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Nonzero flat test field is not a coupled flat solution

Statement refuted

For $Λ = 0$ , every smooth source-free Maxwell field on a prescribed Minkowski metric is an electrovacuum solution with that same flat metric.

Facts & Assumptions

Given: The tensor system and constraint conventions of Electrovacuum tensor Cauchy system and Constrained electrovacuum initial data, with $κ > 0$ , $μ_{0} > 0$ , $Λ = 0$ , and $B_{0} \neq 0$ .

[F1]

Electrovacuum tensor Cauchy system: a coupled solution must satisfy both Maxwell equations and $G + Λ g = κ T^{E M}$ .

[F2]

Maxwell stress algebra and covariant conservation: for $e = 0$ and $b = (0, 0, B_{0})$ , $T^{E M} (n, n) = B_{0}^{2} / (2 μ_{0})$ .

[F3]

Gauss-Codazzi and Maxwell initial constraints: any smooth coupled solution induces data satisfying the Hamiltonian constraint.

Counterexample

Given: The system and constraint conventions above; $κ, μ_{0} > 0$ , $Λ = 0$ , and a fixed $B_{0} \neq 0$ (tesla in the SI interpretation).

1.1F1algebra

Take $M = R^{4}$ with length coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}$ , orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , and future field $N = \partial_{x^{0}}$ ; in the SI interpretation $x^{0} = c t$ . Set $F = B_{0} d x^{1} \land d x^{2}$ . The metric coefficients and those of $F$ and $⋆_{g} F$ are constant, so the curvature vanishes and $d F = d (⋆_{g} F) = 0$ . Hence this is a smooth source-free Maxwell field on the prescribed Minkowski background.

2.1F1F2F3algebra∎

On the slice $x^{0} = 0$ , the data are $h = δ$ , $K = 0$ , $e = 0$ , and $b = (0, 0, B_{0})$ . The Hamiltonian constraint from [F3] would require $0 = (κ / μ_{0}) B_{0}^{2}$ , because $R (h) = 0$ and $Λ = 0$ . This is impossible since $κ, μ_{0} > 0$ and $B_{0} \neq 0$ . Hence the given flat Maxwell test field is not a coupled electrovacuum solution with that metric.

Nonzero flat test field is not a coupled flat solution

Statement refuted

Facts & Assumptions

Counterexample

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