Lorentz covariant formulation

Let the laboratory region $U\cong I\times \mathrm{\Omega }$ with its inertial chart $x$, the electric and magnetic fields $E,B$ of The electric and magnetic fields as fixed-time spatial vector fields, the source fields $(\rho ,J)$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and the SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values with ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ be those of the framework. Let $F$ be the electromagnetic two-form of The electromagnetic two-form, its observer decomposition and the four-current with the components fixed there, $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$, $F^{0i}=E_i\mathrm{/}c$, $F^{ij}={ϵ}_{ijk}{B}_k$ and $\eta =\mathrm{diag}(-1,1,1,1)$, and let $j=(c\rho ,J)$ carry the tensor-component identification adopted in Charge parameter, frame charge and current densities, volume charge and flux, and the four-current. Write ${\mathrm{\partial }}_0=c^{-1}{\mathrm{\partial }}_t$ and ${\mathrm{\partial }}_i$ for the coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and $\mathrm{div},\mathrm{curl}$ for those of Divergence and curl of a $C^1$ vector field. Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches: $E,B$ are jointly $C^1$ on $\mathrm{\Omega }\times I$ with the additional derivatives required by each displayed equation existing and continuous, and $\rho ,J$ are continuous. Then:

(a) [Equivalence] On $I\times \mathrm{\Omega }$ the four smooth-branch equations

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

hold if and only if the two systems

$(\mathrm{T}\mathrm{M}1){\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }(\nu =0,1,2,3),$

$(\mathrm{T}\mathrm{M}2){\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=0(\lambda ,\mu ,\nu \in \{0,1,2,3\})$

hold on $I\times \mathrm{\Omega }$. Every instance of (TM2) with a repeated index vanishes identically because $F$ is alternating, so (TM1) and (TM2) carry exactly eight independent scalar equations: (TM1) with $\nu =0$ is $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$ (using ${\mu }_0{c}^2{\epsilon }_0=1$), (TM1) with $\nu =i$ is the $i$-th component of $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$, (TM2) with $(\lambda ,\mu ,\nu )=(1,2,3)$ is $\mathrm{div}B=0$, and the three cyclic triples $(0,i,j)$ are the three components of $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$; conversely these eight scalar equations reproduce the two systems.

(b) [Form invariance under a change of inertial description] Let $x^{\mathrm{\prime }}$ be another inertial chart with transition $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+b$, $\mathrm{\Lambda }$ a proper orthochronous Lorentz matrix (the changes of inertial description adopted in Adopted inertial covariance of the model equations), and let the primed components of the same objects be the tensor components of Transformation of tensor, covector, derivative and current components under affine Lorentz maps (iii): $F^{\mathrm{\prime }ab}={\mathrm{\Lambda }}^a{}_c{\mathrm{\Lambda }}^b{}_d{F}^{cd}$, $F_{ab}^{\mathrm{\prime }}=({\mathrm{\Lambda }}^{-1}{)}^c{}_a({\mathrm{\Lambda }}^{-1}{)}^d{}_b{F}_{cd}$, $j^{\mathrm{\prime }a}={\mathrm{\Lambda }}^a{}_b{j}^b$. Then (TM1) holds in $x$ if and only if (TM1) holds in $x^{\mathrm{\prime }}$ with $j^{\mathrm{\prime }}$, and (TM2) holds in $x$ if and only if it holds in $x^{\mathrm{\prime }}$; by (a) applied in both charts the four smooth-branch equations are likewise form-invariant, the primed fields satisfying the primed equations with the primed sources.

(c) [Hypotheses actually used] Only first derivatives of $E,B$ occur; the equivalence uses exactly the component dictionary, the alternating character of $F$, the constants ${\mu }_0,c,{\epsilon }_0$ and the identification $j=(c\rho ,J)$, and the covariance clause uses the constant-coefficient component law and the invertibility of $\mathrm{\Lambda }$. No potential, no second derivative, no boundary, decay or smoothness hypothesis beyond the stated class, and no converse from the wave equations to the Maxwell equations is used or asserted.

(d) [Scope] Only the smooth branch is covered; the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches and field or source points of singular support are excluded. The conventions $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$ fix the signs of (TM1), and the alternative sign convention, which reverses the form of the sourced equation, is not mixed with them. The statement is conditional on the adopted postulates, uses no choice principle and makes no empirical claim.

The three cited treatments present the antisymmetric field tensor, the inhomogeneous equation in the form ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ (in suitable conventions) and the homogeneous cyclic identity, and derive the component dictionary from the laboratory fields with sign and index conversions recorded in the locators. The deduction above is the component check that the two systems say exactly the same thing as the four laboratory equations in the framework's conventions, together with the constant-coefficient form invariance; the textbook treatments state these facts for their own conventions, and the conversion to the framework's dictionary is carried out here rather than assumed.

Let $E,B$ be jointly $C^k$ fields of The electric and magnetic fields as fixed-time spatial vector fields on the laboratory region $U\cong I\times \mathrm{\Omega }$ of an inertial chart $x$ of the framework (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields), $k\ge 1$, with the SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $F$ be the electromagnetic two-form of The electromagnetic two-form, its observer decomposition and the four-current with its fixed components $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$, $F^{0i}=E_i\mathrm{/}c$, $F^{ij}={ϵ}_{ijk}{B}_k$. Let $0\le \beta <1$, $\gamma :=(1-{\beta }^2{)}^{-1\mathrm{/}2}$, $v:=\beta c{e}_1$, and let ${\mathrm{\Lambda }}_{\beta }$ be the explicit boost of The explicit Lorentz boost preserves $\eta$, has determinant $1$ and preserves the future cone with ${\mathrm{\Lambda }}_{\beta }{}^0{}_0={\mathrm{\Lambda }}_{\beta }{}^1{}_1=\gamma$, ${\mathrm{\Lambda }}_{\beta }{}^0{}_1={\mathrm{\Lambda }}_{\beta }{}^1{}_0=-\gamma \beta$, ${\mathrm{\Lambda }}_{\beta }{}^2{}_2={\mathrm{\Lambda }}_{\beta }{}^3{}_3=1$ and all other entries zero. Let $x^{\mathrm{\prime }}={\mathrm{\Lambda }}_{\beta }x+b$ be the corresponding inertial chart, and for every event $p$ in the overlap of the two chart restrictions define the primed field components by reading the components of the same $F$ in $x^{\mathrm{\prime }}$ through the dictionary of The electromagnetic two-form, its observer decomposition and the four-current, so that $F_{0i}^{\mathrm{\prime }}=-E_i^{\mathrm{\prime }}\mathrm{/}c$, $F_{ij}^{\mathrm{\prime }}={ϵ}_{ijk}{B}_k^{\mathrm{\prime }}$ at $p$. Then:

(a) [Boosted fields] at every such $p$, with $\parallel$ and $\perp$ the components along and orthogonal to $e_1$ in the spatial Euclidean structure of the chart, $E_1^{\mathrm{\prime }}=E_1,B_1^{\mathrm{\prime }}=B_1,E_{\parallel }^{\mathrm{\prime }}=E_{\parallel },B_{\parallel }^{\mathrm{\prime }}=B_{\parallel },E_{\perp }^{\mathrm{\prime }}=\gamma (E+v\times B{)}_{\perp },B_{\perp }^{\mathrm{\prime }}=\gamma (B-v\times E\mathrm{/}{c}^2{)}_{\perp },$ equivalently, componentwise, $E_2^{\mathrm{\prime }}=\gamma (E_2-\beta c{B}_3)$, $E_3^{\mathrm{\prime }}=\gamma (E_3+\beta c{B}_2)$, $B_2^{\mathrm{\prime }}=\gamma (B_2+\beta E_3\mathrm{/}c)$, $B_3^{\mathrm{\prime }}=\gamma (B_3-\beta E_2\mathrm{/}c)$.

(a′) [Arbitrary boost direction] For a velocity vector $v\in {\mathbb{R}}^3$ with $\mathrm{\mid }v\mathrm{\mid }<c$, use the actual vector boost $B(v\mathrm{/}c)$ of The passive Lorentz boost with parameter beta, with ${\gamma }_v=(1-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$. If $v\mathrm{\ne }0$, decompose the fields along $\widehat{v}=v\mathrm{/}\mathrm{\mid }v\mathrm{\mid }$ and its orthogonal complement. Then $E_{\parallel }^{\mathrm{\prime }}=E_{\parallel }$, $B_{\parallel }^{\mathrm{\prime }}=B_{\parallel }$, $E_{\perp }^{\mathrm{\prime }}={\gamma }_v(E+v\times B{)}_{\perp }$ and $B_{\perp }^{\mathrm{\prime }}={\gamma }_v(B-v\times E\mathrm{/}{c}^2{)}_{\perp }$. Equivalently, for every $v$, including $v=0$, $E^{\mathrm{\prime }}={\gamma }_v(E+v\times B)-\frac{{\gamma }_v^2}{{\gamma }_v+1}\frac{v(v\cdot E)}{c^2},B^{\mathrm{\prime }}={\gamma }_v(B-v\times E\mathrm{/}{c}^2)-\frac{{\gamma }_v^2}{{\gamma }_v+1}\frac{v(v\cdot B)}{c^2}\mathrm{.}$ At $v=0$ these are the identity and no preferred parallel direction is assigned.

(b) [Invariants] at every such $p$, $\mathrm{\mid }{B}^{\mathrm{\prime }}{\mathrm{\mid }}^2-\frac{\mathrm{\mid }{E}^{\mathrm{\prime }}{\mathrm{\mid }}^2}{c^2}=\mathrm{\mid }B{\mathrm{\mid }}^2-\frac{\mathrm{\mid }E{\mathrm{\mid }}^2}{c^2},E^{\mathrm{\prime }}\cdot B^{\mathrm{\prime }}=E\cdot B;$ equivalently the scalar fields $I_1:=\frac{1}{2}{F}_{ab}{F}^{ab}=\mathrm{\mid }B{\mathrm{\mid }}^2-\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}{c}^2$ and $I_2:=F_{ab}(\ast F{)}^{ab}=4E\cdot B\mathrm{/}c$ of The electromagnetic two-form, its observer decomposition and the four-current (b) take the same values whether computed from the components in $x$ or in $x^{\mathrm{\prime }}$.

(c) [General proper orthochronous changes] for any proper orthochronous Lorentz matrix $\mathrm{\Lambda }$ with chart $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+b$, the primed components of $F$ are given by the component law of Transformation of tensor, covector, derivative and current components under affine Lorentz maps (iii), $F^{\mathrm{\prime }ab}={\mathrm{\Lambda }}^a{}_c{\mathrm{\Lambda }}^b{}_d{F}^{cd}$; in particular, if $\mathrm{\Lambda }=\widehat{R}$ is the extension of a spatial rotation $R$ of the frame (Spatial rotations of an inertial frame), then $E^{\mathrm{\prime }}=RE$ and $B^{\mathrm{\prime }}=RB$, and by the unique factorization $\mathrm{\Lambda }=B(-\beta )\widehat{R}$ of The proper orthochronous Lorentz group: boosts, rotations, unique factorization and invariants (iii) every proper orthochronous change acts first by $(E,B)\mapsto (RE,RB)$ and then by the arbitrary-direction boost law (a′) with velocity $-c\beta$, where $\beta$ is the vector parameter of that factorization. The axis-only law (a) is not substituted for a general vector boost.

(d) [Form invariance of the Maxwell branch] if $(E,B,\rho ,J)$ satisfy the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches in $x$ on the overlap, and $({\rho }^{\mathrm{\prime }},J^{\mathrm{\prime }})$ are the components of the same four-current in $x^{\mathrm{\prime }}$, then $(E^{\mathrm{\prime }},B^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }},J^{\mathrm{\prime }})$ satisfy the same four equations in $x^{\mathrm{\prime }}$, by Tensor Maxwell equations equal the four laboratory Maxwell equations.

(e) [Scope] All clauses are pointwise on the overlap $U\cap U^{\mathrm{\prime }}$ of the two chart restrictions, are conditional on the adopted postulates, and concern the smooth classical flat branch; no accelerating coordinates, curved spacetime, material medium, apparatus or empirical claim is involved; the transformation is exact algebra, and the primed component functions are $C^k$ wherever the overlap is open. The invariants clause uses only the proper orthochronous changes of description adopted in Adopted inertial covariance of the model equations; orientation-reversing changes are outside the framework's field conventions (The electric and magnetic fields as fixed-time spatial vector fields).