g$ and $x\times \mathrm{\Pi }+a\times \mathrm{\Pi }$ respectively, so the definition fixes one origin in the laboratory frame and every statement about $\mathrm{\ell }$ or $x\times \mathrm{\Pi }$ is relative to that fixed origin.

(f) [Status] This item fixes objects and index conventions only. No balance identity and no interpretation of $\sigma$ or $\mathrm{\Pi }$ as force or momentum flux is asserted here; every balance identity involving these objects is the content of the separate theorem items of this page.

(g) [Quadratic character and singular support] All five objects are quadratic in the fields $E,B$. For a locally integrable branch of The microscopic vacuum Maxwell equations: smooth and weak branches the exclusions recorded in Vacuum electromagnetic energy density and energy flux (the Poynting vector) apply to each of them: the products $E_i{E}_j$, $B_i{B}_j$ and the components of $E\times B$ need not be locally integrable, no product of distributions is defined, and the displayed formulas then define the objects pointwise almost everywhere, the framework using them only where the local integrability of the products is separately established. No value is assigned at a point of singular source support.

(h) [No covariant object] No four-dimensional or covariant interpretation of $(u,S,g,\sigma )$ is defined here; in particular $g$ is not identified with the spatial components of a covariant momentum density.

• Sign convention. With $\mathrm{\Pi }=-\sigma$ the local balance of the next items reads ${\mathrm{\partial }}_{t}g+\mathrm{div}\mathrm{\Pi }=-f$ for the force density $f$ of The Lorentz coupling: external force on a test charge and the continuum force and power densities; writing $\sigma$ in place of $\mathrm{\Pi }$ changes the sign of the divergence term. The gradient convention $(\mathrm{div}\sigma {)}_i={\sum }_j{\mathrm{\partial }}_j{\sigma }_{ij}$ and the component order ${\sigma }_{ij}$ (first index: component of the transported momentum; second index: direction of transport) are fixed by the tensor definition above and are used consistently by the balance theorem of this page.
• The Levi-Civita convention. The symbol ${ϵ}_{ijk}$ is the one fixed in clause (d); it is not a tensor density and no orientation-reversing change of basis is used in this framework, the laboratory basis $(e_1,e_2,e_3)$ being positively oriented by Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.