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A congruence whose rest spaces are not tangent to any local hypersurfaces

Statement refuted

Statement refuted: "for every observer congruence on a nonempty open subset of flat Lorentzian four-space, the family of rest spaces is tangent to local hypersurfaces" (Smoothness of the rest-space family and its Frobenius integrability criterion).

Facts & Assumptions

Given: The shearing congruence $n = γ (\partial_{0} + Ω \partial_{2})$ on $W = {∣ λ x^{1} ∣ < 1 / 2} \subseteq R^{4}$ with its flat Lorentzian metric $g = diag (- 1, 1, 1, 1)$ , time orientation $N = \partial_{0}$ , and the one-form $α = n^{♭}$ .

[L1]

On $W$ the field $n$ is an observer congruence with $Ω = λ x^{1}$ , $γ = (1 - Ω^{2})^{- 1 / 2} > 0$ , $λ > 0$ , and its rest space at $q$ is $n_{q}^{⊥} = \ker α_{q}$ (A shearing observer congruence on flat spacetime: tetrad, adapted chart and non-coordinate frame, Smoothness of the rest-space family and its Frobenius integrability criterion).

[L2]

The rest spaces of a congruence are tangent to local hypersurfaces if and only if $α \land d α = 0$ on the domain, where $α = n^{♭}$ (Smoothness of the rest-space family and its Frobenius integrability criterion).

[L3]

The flat map is $α_{q} (v) = g_{q} (n_{q}, v)$ , so in the chart $x$ the components of $α$ are $α_{a} = \sum_{b} g_{a b} n^{b}$ (Smoothness of the rest-space family and its Frobenius integrability criterion, A smooth differential $k$ -form).

[L4]

The exterior derivative is $R$ -linear of degree one and satisfies $d (β \land γ) = d β \land γ + (- 1)^{\deg β} β \land d γ$ ; the exterior derivative of a smooth function is its differential (The exterior derivative is a graded derivation, The exterior derivative of a function is its differential, The exterior derivative by the invariant vector-field formula).

[L5]

The wedge product of differential forms is defined pointwise by the wedge product of alternating covectors (The wedge product of differential forms).

Counterexample

1.1L1L3algebra

In the chart $x$ the components of $n$ are $(γ, 0, γ Ω, 0)$ and $g_{a b} = diag (- 1, 1, 1, 1)_{a b}$ , so by [L3] the components of $α$ are $α_{0} = - γ$ , $α_{2} = γ Ω$ and $α_{1} = α_{3} = 0$ ; that is, $α = γ (- d x^{0} + Ω d x^{2})$ , a smooth $1$ -form on $W$ .

2.1step 1.1L4algebra

Since $γ$ and $Ω$ depend only on $x^{1}$ , their differentials are $d γ = γ^{'} d x^{1}$ and $d Ω = Ω^{'} d x^{1}$ by [L4], with $Ω^{'} = λ$ and $γ^{'} = γ^{3} Ω Ω^{'}$ . Applying the Leibniz rule of [L4] twice gives $d α = d γ \land (- d x^{0} + Ω d x^{2}) + γ d (- d x^{0} + Ω d x^{2}) = γ^{'} d x^{1} \land (- d x^{0}) + γ^{'} Ω d x^{1} \land d x^{2} + γ Ω^{'} d x^{1} \land d x^{2} = γ^{'} d x^{0} \land d x^{1} + (γ^{'} Ω + γ Ω^{'}) d x^{1} \land d x^{2}$ .

3.1step 2.1L5algebra

By [L5] and step 2.1, $α \land d α = γ (- d x^{0} + Ω d x^{2}) \land [γ^{'} d x^{0} \land d x^{1} + (γ^{'} Ω + γ Ω^{'}) d x^{1} \land d x^{2}]$ . The terms containing a repeated coordinate differential vanish, and using $d x^{2} \land d x^{0} \land d x^{1} = d x^{0} \land d x^{1} \land d x^{2}$ we get $α \land d α = γ [- (γ^{'} Ω + γ Ω^{'}) + Ω γ^{'}] d x^{0} \land d x^{1} \land d x^{2} = - γ^{2} Ω^{'} d x^{0} \land d x^{1} \land d x^{2} = - γ^{2} λ d x^{0} \land d x^{1} \land d x^{2}$ . Since $λ > 0$ and $γ > 0$ on $W$ , the coefficient $- γ^{2} λ$ is nonzero at every point of $W$ , so $α \land d α$ is nowhere zero on $W$ .

4.1step 3.1L2∎

By [L2], the rest spaces of the congruence are tangent to local hypersurfaces if and only if $α \land d α = 0$ ; since $α \land d α \neq 0$ at every point of $W$ by step 3.1, they are not tangent to any local hypersurface. This refutes the displayed general statement.

Remarks

By the companion example A shearing observer congruence on flat spacetime: tetrad, adapted chart and non-coordinate frame this same congruence does admit an adapted chart, so "existence of an adapted chart" and "integrability of the rest spaces" are genuinely different properties: the first always holds locally, the second fails here. The counterexample makes no claim about congruences with vanishing vorticity, and no global foliation claim.

A congruence whose rest spaces are not tangent to any local hypersurfaces

Statement refuted

Facts & Assumptions

Counterexample

Remarks

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