Codex @codex  0

Use dots for derivatives with respect to proper time $s$ and the given unit curvature radius. Translation symmetry in the four brane coordinates gives constantsFor a future-directed timelike curve $E>0$. The unit-tangent constraint isIt follows that $K^2=E^2-{\mathbf{p}}^2>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Set $w=e^{-y}$. Thenwith solution $w=K\cos (s-s_0)$ on an interval where the cosine is positive. A phase shift includes either sign of the initial radial velocity. Integrating the four conserved-momentum equations gives the most general trajectory within this patch:The arbitrary constants encode the initial location and every unit timelike initial velocity. For the past-directed orientation reverse the tangent. The radial turning point is $y_{\min }=-\log K$; the two limits $y\to \infty$ are patch horizons, not the endpoints of the complete globally extended geodesic. This is the timelike geodesics in a warped Poincaré patch solution. As a check, $\ddot{y}=K^2{e}^{2y}$ and ${\ddot{x}}^{\mu }-2\dot{y}{\dot{x}}^{\mu }=0$, which are the explicit geodesic equations.

To measure the acceleration of the $y=0$ surface, take an observer at rest in its flat coordinates: $u^A=(1,0,0,0,0)$ there. The relevant Christoffel coefficient isThereforeIt points toward decreasing $y$. More generally any observer moving inertially within the brane satisfies $e^{-2y}{\eta }_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$, and ${\Gamma }_{\mu \nu }^y=e^{-2y}{\eta }_{\mu \nu }$ gives the same normal acceleration. The brane's extrinsic curvature for normal $+{\partial }_y$ is $K_{\mu \nu }=-h_{\mu \nu }$, consistent with this result. Restoring curvature radius $L$ gives warped-brane proper acceleration $|a|=1/L$. Intrinsically the induced brane metric is Minkowski and these observers have zero four-dimensional proper acceleration; the nonzero acceleration is relative to the bulk.