Worked vertical geodesics of the open flat slab: affine parameter, limit point and proper time

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

Let $(M,g,N)$ be the open flat slab $M=x^{-1}((0,1)\times {\mathbb{R}}^3)$ of The open flat slab: an incomplete timelike geodesic, a Cauchy slice and global hyperbolicity with restricted inertial chart $x=(x^0,z)$. Fix $z_0\in {\mathbb{R}}^3$ and let $\gamma :(0,1)\to M$, $\gamma (s):=x^{-1}(s,z_0)$. This example records the explicit numbers behind the incompleteness claim of that theorem:

(i) $\gamma$ is a smooth future timelike geodesic with affine parameter $s$ and unit tangent ${\gamma }^{\mathrm{\prime }}(s)={\mathrm{\partial }}_0$; in the chart the geodesic equation is $\ddot{y}=0$.

(ii) The affine separation and the Lorentzian length of the segment from $s_1$ to $s_2$, $0<s_1<s_2<1$, are $\mathrm{\Delta }s=s_2-s_1,L(\gamma {\mathrm{\mid }}_{[s_1,s_2]})={\int }_{s_1}^{s_2}\sqrt{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}ds=s_2-s_1;$ with the fixed mathematical scaling constant $c>0$ of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, the metric-defined proper-time increment is $\tau (s_2)-\tau (s_1)=(s_2-s_1)\mathrm{/}c$.

(iii) The maximal domain of a geodesic with the data of $\gamma$ is exactly $(0,1)$; the image converges in the ambient affine space to the event $x^{-1}(1,z_0)$, which does not belong to $M$. Hence $\gamma$ is future-inextendible in $M$ with bounded affine parameter, and the slab is timelike geodesically incomplete.

(iv) The slice ${\mathrm{\Sigma }}_{1\mathrm{/}2}=\{x^0=1\mathrm{/}2\}$ is a Cauchy hypersurface and the slab is globally hyperbolic, so the example exhibits incompleteness in a globally hyperbolic model.

• Conditional physical interpretation. Under the framework's SI length-coordinate convention SI unit and coordinate conventions of the classical gravitational framework clause (b), the coordinates $x^0=s$ and the affine parameter values $s_1,s_2$ are lengths in metres, while $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$; thus $(s_2-s_1)\mathrm{/}c$ is a duration in seconds. If $\gamma$ is also interpreted as an ideal clock's worldline, the ideal-clock assumption of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (i) identifies that metric-defined proper-time increment with the clock-reading difference. This is contextual physical interpretation, outside the mathematical claims and their Verification. The postulate restricts it to ideal clocks in the smooth classical test-field regime without back-reaction; it asserts no real device is exactly ideal and reports no measured reading. The mathematical example itself supplies neither this assumption nor empirical evidence.

• Numbers used. The affine parameter interval has length $1$ in these coordinates; the value is a coordinate statement about the chart $x$ and changes under affine reparametrizations of $x^0$ by a constant factor, under which the incompleteness verdict is unchanged.

• Not a singularity. Curvature vanishes identically on the slab; the incompleteness is caused by the absent boundary of the region, not by curvature blow-up.