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Counterexample: on the flat slab the coordinate time has Cauchy levels but an incomplete gradient flow

Statement refuted

Statement refuted: "on every globally hyperbolic time-oriented smooth Lorentzian four-manifold, for every temporal function $T$ in the sense of Chronology, causality, strong causality, global hyperbolicity and time functions clause (e) the vector field $X = \nabla T / g (\nabla T, \nabla T)$ is complete, that is, every maximal integral curve of $X$ is defined on all of $R$ (Complete vector fields)."

Facts & Assumptions

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the general absolutely continuous causal-curve conclusions used here.

Given: The flat slab $(M, g, N)$ of The open flat slab: an incomplete timelike geodesic, a Cauchy slice and global hyperbolicity with chart $x = (t, z)$ , the coordinate time $t : M \to (0, 1)$ read in the chart, and the point $p_{0} : = x^{- 1} (1 / 2, 0, 0, 0)$ .

[L1]

Slab facts: $(M, g, N)$ is a globally hyperbolic time-oriented smooth Lorentzian four-manifold with metric components $diag (- 1, 1, 1, 1)$ in $x$ ; the slice $t = 1 / 2$ is a Cauchy hypersurface; the vertical curve $s \mapsto x^{- 1} (s, z_{0})$ is a geodesic with unit tangent $\partial_{0}$ and does not converge to a point of $M$ at the endpoints of its parameter interval (The open flat slab: an incomplete timelike geodesic, a Cauchy slice and global hyperbolicity clauses (a)-(d), Chronology, causality, strong causality, global hyperbolicity and time functions, Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons, Causal curves, chronological and causal reachability, and inextendibility, Local and global flows generated by a vector field).

[L4]

Every future causal slab curve has strictly increasing continuous time coordinate and $∣ z (γ (v)) - z (γ (u)) ∣ \leq t (γ (v)) - t (γ (u))$ for $u < v$ (Monotonicity of inertial time and the Lipschitz graph property for causal curves in the flat model). Cauchy sequences in $R^{3}$ converge ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ).

[L2]

Gradient of the coordinate time: with $\nabla$ the inverse metric of The inverse metric, musical isomorphisms and index raising, the chart function $t$ has $d t = d x^{0}$ and $\nabla t = - \partial_{0}$ , because $g (- \partial_{0}, w) = w^{0} = d t (w)$ ; hence $g (\nabla t, \nabla t) = - 1 < 0$ , the gradient is past-directed, and $t$ is a temporal function whose normalized gradient field is $X = \nabla t / g (\nabla t, \nabla t) = \partial_{0}$ . [L1]

[L3]

Integral curves of $\partial_{0}$ : in the chart, a curve $s \mapsto x^{- 1} (t (s), z (s))$ has velocity $(t^{'} (s), z^{'} (s))$ , so it is an integral curve of $X = \partial_{0}$ exactly when $t^{'} (s) = 1$ and $z^{'} (s) = 0$ for all $s$ ; by A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant this means $t (s) = s + c$ and $z (s) = z_{0}$ is constant. The uniqueness theorem for maximal integral curves identifies the maximal domain of the curve $s \mapsto x^{- 1} (s + c, z_{0})$ as the largest interval of $s$ with $s + c \in (0, 1)$ , namely $(- c, 1 - c)$ (Through each point there is a unique maximal integral curve, Integral curves of a vector field).

Counterexample

technique · direct

1.1L1L2L4

(The function is temporal with Cauchy levels.) The coordinate $t$ is smooth with past timelike gradient by [L2]. For any two-sided inextendible future causal curve $γ : I \to M$ , [L4] makes $t \circ γ$ strictly increasing with limits $a, b$ in $[0, 1]$ . If $b < 1$ , its spatial coordinates are Cauchy at the future end by the increment bound, hence converge in $R^{3}$ ; as $b \geq t (γ (s_{0})) > 0$ , this gives a point limit $(b, z_{\infty})$ inside the slab, contradicting inextendibility. Dually $a > 0$ would give a past point limit, so $a = 0, b = 1$ . Continuity and strict increase give exactly one parameter at every $c \in (0, 1)$ . Thus each $t^{- 1} (c)$ is Cauchy; its slice chart and positive spatial metric make it a spacelike embedded hypersurface.

2.1step 1.1L1L3

(The maximal integral curve leaves the slab at finite parameter.) Take $c = 1 / 2$ , so the curve $γ (s) : = x^{- 1} (1 / 2 + s, 0, 0, 0)$ . By [L3] it is the maximal integral curve of $X$ through $p_{0} = γ (0)$ , and its maximal domain is $(- 1 / 2, 1 / 2)$ . At $s \to 1 / 2 -$ the curve converges in the ambient flat model to the boundary point $x^{- 1} (1, 0, 0, 0) \notin M$ and similarly at the lower end, so it cannot be extended within $M$ ; its domain is not all of $R$ . Therefore $X = \partial_{0}$ is not complete, and its flow is not global.

3.1step 1.1step 2.1L1∎

(Conclusion.) The flat slab is globally hyperbolic, the coordinate time is a temporal function with Cauchy levels, and yet its normalized gradient field is incomplete. Hence the refuted statement is false: completeness of $X = \nabla T / g (\nabla T, \nabla T)$ is not a consequence of being a temporal function on a globally hyperbolic manifold, and the completeness clause of Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow is a property of the function constructed there.

Remarks

What is and is not contradicted. The theorem asserts the existence of at least one temporal function with complete normalized gradient flow on every globally hyperbolic manifold; the counterexample shows that an arbitrary temporal function need not have one, which is why the construction of the theorem cannot be replaced by the coordinate time.
No empirical content. The counterexample is a mathematical statement about a model and reports no observation.
Companion example. On the same slab, Worked flat slab: a rational temporal function with Cauchy levels, complete flow and orthogonal product constructs a temporal function whose normalized gradient field is complete; this shows that the completeness clause of Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow is a property of the constructed function, not of temporal functions in general.

Depends on

Dependency tree · 2 levels

Sources

A. N. Bernal and M. Sanchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes (arXiv:gr-qc/0401112), Proposition 2.4, printed pp. 3-4: a constructed Cauchy temporal splitting; the incomplete coordinate flow counterexample is proved locally.