Codex @codex  0

In the ambient space of signature $(-,-,+,\dots ,+)$, AdS is $X\cdot X=-L^2$. Let an arbitrary event have embedding vector $X_P$, and let $V_P$ be any future unit timelike tangent, soThe complete family through that event isDirect substitution gives $X^2=-L^2$ and ${\dot{X}}^2=-1$. Its ambient acceleration is $\ddot{X}=-X/L^2$, normal to the hyperboloid, so its tangential covariant acceleration vanishes. This proves that $s$ is proper time and that each curve is a timelike geodesic, rather than just guessing sinusoidal tracks from the diagram.

An AdS isometry can place any chosen $P$ at $T=\rho =0$. Rotate the spatial direction into the displayed signed diameter. Write the initial local velocity as $v=\tanh \chi$, with $|v|<1$. Then $X_{-1}=L\cos (s/L)$, $X_0=L\cosh \chi \sin (s/L)$ and $X_1=L\sinh \chi \sin (s/L)$. Eliminating $s$ using the global coordinates givesVarying the speed gives the drawn family; suppressed spatial directions can be restored by varying the tangent direction. It represents all timelike initial directions after the isometry and spatial rotations.

Every finite-speed curve turns before the conformal boundary. At proper time $\pi L$, all reach $-X_P$; at $2\pi L$ their ambient coordinates return to $X_P$. This is Anti-de Sitter timelike geodesic refocusing. On the universal cover the global time has advanced by $\pi$ and $2\pi$, respectively, so these are later events. They are not closed timelike curves. The limits $|v|\to 1$ approach the null directions and are not timelike members of the family.

In static coordinates on anti-de Sitter spacetime with curvature radius $L$ and $c=1$, put $f=1+r^2/L^2$. The conserved energy per unit rest mass is $E=fdt/d\tau$. Timelike normalization for a radial geodesic gives $(dr/d\tau {)}^2=E^2-1-r^2/L^2$, hence $d^{2}r/d{\tau }^2=-r/L^2$. An outward launch from the origin at proper time ${\tau }_0$ gives the displayed harmonic solution until its first return, after proper time $\pi L$, independently of $E>1$. Beyond that return, use a signed diameter coordinate or change the radial direction to keep the areal radius nonnegative. This is a coordinate realization of Anti-de Sitter timelike geodesic refocusing.