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Conformal Maxwell symmetry fails to preserve electrovac geometry

Statement refuted

Fix positive constants $c, G, μ_{0}$ and $κ = 8 π G / c^{4}$ as in Exact geometric electrovac normalization. On the smooth oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions, a source-free electrovac solution with $Λ = 0$ here means a metric $g$ and smooth real two-form $F$ satisfying

$G_{a b} (g) = \frac{κ}{μ_{0}} Q_{a b} (g, F), d F = 0, d (⋆_{g} F) = 0,$

where $Q_{a b} (g, F) = F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}$ with indices raised by $g^{- 1}$ , as in Exact geometric electrovac normalization, and $d, ⋆_{g}$ are the operators used in Four-dimensional conformal Maxwell covariance. These equations define a mathematical solution class, without an assertion about nature.

Claim: in four dimensions, conformal covariance of the source-free Maxwell equations implies that every positive conformal rescaling of a source-free electrovac solution is again a solution of the same electrovac Einstein equations with $Λ = 0$ , keeping $F$ and the numerical constants fixed.

Facts & Assumptions

Given: The conformal Maxwell covariance and Einstein-tensor transformation identities of Four-dimensional conformal Maxwell covariance and Conformal Einstein tensor transformation, on the smooth Lorentzian geometry of Oriented Lorentzian fields, observers and coordinate conventions. The inherited geometric setup carries ${A C}_{ω}$ as in [A1].

[F1]

A smooth positive conformal rescaling in four dimensions preserves the source-free Maxwell equations when $F = 0$ and $j = 0$ . (Four-dimensional conformal Maxwell covariance)

[F2]

For $\tilde{g} = Ω^{2} g$ and $u = Ω^{- 1} d Ω$ , the scalar curvature is $\tilde{S} = Ω^{- 2} (S - 6 θ - 6 s)$ , with $θ = g^{a b} \nabla_{a} u_{b}$ and $s = g^{a b} u_{a} u_{b}$ . (Conformal Einstein tensor transformation)

[F3]

The metric-selected connection has the Christoffel formula $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ . (Metric compatibility, torsion freedom and the metric-selected connection class)

[F4]

The Einstein tensor is $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ , so $g^{a b} G_{a b} = - S$ in four dimensions. (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors)

[F5]

The Lorentzian spacetime is a smooth oriented and time-oriented four-manifold with arbitrary smooth real two-forms, so it admits the zero field $F = 0$ on the stated domain. (Oriented Lorentzian fields, observers and coordinate conventions)

[F6]

The coupled equation and quadratic tensor are those of Exact geometric electrovac normalization: $G_{a b} = (κ / μ_{0}) Q_{a b} (g, F)$ at $Λ = 0$ ; its quadratic definition gives $Q (g, 0) = 0$ for either metric.

[A1]

The inherited choice assumption is Countable Choice ${A C}_{ω}$ , not full Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Counterexample

Given: $R^{4}$ with coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , Minkowski metric $η = diag (- 1, 1, 1, 1)$ , standard orientation and time orientation, zero field and current, and $Λ = 0$ . Choose a real $k \neq 0$ with inverse-length units, and define $Ω (x) = e^{k x^{1}}$ and $\tilde{g} = Ω^{2} η$ .

1.1givenF3F4F5F6A1algebra

The constant components of $η$ have zero Christoffel symbols by [F3], hence zero curvature and $G (η) = 0$ by [F4]. Also $Q (η, 0) = 0$ , while [F5] admits $F = 0$ , which satisfies both source-free Maxwell equations. Thus $(η, 0, 0)$ is an exact source-free electrovac solution.

2.1givenF2step 1.1algebra

For this $Ω$ , $u = Ω^{- 1} d Ω = k d x^{1}$ . Since the background connection is flat with constant coefficients, $H_{a b} = \nabla_{a} u_{b} = 0$ and $θ = 0$ ; because $x^{1}$ is spacelike, $s = η^{a b} u_{a} u_{b} = k^{2}$ . The scalar-curvature formula [F2] therefore gives $\tilde{S} = - 6 k^{2} e^{- 2 k x^{1}}$ , which is nonzero at every point.

3.1F1F4F6step 2.1algebra∎

Maxwell remains satisfied because the field and current are still zero. If the rescaled pair solved the electrovac Einstein equation with $Λ = 0$ , it would have $\tilde{G} = 0$ ; tracing this equation and using [F4] would force $\tilde{S} = 0$ , contradicting step 2.1. Hence conformal Maxwell covariance does not imply conformal electrovac Einstein covariance.

Conformal Maxwell symmetry fails to preserve electrovac geometry

Statement refuted

Facts & Assumptions

Counterexample

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