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Zero quadratic invariants need not mean flatness

Statement refuted

For every smooth source-free geometric electrovac datum, if $F_{a b} F^{a b}$ , the scalar curvature $S$ , ${Ric}_{a b} {Ric}^{a b}$ , and $R_{a b c d} R^{a b c d}$ all vanish, then $F = 0$ and the metric is flat, meaning its Riemann tensor vanishes identically as fixed in Null wave curvature and zero quadratic invariants.

Facts & Assumptions

Given: The smooth null-wave electrovac family and its quadratic invariants.

[F1]

Smooth null electrovac wave candidate data defines the candidate data on $R^{4}$ from smooth profiles $a, b$ and a transverse harmonic function $h$ .

[F2]

Geometric electrovac equations and flux normalization defines the source-free Einstein-Maxwell equations and curvature convention.

[F3]

Exact electrovac plane-wave family proves the defined family is a source-free geometric electrovac.

[F4]

Null wave curvature and zero quadratic invariants proves the four scalar invariants in the Statement vanish identically and gives $R_{u x u x} = - P_{x x} / 2$ in the defined coordinates.

[F5]

Null wave curvature and zero quadratic invariants fixes “flat” to mean that the Riemann tensor vanishes identically on the four-manifold.

[A1]

Fix a constant $a_{0} \neq 0$ with inverse-length units, and set $a (u) = a_{0}$ , $b (u) = 0$ , and $h (u, x, y) = 0$ .

Counterexample

1.1givenF1F2F3A1construct

The choices in [A1] satisfy the smoothness and transverse-harmonic hypotheses in [F1]. They give $P = - a_{0}^{2} (x^{2} + y^{2})$ and $F = a_{0} d u \land d x$ , which is nonzero. Under the definition in [F2], [F3] proves this is a smooth source-free geometric electrovac datum.

2.1givenF1F3F4F5A1step 1.1algebra∎

Applying [F4] gives $F_{a b} F^{a b} = S = {Ric}_{a b} {Ric}^{a b} = R_{a b c d} R^{a b c d} = 0$ . For the constant profile in [A1], [F3] also gives ${Ric}_{u u} = 2 a_{0}^{2} \neq 0$ . But $P_{x x} = - 2 a_{0}^{2}$ , so $R_{u x u x} = - P_{x x} / 2 = a_{0}^{2} \neq 0$ . Thus the field and Ricci tensor are nonzero and, by [F5], the metric is not flat although all four displayed scalar invariants vanish.

Zero quadratic invariants need not mean flatness

Statement refuted

Facts & Assumptions

Counterexample

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