billiard ball moves to infinity in hyperbolic space.

Conventions
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We adopt a “mostly plus” signature ($-+++ \ldots$). The spacetime dimension is $ D \equiv d + 1$. Greek indices range from $0$ to $d$, while Latin indices $\in \{1,\ldots,d\}$. The spatial Ricci tensor is labeled $R$ and the spacetime Ricci tensor is labeled $^{(D)}R$. Our curvature conventions are such that the Ricci tensor of a sphere is positive definite. Einstein’s equations read $G_{\alpha \beta} = T_{\alpha \beta}$, where $G_{\alpha \beta} = R_{\alpha \beta} - R g_{\alpha \beta}/2$ denotes the Einstein tensor and $T_{\alpha \beta}$ denotes the matter stress-energy tensor, $T_{\alpha \beta}
= - (2/ \sqrt{-g}) \delta S_{\rm matter}/ \delta g^{\alpha \beta}$, and units such that $ 8 \pi G =1$. The spatial metric compatible covariant derivative is labeled $\nabla_a$ and the spacetime metric compatible covariant derivative is labeled $^{(D)} \nabla_\alpha$. The velocity-dominated metric compatible covariant derivative is labeled $^0 \nabla_a$. According to the context, $g$ denotes the (positive) determinant of $g_{ab}$ in $d +1$-decomposed expressions, and the (negative) determinant of $g_{\mu \nu}$ in spacetime expressions. Whenever $t^\delta$ or $t^{-\delta}$ appears, $\delta$ denotes a strictly positive number, arbitrarily small. We use Einstein’s summation convention on repeated tensor indices of different 