The boundary of the upper half-plane $\mathbb{H}$ model of the hyperbolic plane is the extended real line $\overline{\mathbb{R}}= \mathbb{R} \cup \{ \infty \}$.
What is the boundary of the hyperboloid model of the hyperbolic plane? Is it the positive light cone $L^+$?
The boundary is not the light cone $L_+$ itself, but the set of directions of the light cone, or equivalently the projectivization $L_+ / \mathbb R_+ \cong S^{n-1}$.