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Tensor Maxwell equations equal the four laboratory Maxwell equations

Physical scope

Conditional statement about the smooth branch of the adopted vacuum Maxwell postulate on the laboratory region: the four SI Maxwell equations are equivalent to the two tensor systems (TM1) and (TM2) for the components of the electromagnetic two-form and four-current fixed by the framework's definitions, with eta=diag(-1,1,1,1), x^0=ct, mu_0 c^2 eps_0=1, and form-invariance under proper orthochronous changes of inertial chart. Excluded: the weak branch, singular source support, magnetic charge, potentials, and any dynamics beyond the adopted equations; no choice principle and no empirical claim.

Statement

Let the laboratory region $U ≅ I \times Ω$ 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 $(ρ, J)$ 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 $ε_{0} : = (μ_{0} c^{2})^{- 1}$ 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, $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , $F^{0 i} = E_{i} / c$ , $F^{i j} = ϵ_{i j k} B_{k}$ and $η = diag (- 1, 1, 1, 1)$ , and let $j = (c ρ, J)$ carry the tensor-component identification adopted in Charge parameter, frame charge and current densities, volume charge and flux, and the four-current. Write $\partial_{0} = c^{- 1} \partial_{t}$ and $\partial_{i}$ for the coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , and $div, curl$ 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 $Ω \times I$ with the additional derivatives required by each displayed equation existing and continuous, and $ρ, J$ are continuous. Then:

(a) [Equivalence] On $I \times Ω$ the four smooth-branch equations

$div E = ρ / ε_{0}, div B = 0, curl E = - \partial_{t} B, curl B = μ_{0} J + c^{- 2} \partial_{t} E$

hold if and only if the two systems

$(T M 1) \partial_{μ} F^{μ ν} = - μ_{0} j^{ν} (ν = 0, 1, 2, 3),$

$(T M 2) \partial_{λ} F_{μ ν} + \partial_{μ} F_{ν λ} + \partial_{ν} F_{λ μ} = 0 (λ, μ, ν \in {0, 1, 2, 3})$

hold on $I \times Ω$ . 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 $ν = 0$ is $div E = ρ / ε_{0}$ (using $μ_{0} c^{2} ε_{0} = 1$ ), (TM1) with $ν = i$ is the $i$ -th component of $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , (TM2) with $(λ, μ, ν) = (1, 2, 3)$ is $div B = 0$ , and the three cyclic triples $(0, i, j)$ are the three components of $curl E = - \partial_{t} B$ ; conversely these eight scalar equations reproduce the two systems.

(b) [Form invariance under a change of inertial description] Let $x^{'}$ be another inertial chart with transition $x^{'} = Λ x + b$ , $Λ$ 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): $F^{' a b} = Λ^{a}_{c} Λ^{b}_{d} F^{c d}$ , $F_{a b}^{'} = (Λ^{- 1})^{c}_{a} (Λ^{- 1})^{d}_{b} F_{c d}$ , $j^{' a} = Λ^{a}_{b} j^{b}$ . Then (TM1) holds in $x$ if and only if (TM1) holds in $x^{'}$ with $j^{'}$ , and (TM2) holds in $x$ if and only if it holds in $x^{'}$ ; 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 $μ_{0}, c, ε_{0}$ and the identification $j = (c ρ, J)$ , and the covariance clause uses the constant-coefficient component law and the invertibility of $Λ$ . 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 $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ 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.

Facts & Assumptions

Given: The laboratory region $U ≅ I \times Ω$ with its inertial chart $x$ ( $x^{0} = c t$ ); the jointly $C^{1}$ fields $E, B$ with the additional derivatives required by each Maxwell equation existing and continuous; the continuous source fields $(ρ, J)$ ; the electromagnetic two-form $F$ with the component dictionary and index conventions of The electromagnetic two-form, its observer decomposition and the four-current; the four-current $j = (c ρ, J)$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current; the constants $μ_{0}, c > 0$ and $ε_{0} = (μ_{0} c^{2})^{- 1}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values; the smooth-branch Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches; and a second inertial chart $x^{'} = Λ x + b$ with $Λ$ a proper orthochronous Lorentz matrix as adopted in Adopted inertial covariance of the model equations.

[F1]

In the chart $x$ the electromagnetic two-form has components $F_{00} = 0$ , $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , $F_{a b} = - F_{b a}$ , $F^{0 i} = E_{i} / c$ , $F^{i 0} = - E_{i} / c$ , $F^{i j} = ϵ_{i j k} B_{k}$ , and $η = diag (- 1, 1, 1, 1)$ ; here $ϵ_{i j k}$ is the spatial permutation symbol, so $ϵ_{i j k} = + 1$ for even and $- 1$ for odd permutations of $(1, 2, 3)$ and $ϵ_{i j k} = 0$ if two indices coincide (The electromagnetic two-form, its observer decomposition and the four-current (a), Framework-local SI units, the vacuum constants and the exact/measured status of their values). The four-current components are $j^{0} = c ρ$ and $j^{i} = J^{i}$ (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (c)).

[F2]

On $U$ the smooth branch of the vacuum Maxwell postulate holds: $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , for jointly $C^{1}$ fields with the additional derivatives required by each equation existing and continuous and continuous sources (The microscopic vacuum Maxwell equations: smooth and weak branches (a)).

[F3]

For a $C^{1}$ vector field $G = (G_{1}, G_{2}, G_{3})$ on an open subset of $R^{3}$ , $div G = \partial_{1} G_{1} + \partial_{2} G_{2} + \partial_{3} G_{3}$ and $curl G = (\partial_{2} G_{3} - \partial_{3} G_{2}, \partial_{3} G_{1} - \partial_{1} G_{3}, \partial_{1} G_{2} - \partial_{2} G_{1})$ , so $(curl G)_{i} = ϵ_{i j k} \partial_{j} G_{k}$ for $i = 1, 2, 3$ (Divergence and curl of a $C^{1}$ vector field).

[F4]

The coordinate $x^{0}$ of the chart is $x^{0} = c t$ , so $\partial_{0} = c^{- 1} \partial_{t}$ , where $\partial_{t}$ is the partial derivative at fixed position and $\partial_{i}$ are the spatial partial derivatives of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields (a), (c), Framework-local SI units, the vacuum constants and the exact/measured status of their values (a)).

[F5]

The constants satisfy $ε_{0} = (μ_{0} c^{2})^{- 1}$ , hence $μ_{0} c^{2} ε_{0} = 1$ and $c^{- 2} = μ_{0} ε_{0}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values (a)).

[F6]

A change of inertial description is an affine map $x^{'} = Λ x + b$ whose linear part $Λ$ is a proper orthochronous Lorentz matrix; such a $Λ$ is invertible, with inverse matrix $(Λ^{- 1})^{a}_{b}$ satisfying $Λ^{a}_{b} (Λ^{- 1})^{b}_{c} = δ_{c}^{a}$ and $(Λ^{- 1})^{a}_{b} Λ^{b}_{c} = δ_{c}^{a}$ (Adopted inertial covariance of the model equations (ii)).

[F7]

For the same tensor field $F$ and vector field $j$ read in the two charts, the components transform as $F^{' a b} = Λ^{a}_{c} Λ^{b}_{d} F^{c d}$ , $F_{a b}^{'} = (Λ^{- 1})^{c}_{a} (Λ^{- 1})^{d}_{b} F_{c d}$ , $j^{' a} = Λ^{a}_{b} j^{b}$ , and the coordinate partial derivatives satisfy $\partial_{a}^{'} = (Λ^{- 1})^{c}_{a} \partial_{c}$ wherever the derivatives exist (Transformation of tensor, covector, derivative and current components under affine Lorentz maps (i), (iii), (iv)).

[L1]

Because $F$ is alternating, $F_{a b} = - F_{b a}$ and $F_{a a} = 0$ for all $a, b \in {0, 1, 2, 3}$ (The electromagnetic two-form, its observer decomposition and the four-current (a)).

Proof

technique · direct component computation in the fixed dictionary, followed by the constant-coefficient transformation law

1.1givenF1F3F5algebra

(Sourced equation, $ν = 0$ ) On $I \times Ω$ the equation (TM1) with $ν = 0$ holds if and only if $div E = ρ / ε_{0}$ . Indeed $\partial_{μ} F^{μ 0} = \partial_{0} F^{00} + \partial_{i} F^{i 0} = 0 + \partial_{i} (- E_{i} / c) = - div E / c$ by [F1] and [F3], using $F^{00} = 0$ , $F^{i 0} = - E_{i} / c$ ; the right-hand side is $- μ_{0} j^{0} = - μ_{0} c ρ$ by [F1]; the identity $- div E / c = - μ_{0} c ρ$ is equivalent to $div E = μ_{0} c^{2} ρ = ρ / ε_{0}$ by [F5] and $c > 0$ .

1.2givenF1F3F4L1algebra

(Sourced equations, $ν = i$ ) On $I \times Ω$ the three equations (TM1) with $ν = i$ , $i = 1, 2, 3$ , hold if and only if $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ . Indeed $\partial_{μ} F^{μ i} = \partial_{0} F^{0 i} + \partial_{j} F^{j i} = c^{- 2} \partial_{t} E_{i} + ϵ_{j i k} \partial_{j} B_{k} = c^{- 2} \partial_{t} E_{i} - (curl B)_{i}$ : the first term uses $F^{0 i} = E_{i} / c$ and $\partial_{0} = c^{- 1} \partial_{t}$ from [F4], the second uses $F^{j i} = - ϵ_{i j k} B_{k} = ϵ_{j i k} B_{k}$ by [F1] and [L1], and the last equality is $(curl B)_{i} = ϵ_{i j k} \partial_{j} B_{k}$ from [F3] together with $ϵ_{j i k} = - ϵ_{i j k}$ . Equating with $- μ_{0} j^{i} = - μ_{0} J_{i}$ by [F1] gives $(curl B)_{i} = μ_{0} J_{i} + c^{- 2} \partial_{t} E_{i}$ for each $i$ , that is $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ .

1.3givenF1L1algebra

(Repeated-index instances of (TM2) vanish) If two of $λ, μ, ν$ coincide, the cyclic sum in (TM2) vanishes identically on $U$ . For $λ = μ$ the sum is $\partial_{λ} F_{λ ν} + \partial_{λ} F_{ν λ} + \partial_{ν} F_{λ λ} = \partial_{λ} (F_{λ ν} + F_{ν λ}) + 0 = 0$ by [L1]; the cases $λ = ν$ and $μ = ν$ are the same computation with the terms relabelled, since in each case two of the three terms are $\pm$ the same derivative and the third contains the vanishing diagonal entry $F_{a a} = 0$ .

1.4givenF1F3algebra

(Instance $(1, 2, 3)$ is the Gauss law for $B$ ) The instance $(λ, μ, ν) = (1, 2, 3)$ of (TM2) reads $\partial_{1} F_{23} + \partial_{2} F_{31} + \partial_{3} F_{12} = \partial_{1} B_{1} + \partial_{2} B_{2} + \partial_{3} B_{3} = div B$ , because $ϵ_{231} = ϵ_{312} = ϵ_{123} = + 1$ gives $F_{23} = B_{1}$ , $F_{31} = B_{2}$ and $F_{12} = B_{3}$ in [F1], and [F3] identifies the sum with $div B$ . Hence this instance is equivalent to $div B = 0$ .

1.5givenF1F3F4L1algebra

(Cyclic instances $(0, i, j)$ are the Faraday components) For each cyclic ordering $(i, j, k)$ of $(1, 2, 3)$ the instance $(λ, μ, ν) = (0, i, j)$ of (TM2) reads $\partial_{0} F_{i j} + \partial_{i} F_{j 0} + \partial_{j} F_{0 i} = c^{- 1} \partial_{t} B_{k} + c^{- 1} \partial_{i} E_{j} - c^{- 1} \partial_{j} E_{i} = c^{- 1} (\partial_{t} B_{k} + (curl E)_{k})$ , using $\partial_{0} = c^{- 1} \partial_{t}$ [F4], $F_{i j} = B_{k}$ for the cyclic index order [F1], $F_{j 0} = - F_{0 j} = E_{j} / c$ and $F_{0 i} = - E_{i} / c$ [F1], [L1], and $(curl E)_{k} = ϵ_{k i j} \partial_{i} E_{j} = \partial_{i} E_{j} - \partial_{j} E_{i}$ for the cyclic order [F3]. Hence these three instances are equivalent to the three components of $curl E = - \partial_{t} B$ .

1.6givenF6F7algebra

(Transformation data) In the second chart $x^{'} = Λ x + b$ the component laws of [F7] hold, with the invertible matrix $Λ$ and its inverse entries $(Λ^{- 1})^{a}_{b}$ as in [F6]. The primed component functions are constant-coefficient finite linear combinations of the unprimed components, so on the overlap of the two chart restrictions they are again jointly $C^{1}$ with the same additional derivatives existing and continuous.

2.1givenstep 1.3step 1.4step 1.5F1L1algebra

(Reduction of all instances of (TM2)) Every instance of (TM2) is either a repeated-index instance, which vanishes identically by step 1.3, or an instance with distinct indices. For distinct indices, interchanging two of $λ, μ, ν$ changes the cyclic sum's sign: with $S (λ, μ, ν) : = \partial_{λ} F_{μ ν} + \partial_{μ} F_{ν λ} + \partial_{ν} F_{λ μ}$ one has $S (λ, ν, μ) = \partial_{λ} F_{ν μ} + \partial_{ν} F_{μ λ} + \partial_{μ} F_{λ ν} = - (\partial_{λ} F_{μ ν} + \partial_{μ} F_{ν λ} + \partial_{ν} F_{λ μ}) = - S (λ, μ, ν)$ by [L1], and the other two transpositions are the same computation after relabelling. Since transpositions generate the symmetric group, each of the $24$ distinct-index instances equals $\pm S$ of one of the four index sets ${1, 2, 3}$ , ${0, 1, 2}$ , ${0, 2, 3}$ , ${0, 3, 1}$ ; by steps 1.4 and 1.5 the system (TM2) is therefore equivalent to the four scalar equations $div B = 0$ and the three components of $curl E = - \partial_{t} B$ .

2.2givenstep 1.6F6F7algebra

((TM1) is form-invariant) Using [F7], whose $\partial_{a}^{'} = (Λ^{- 1})^{c}_{a} \partial_{c}$ has constant coefficients, $\partial_{a}^{'} F^{' a b} = (Λ^{- 1})^{c}_{a} \partial_{c} (Λ^{a}_{e} Λ^{b}_{f} F^{e f}) = (Λ^{- 1})^{c}_{a} Λ^{a}_{e} Λ^{b}_{f} \partial_{c} F^{e f} = δ_{e}^{c} Λ^{b}_{f} \partial_{c} F^{e f} = Λ^{b}_{f} \partial_{c} F^{c f}$ , the third equality by the inverse-matrix identity of [F6]. Since also $j^{' b} = Λ^{b}_{f} j^{f}$ by [F7], the equation (TM1) in $x^{'}$ is $Λ^{b}_{f} (\partial_{c} F^{c f} + μ_{0} j^{f}) = 0$ for every $b$ ; multiplying by $(Λ^{- 1})^{g}_{b}$ and summing over $b$ gives $\partial_{c} F^{c g} = - μ_{0} j^{g}$ for every $g$ , and substitution gives the converse. Hence (TM1) holds in $x$ if and only if it holds in $x^{'}$ with $j^{'}$ .

2.3givenstep 1.6F6F7algebra

((TM2) is form-invariant) Write $S_{a b c} : = \partial_{a} F_{b c} + \partial_{b} F_{c a} + \partial_{c} F_{a b}$ . By [F7] the three terms of the primed cyclic sum are $\partial_{λ}^{'} F_{μ ν}^{'} = (Λ^{- 1})^{a}_{λ} (Λ^{- 1})^{b}_{μ} (Λ^{- 1})^{c}_{ν} \partial_{a} F_{b c}$ , $\partial_{μ}^{'} F_{ν λ}^{'} = (Λ^{- 1})^{a}_{λ} (Λ^{- 1})^{b}_{μ} (Λ^{- 1})^{c}_{ν} \partial_{b} F_{c a}$ and $\partial_{ν}^{'} F_{λ μ}^{'} = (Λ^{- 1})^{a}_{λ} (Λ^{- 1})^{b}_{μ} (Λ^{- 1})^{c}_{ν} \partial_{c} F_{a b}$ , the second and third after relabelling the dummy summation indices $a, b, c$ . Their sum is $S_{λ μ ν}^{'} = (Λ^{- 1})^{a}_{λ} (Λ^{- 1})^{b}_{μ} (Λ^{- 1})^{c}_{ν} S_{a b c}$ ; hence $S = 0$ implies $S^{'} = 0$ , and conversely $S_{a b c} = Λ^{λ}_{a} Λ^{μ}_{b} Λ^{ν}_{c} S_{λ μ ν}^{'}$ by the inverse-matrix identity of [F6], so $S^{'} = 0$ implies $S = 0$ . Thus (TM2) holds in $x$ if and only if it holds in $x^{'}$ .

3.1givenstep 1.1step 1.2step 1.4step 1.5step 2.1F2algebra

(Equivalence of clause (a)) If the four smooth-branch equations hold on $I \times Ω$ , then (TM1) holds by steps 1.1 and 1.2 and (TM2) holds by steps 1.3, 1.4, 1.5 and 2.1: the repeated-index instances vanish, the instance $(1, 2, 3)$ is $div B = 0$ , and the three cyclic instances vanish because $curl E = - \partial_{t} B$ . Conversely, if (TM1) and (TM2) hold, then steps 1.1 and 1.2 give $div E = ρ / ε_{0}$ and the three components of $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , while steps 1.4 and 1.5 give $div B = 0$ and the three components of $curl E = - \partial_{t} B$ . This proves the biconditional and identifies the eight scalar equations: (TM1)'s four components and, from the reduction of step 2.1, the distinct group $div B = 0$ and the three Faraday components; all remaining instances of (TM2) are repeats of these up to sign.

4.1givenstep 3.1step 1.6step 2.2step 2.3F6

(Form invariance of the four equations) The computation of steps 1.1-3.1 is a statement about an inertial chart and its component dictionary and applies verbatim in the chart $x^{'}$ , whose primed fields and sources are again of the required class by step 1.6: the four primed smooth-branch equations hold if and only if (TM1) and (TM2) hold in $x^{'}$ with $j^{'}$ . By steps 2.2 and 2.3 the primed tensor systems hold if and only if the unprimed ones do; by step 3.1 the latter hold if and only if the four equations hold in $x$ . Therefore the four smooth-branch equations hold in $x$ if and only if the same four equations, with the primed fields and primed sources, hold in $x^{'}$ . This is conditional on the adopted covariance premise [F6] and proves no physical equivalence of descriptions beyond it.

5.1givenstep 1.1step 4.1F1F2F5F6F7∎

(Hypotheses and scope) Steps 1.1-4.1 used only first derivatives of $E, B$ ; the equivalence used the component dictionary [F1], the alternating character of $F$ [L1], the constants $μ_{0}, c, ε_{0}$ [F5] and the identification $j = (c ρ, J)$ ; the covariance clauses used the constant-coefficient component law [F7] and the invertibility of $Λ$ [F6]. No potential, no second derivative, no boundary, decay or additional smoothness hypothesis was used, no converse from the wave equations to the Maxwell equations is asserted, and no step covers the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches or singular source support; the signs are fixed by the conventions $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ and no alternative dictionary is mixed in. The argument is finite linear algebra over the fixed component functions; it uses no choice principle and makes no empirical claim.

Source notes

The three cited treatments present the antisymmetric field tensor, the inhomogeneous equation in the form $\partial_{μ} F^{μ ν} = - μ_{0} j^{ν}$ (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.

Tensor Maxwell equations equal the four laboratory Maxwell equations

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

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