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A flat time-oriented Lorentzian four-manifold with a closed timelike curve

Statement refuted

Statement refuted: "every smooth time-oriented flat Lorentzian four-manifold is chronological, that is, contains no closed future timelike curve" (Chronology, causality, strong causality, global hyperbolicity and time functions clause (a), Causal curves, chronological and causal reachability, and inextendibility clause (c)).

Facts & Assumptions

Given: The flat cylinder model $(M, g, N) = A / Λ$ of The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set, built from the flat model $(A, η, N)$ of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets with $Λ = {n L e_{0} : n \in Z}$ , $L > 0$ , orbit map $q$ , a fixed point $o \in A$ , and the curve $γ : R \to M$ , $γ (s) = q (o + s e_{0})$ .

[L1]

$M$ is a smooth four-manifold, $g$ is a smooth Lorentzian metric obtained by descent so that $g_{q (p)} (d q_{p} V, d q_{p} W) = η_{p} (V, W)$ , $N = d q (\partial_{0})$ is a time orientation, the model is flat, and the curve $γ$ is a nonconstant smooth future timelike curve with $γ (s + L) = γ (s)$ for all $s$ , hence a closed future timelike curve (The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set clauses (a), (b), The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets clauses (b), (c)).

[L2]

A model is chronological when no future timelike curve $γ : [a, b] \to M$ with $a < b$ and $γ (a) = γ (b)$ exists, and nonconstant on $[a, b]$ (Chronology, causality, strong causality, global hyperbolicity and time functions clause (a), Causal curves, chronological and causal reachability, and inextendibility clauses (b), (c)).

Counterexample

1.1givenL1

(The witness is a future timelike curve.) By [L1] $γ$ is a smooth mapping into $M$ with velocity $γ^{'} (s) = d q_{o + s e_{0}} (\partial_{0})$ ; the local isometry identity gives $g (γ^{'}, γ^{'}) = η (\partial_{0}, \partial_{0}) = - 1 < 0$ and $g (γ^{'}, N) = η (\partial_{0}, \partial_{0}) = - 1 < 0$ , so every tangent is future unit timelike and $γ$ is a future timelike curve in the class of Causal curves, chronological and causal reachability, and inextendibility.

2.1step 1.1givenL1

(The witness is closed and nonconstant on one period.) By [L1] $γ (s + L) = q (o + s e_{0} + L e_{0}) = q (o + s e_{0}) = γ (s)$ because $L e_{0} \in Λ$ . It is nonconstant on $[0, L]$ : if $γ (s) = γ (0)$ for some $0 < s < L$ , then $s e_{0} \in Λ$ , so $s = n L$ for some nonzero integer $n$ , which is impossible for $0 < s < L$ . Hence $γ ∣_{[0, L]}$ is a nonconstant future timelike curve with equal endpoints.

3.1step 2.1givenL1L2∎

(Conclusion.) By step 2.1 and [L2] the model $(M, g, N)$ is not chronological: it contains a closed future timelike curve. Since $M$ is flat and time-oriented by [L1], this refutes the displayed statement. It also shows that a model with a closed timelike curve is not causal, and by The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set clause (d) it admits no Cauchy set at all.

Remarks

Where flatness survives. The identification acts by isometries and the quotient is locally isometric to the flat model; the pathology is global and invisible to curvature. This is the flat analogue of the Misner-space warning recorded by Carroll.
Scope. The model is a mathematical quotient of the flat model of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets; the counterexample says nothing about the physical realizability of such an identification.

A flat time-oriented Lorentzian four-manifold with a closed timelike curve

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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