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The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set

Statement

Let $A : = R^{4}$ carry the flat structure $(η, N)$ of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets, with global inertial chart $x = (x^{0}, z)$ and translations $τ_{λ} (p) = p + λ$ of clause (d), and write $e_{0} : = (1, 0, 0, 0) \in A$ . Fix $L > 0$ , let $Λ : = {n L e_{0} : n \in Z} \subseteq A$ act on $A$ by the translations $τ_{λ} (p) = p + λ$ and fix $o \in A$ (the action of the discrete group $Z$ , equipped with its discrete smooth structure, is a special case of the actions of Free and proper Lie-group actions). Let $(M, g, N)$ be the quotient time-oriented Lorentzian four-manifold produced by Quotients of Lorentzian manifolds by free proper isometric discrete actions, with orbit map $q : A \to M$ . Then:

(a) $q$ is a local isometry: $g_{q (p)} (d q_{p} V, d q_{p} W) = η_{p} (V, W)$ for all $p \in A$ and $V, W \in T_{p} A$ ; and $(M, g)$ is flat, in the sense that every point has a chart in which the metric components are the constants $diag (- 1, 1, 1, 1)$ and the curvature tensor of the metric-selected connection vanishes identically.

(b) The curve $γ : R \to M$ , $γ (s) : = q (o + s e_{0})$ , is a nonconstant smooth future timelike curve with $γ (s + L) = γ (s)$ for all $s$ ; hence $(M, g, N)$ is neither chronological nor causal, and it possesses a closed future timelike curve.

(c) Every two points $p, q \in M$ (equal or distinct) satisfy $p ≪ q$ : the model is totally vicious. Consequently no subset of $M$ with two distinct points is achronal or acausal.

(d) $M$ admits no Cauchy set and hence no Cauchy hypersurface in the sense of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons.

Facts & Assumptions

Given: The flat model $(A, η, N)$ of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets, the vector $e_{0}$ , the number $L > 0$ , the translation group $Λ$ , and the quotient of the descent lemma.

[L1]

A translation $τ_{λ}$ of $A$ preserves $η$ : $η_{p + λ} (V, W) = η_{p} (V, W)$ for all $p, V, W$ . Indeed, in the global chart $x$ the components of $η$ are the constants $diag (- 1, 1, 1, 1)$ , and the coordinate representative $u \mapsto u + x (λ)$ of $τ_{λ}$ has the identity as its Jacobian, so the components of $τ_{λ}^{*} η$ equal those of $η$ (The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets clauses (a), (b), (d)).

[L2]

The action of $Λ$ on $A$ is smooth, free (no $n \neq 0$ fixes a point, since $n L e_{0} \neq 0$ ) and proper (for a compact $K$ , the finitely many $λ$ with $(K + λ) \cap K \neq \emptyset$ are bounded by $diam K$ ), and the field $N = \partial_{0}$ of the flat model is invariant under it (Free and proper Lie-group actions, The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets clauses (a) and (c)).

[L3]

The descent lemma applies: $M = A / Λ$ is a Hausdorff second-countable smooth four-manifold, $q$ is a smooth surjective local diffeomorphism and a local isometry for the descended metric, the descended field $N : = d q (\partial_{0})$ is a time orientation, and smooth local sections of $q$ provide charts with $g = s^{*} η$ (Quotients of Lorentzian manifolds by free proper isometric discrete actions).

[L4]

Metric-selected connection in a constant-component chart. Let $(V, y)$ be a chart in which all metric components $g_{a b}$ are constant. On the open submanifold $V$ with the restricted nondegenerate metric, the coordinate connection $\nabla^{c}$ with all Christoffel symbols zero is torsion-free and metric-compatible: its Christoffel symbols satisfy the symmetry $Γ^{a}_{b c} = Γ^{a}_{c b}$ , and the compatibility identity $\partial_{a} g_{b c} = Γ^{d}_{a b} g_{d c} + Γ^{d}_{a c} g_{b d}$ has both sides zero (Torsion freedom and metric compatibility in chart components, Christoffel symbols of an affine connection). The metric-selected connection of a nondegenerate metric is the unique torsion-free metric-compatible affine connection (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (b), Metric compatibility, torsion freedom and the metric-selected connection class), so the Christoffel symbols of the metric-selected connection in $(V, y)$ all vanish. The coordinate formula for the curvature then gives $R^{a}_{b c d} = \partial_{c} Γ^{a}_{b d} - \partial_{d} Γ^{a}_{b c} + Γ^{e}_{b d} Γ^{a}_{c e} - Γ^{e}_{b c} Γ^{a}_{d e} = 0$ on $V$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L5]

A straight path with displacement in $C^{+} = {v : v^{0} > ∣ v_{s} ∣_{e}}$ has constant nonzero future timelike tangent by The causal cones of the flat model in inertial coordinates, and closure of the future cone under sums clause (a). Its smooth coordinates are locally absolutely continuous by the bounded-derivative estimate, so it is a future timelike curve in Causal curves, chronological and causal reachability, and inextendibility clause (b). No global-hyperbolicity result is needed.

Proof

technique · direct

1.1givenL1L2L3L4

(The quotient model.) By [L1] and [L2] the hypotheses of the descent lemma are satisfied, so [L3] gives the time-oriented Lorentzian four-manifold $(M, g, N)$ , the local isometry identity of (a) and the local charts with constant components. In each such chart, [L4] shows that the Christoffel symbols of the metric-selected connection vanish on the chart and hence that its curvature vanishes there; since the charts of the descent lemma cover $M$ , the curvature vanishes on $M$ and the model is flat in the asserted sense. This proves (a).

2.1step 1.1givenL3

(The closed timelike curve.) Define $γ (s) : = q (o + s e_{0})$ . Since $s \mapsto o + s e_{0}$ is smooth and $q$ is smooth, $γ$ is smooth, and $γ (s + L) = q (o + (s + L) e_{0}) = q (o + s e_{0} + L e_{0}) = q (o + s e_{0}) = γ (s)$ because $L e_{0} \in Λ$ . Its velocity is $γ^{'} (s) = d q_{o + s e_{0}} (\partial_{0})$ by the chain rule, and by the local isometry identity $g (γ^{'}, γ^{'}) = η (\partial_{0}, \partial_{0}) = - 1 < 0$ and $g (γ^{'}, N) = g (γ^{'}, d q (\partial_{0})) = η (\partial_{0}, \partial_{0}) = - 1 < 0$ ; so every tangent is future timelike and the curve is a future timelike causal curve in the class of Causal curves, chronological and causal reachability, and inextendibility. It is nonconstant: if $γ (s) = γ (0) = q (o)$ then $s e_{0} \in Λ$ , so $s = n L$ for some integer $n$ , and for $0 < s < L$ no such integer exists. Hence $γ ∣_{[0, L]}$ is a nonconstant closed future timelike curve, and by Chronology, causality, strong causality, global hyperbolicity and time functions clause (a) the model is neither chronological nor causal. This proves (b).

3.1step 2.1givenL5

(Total viciousness.) Let $p, q \in M$ and choose lifts $\tilde{p}, \tilde{q} \in A$ of them, so that $q (\tilde{p}) = p$ and $q (\tilde{q}) = q$ ; such lifts exist because $q$ is surjective. Choose $n \in Z$ with $n L > ∣ z (\tilde{q}) - z (\tilde{p}) ∣_{e} - (x^{0} (\tilde{q}) - x^{0} (\tilde{p}))$ in the global inertial chart $x$ ; possible because $L > 0$ and the left side tends to $+ \infty$ . The displacement $(\tilde{q} + n L e_{0}) - \tilde{p}$ has time coordinate larger than the Euclidean norm of its spatial part, so it lies in the open future cone $C^{+}$ ; by [L5] the straight path $s \mapsto \tilde{p} + s ((\tilde{q} + n L e_{0}) - \tilde{p})$ , $s \in [0, 1]$ , is a future timelike curve of the flat model, and composing with $q$ gives a smooth future timelike curve from $p$ to $q (\tilde{q} + n L e_{0}) = q (\tilde{q}) = q$ , since $n L e_{0} \in Λ$ . Hence $p ≪ q$ ; taking $q = p$ exhibits the closed orbits, and taking $p \neq q$ shows that no two distinct points are achronal or acausal. This proves (c).

3.2step 2.1givenalgebra

(No Cauchy set.) The periodic timelike curve $γ (s) = q (o + s e_{0})$ of step 2.1 is inextendible. Indeed, for integers $n \to + \infty$ the sequences of parameter values $n L$ and $n L + L / 2$ both tend to $+ \infty$ , while their curve values are the distinct fixed points $q (o)$ and $q (o + (L / 2) e_{0})$ ; the points are distinct because $(L / 2) e_{0} \notin Λ$ . The same two phase sequences with $n \to - \infty$ show that $γ$ has no limit as $s \to - \infty$ . By clause (d) of Causal curves, chronological and causal reachability, and inextendibility, it is inextendible. Let $S \subseteq M$ be a Cauchy set. The inextendible curve $γ$ must meet $S$ at a parameter $s_{0}$ . Periodicity gives $γ (s_{0} + L) = γ (s_{0}) \in S$ , so its preimage of $S$ contains the two distinct parameters $s_{0}$ and $s_{0} + L$ . This contradicts the defining singleton-preimage condition. Thus no Cauchy set exists, and in particular no Cauchy hypersurface exists. This proves (d).

4.1step 1.1step 2.1step 3.1step 3.2∎

(Conclusion.) Part (a) is step 1.1, part (b) is step 2.1, part (c) is step 3.1, and part (d) is step 3.2; this completes the proof.

Remarks

What the example shows. Smoothness, flatness and time orientability of a Lorentzian four-manifold do not imply causality, and a flat model need not admit a Cauchy hypersurface; the identification changes the global causal structure while leaving the local metric flat. This is the flat analogue of the Misner-space warning recorded by Carroll.
What it does not show. The model is a quotient of the flat model by a discrete translation group and is not the flat model itself; the theorem asserts nothing about the physical applicability of such identifications.

The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources