Energy momentum angular balance

Let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, so that $I\subseteq \mathbb{R}$ is a nonempty open interval, $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ is nonempty open, $x=(x^1,x^2,x^3)$ is the position in the spatial chart, $(e_1,e_2,e_3)$ is the frame basis of the spatial slice, and the spatial inner product is the Euclidean form $\delta$; let the units and the constants $c>0$, ${\mu }_0>0$ and ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ be those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; let $k\ge 1$, and let $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be an electric and a magnetic field of class $C^k$ of The electric and magnetic fields as fixed-time spatial vector fields. Write $⟨\cdot ,\cdot ⟩$ and $\mathrm{\mid }\cdot \mathrm{\mid }$ for the Euclidean inner product and norm 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 ${\mathbb{R}}^3$.

(a) [Energy density and flux] The electromagnetic energy density $u:\mathrm{\Omega }\times I\to \mathbb{R}$ and the energy flux density (the Poynting vector) $S:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ are defined by

$u(x,t):=\frac{{\epsilon }_0}{2}⟨E(x,t),E(x,t)⟩+\frac{1}{2{\mu }_0}⟨B(x,t),B(x,t)⟩,S(x,t):=\frac{1}{{\mu }_0}E(x,t)\times B(x,t)\mathrm{.}$

At fixed $t\in I$ the map $u(\cdot ,t):\mathrm{\Omega }\to \mathbb{R}$ is a real-valued function on $\mathrm{\Omega }$ and $S(\cdot ,t):\mathrm{\Omega }\to {\mathbb{R}}^3$ is a spatial vector field on $\mathrm{\Omega }$, read in the basis $(e_1,e_2,e_3)$. The energy density carries $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$ and the energy flux density carries $\mathrm{W}{\mathrm{m}}^{-2}=\mathrm{k}\mathrm{g}{\mathrm{s}}^{-3}$.

(b) [Regularity] If $E,B$ are of class $C^k$, then $u$ and $S$ are of class $C^k$: they are the components of a polynomial expression in the components of $E,B$, and the verification below exhibits the composite map. In particular $u$ and $S$ are continuous, and the definitions use no derivative of the fields.

(c) [Pointwise reading, quadratic character] The definitions use only the values of $E$ and $B$ at the same point $(x,t)$, and they are quadratic in the fields: for every real $\lambda$ one has $u(\lambda E,\lambda B)={\lambda }^{2}u(E,B)$ and $S(\lambda E,\lambda B)={\lambda }^{2}S(E,B)$, and $u\ge 0$. Consequently decompositions of the fields do not make $u$ or $S$ additive; cross terms generally contribute. No convention beyond the fixed laboratory frame enters the definitions.

(d) [Products and singular support] If $E,B$ are only locally integrable (the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches), the products $E_i{E}_j$, $B_i{B}_j$ and the components of $E\times B$ need not be locally integrable, and no product of distributions is defined. The displayed formulas then define $u$ and $S$ pointwise almost everywhere, and the framework uses them only where the local integrability of the products, respectively of the component being integrated, is separately established. No value of $u$ or $S$ is assigned at a point of singular source support.

(e) [Status] This item defines quantities only. The interpretation of $u$ as an energy density and of $S$ as an energy flux density, and every balance identity involving them, is the content of Local and fixed-volume electromagnetic energy balance and of the other balance items of this page; no identification of $(u,S)$ with components of a covariant tensor is made here, and the relation $S=c^{2}g$ to the momentum density $g$ of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum is a consequence of those definitions, not an identification made at this point.

• Non-additivity, explicitly. The quadratic character of clause (c) is not a technicality: at a point where $E=e_1$, $E^{\mathrm{\prime }}=e_1$ and $B=0$ one has $u(E+E^{\mathrm{\prime }},0)=2{\epsilon }_0$ but $u(E,0)+u(E^{\mathrm{\prime }},0)={\epsilon }_0$, because $u(e_1,0)={\epsilon }_0\mathrm{/}2$. Every balance identity of this page is therefore stated for the total fields, never for a sum of separately balanced parts without the cross terms.

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.

Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields: the fields $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ are jointly $C^1$ of The electric and magnetic fields as fixed-time spatial vector fields and satisfy, with the continuous source fields $\rho ,J$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values,

$\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$

pointwise on $\mathrm{\Omega }\times I$, with $\mathrm{div}$ and $\mathrm{curl}$ of Divergence and curl of a $C^1$ vector field applied at fixed times and ${\mathrm{\partial }}_t$ the partial derivative at fixed position of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. Let $u$ and $S$ be the energy density and Poynting vector of Vacuum electromagnetic energy density and energy flux (the Poynting vector). Then:

(a) [Local identity] On $\mathrm{\Omega }\times I$,

${\mathrm{\partial }}_{t}u+\mathrm{div}S=-J\cdot E,$

where $\mathrm{div}S$ is taken at fixed time and $J\cdot E$ is the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

(b) [Fixed-volume identity] Let ${\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathrm{\Omega }$ be open with $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}\subseteq \mathrm{\Omega }$ carrying the finite piecewise $C^1$ presentation of Divergence for finite piecewise C1 presentations, let $t_0<t_1$ with $(t_0,t_1)\subseteq I$, and let $n$ be the outward unit normal of $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$. Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis under which that divergence theorem is stated. Then for every $t\in (t_0,t_1)$,

$\frac{d}{dt}{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}u(x,t)dx=-{\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}J\cdot Edx-{\oint }_{\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}S\cdot ndS,$

the closed-surface integral being the finite face sum of that divergence theorem and the derivative being the derivative of the one-variable function $t\mapsto {\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}u(x,t)dx$ on $(t_0,t_1)$.

(c) [Interpretation and scope] Under clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities, $J\cdot E$ is the power density supplied to the matter model of that coupling, so (a) and (b) read as a local, respectively fixed-volume, energy balance of the field and that matter model. Nothing is claimed for a prescribed current without such a matter model, for the weak branch of the Maxwell postulate, for singular sources, or at field points where the products defining $u$ or $S$ are not locally integrable; the fixed-volume identity retains both the work term and the boundary flux. The field-energy rate can vanish by cancellation of these two terms; vanishing boundary flux alone does not eliminate exchange with matter. A conservation claim for field plus matter additionally requires an explicit matter balance. This is a conditional consequence of the adopted postulates, not an empirical claim.