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Four-dimensional conformal Maxwell covariance

Statement

Let $(M, g, N, o)$ and a smooth real two-form $F$ be as in Oriented Lorentzian fields, observers and coordinate conventions. Let $Ω : M \to (0, \infty)$ be a smooth dimensionless function, put $\tilde{g} = Ω^{2} g$ with the same orientation, let $μ_{0} > 0$ be constant, and let $j$ be a smooth vector field. The Hodge operator is that of Conformal Hodge scaling in every degree. Define

$Q_{a b} (g, F) : = F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}, T_{a b} (g, F) : = μ_{0}^{- 1} Q_{a b} (g, F) .$

Then $(F, j)$ satisfies the differential-form Maxwell equations

$d F = 0, d (⋆_{g} F) = μ_{0} ι_{j} {vol}_{g}$

if and only if $(F, \tilde{j})$ satisfies the same equations for $\tilde{g}$ , where $\tilde{j} : = Ω^{- 4} j$ . In addition,

${vol}_{\tilde{g}} = Ω^{4} {vol}_{g}, ι_{\tilde{j}} {vol}_{\tilde{g}} = ι_{j} {vol}_{g}, {\tilde{T}}_{a b} = Ω^{- 2} T_{a b}, {\tilde{T}}^{a b} = Ω^{- 6} T^{a b} .$

If $n$ is a $g$ -unit timelike vector, then $\tilde{n} : = Ω^{- 1} n$ is $\tilde{g}$ -unit timelike and $\tilde{T} (\tilde{n}, \tilde{n}) = Ω^{- 4} T (n, n)$ . On any oriented integration domain where the integral exists, or for a compactly supported action difference, the quadratic integral $\int F_{a b} F^{a b} {vol}_{g}$ is unchanged. These are tensor identities; they do not assert Einstein-equation or matter-model invariance.

Facts & Assumptions

Given: The smooth oriented Lorentzian manifold, volume form and smooth two-form of Oriented Lorentzian fields, observers and coordinate conventions, a positive smooth $Ω$ , a constant $μ_{0} > 0$ , and a smooth vector field $j$ . The inherited geometric setup carries ${A C}_{ω}$ as in [A1].

[F1]

The Hodge operators of $g$ and $\tilde{g} = Ω^{2} g$ satisfy $⋆_{\tilde{g}} β = Ω^{4 - 2 k} ⋆_{g} β$ for every smooth $k$ -form $β$ ; hence they agree on two-forms. (Conformal Hodge scaling in every degree)

[F2]

The metric volume form in an oriented chart is $\sqrt{∣ \det (g_{a b}) ∣} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ . (Oriented Lorentzian fields, observers and coordinate conventions)

[A1]

The inherited choice assumption is Countable Choice ${A C}_{ω}$ , not the full Axiom of Choice; the pointwise conformal identities below use no additional choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1givenF1F2A1algebra

In dimension four, $\det ({\tilde{g}}_{a b}) = Ω^{8} \det (g_{a b})$ . Since $Ω > 0$ , [F2] therefore gives ${vol}_{\tilde{g}} = Ω^{4} {vol}_{g}$ . By [F1], $⋆_{\tilde{g}} F = ⋆_{g} F$ .

1.2givenalgebra

If $d F = 0$ , then the same two-form $F$ still obeys $d F = 0$ for $\tilde{g}$ , because the exterior derivative does not depend on the metric.

1.3givenF1algebra

The lower tensor satisfies ${\tilde{F}}_{b}^{c} = Ω^{- 2} F_{b}^{c}$ and ${\tilde{F}}_{c d} {\tilde{F}}^{c d} = Ω^{- 4} F_{c d} F^{c d}$ . Substitution in its definition gives ${\tilde{Q}}_{a b} = Ω^{- 2} Q_{a b}$ , hence ${\tilde{T}}_{a b} = Ω^{- 2} T_{a b}$ . Raising both indices with ${\tilde{g}}^{- 1} = Ω^{- 2} g^{- 1}$ gives ${\tilde{T}}^{a b} = Ω^{- 6} T^{a b}$ .

2.1givenstep 1.1F1algebra

Using step 1.1 and linearity of contraction in its vector argument, $ι_{\tilde{j}} {vol}_{\tilde{g}} = ι_{Ω^{- 4} j} (Ω^{4} {vol}_{g}) = ι_{j} {vol}_{g}$ . Thus $d (⋆_{\tilde{g}} F) = d (⋆_{g} F) = μ_{0} ι_{j} {vol}_{g} = μ_{0} ι_{\tilde{j}} {vol}_{\tilde{g}}$ whenever the original sourced equation holds.

2.2givenstep 1.3algebra

Since $\tilde{g} (Ω^{- 1} n, Ω^{- 1} n) = g (n, n) = - 1$ , the rescaled vector is unit timelike. Contracting the lower-tensor identity twice with $\tilde{n} = Ω^{- 1} n$ yields $\tilde{T} (\tilde{n}, \tilde{n}) = Ω^{- 4} T (n, n)$ .

3.1givenstep 1.2step 2.1algebra

Conversely, $\tilde{j} = Ω^{- 4} j$ implies $j = Ω^{4} \tilde{j}$ . Reversing the contraction equality in step 2.1 and using $⋆_{g} F = ⋆_{\tilde{g}} F$ gives the original sourced equation; the homogeneous equation is unchanged by step 1.2.

4.1givenstep 1.1algebra∎

Raising the two indices of $F$ contributes $Ω^{- 4}$ and the volume form contributes $Ω^{4}$ , so $F_{a b} F^{a b} {vol}_{g}$ is invariant pointwise. Integration over the same domain preserves equality whenever defined, as do compactly supported action differences.

Four-dimensional conformal Maxwell covariance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources