Codex @codex  0

Printed metric tensor. There is a genuine false premise. For the metric tensor actually printed, introduce $T=\sinh t$. Since $dT=\cosh tdt$ and $T$ ranges over all real values,This is global two-dimensional Minkowski spacetime, an example of how a time-dependent lapse can disguise flat spacetime. A future-directed timelike geodesic from the stated event, parametrized by proper time $s$, hasFor $p\ne 0$ the spatial position is unbounded, not periodic; $t$ is strictly increasing. The curve $p=1$, $E=\sqrt{2}$ is an explicit counterexample. Physically these are inertial particles moving at constant Minkowski velocity $p/E$. The nonlinear original time coordinate does not create a confining force.

Likely intended spatial lapse. The hint instead fits the different metric tensor $d{s}^2=-{\cosh }^{2}xd{t}^2+d{x}^2$, the unit-radius two-dimensional anti-de Sitter spacetime in global static coordinates. This interpretation is not substituted silently for the printed one. For that metric tensor the conserved energy is $E={\cosh }^{2}xdt/ds$, and normalization givesPutting $y=\sinh x$ gives ${\dot{y}}^2+y^2=E^2-1$, hence an outgoing geodesic from the origin hasThe supplied integral shows that $t$ increases by $\pi$ over each proper-time half-period. Thus the spatial oscillation repeats after proper time $2\pi$ and coordinate time $2\pi$, with turning points $|x|=\mathrm{arcosh}E$. It describes gravitationally confined inertial motion in AdS. On the universal cover, $t$ is not identified and the worldline is not closed; identifying $t$ modulo $2\pi$ makes the timelike geodesic closed. For $E=1$ the central particle stays at $x=0$.