Light cone as the space of horospheres of $\mathbb{H}^n$ Let $\mathbb{H}^n$ be identified with the hyperboloid model in $\mathbb{R}^{n+1}$ $$I^n=\{x|\langle x,x \rangle =-1\},$$ where $\langle x,y \rangle =x_1y_1+\ldots x_ny_n-x_{n+1}y_{n+1}$.
Let also $L$ be the positive light cone $$L=\{x|\langle x,x\rangle =0 \text{ and } x_{n+1}\ge 0\}.$$
A point $x\in L$ defines a hyperplane $H_x=\{y\in \mathbb{R}^{n+1}|\langle x,y\rangle= -1\}$ which is  parallel to the tangent space $T_xL=\{y\in \mathbb{R}^{n+1}|\langle x,y\rangle= 0\}$. The rays of $L$ are in correspondence with the points of $\partial I^{n+1}$. We denote be $x_\infty$ the point in $\partial I^{n+1}$ corresponding to the ray containing $x$.
I would like to show that $H_x\cap I^n$ is a horosphere centered in $x_{\infty}$, i.e. every geodesic towards $x_{\infty}$ is perpendicular to $H_x\cap I^n$.
 A: Let $\gamma$ be a hyperbolic geodesic in $\mathbb{H}^n$ which tends to $x_{\infty}$ at one of its ends. Thus $\gamma$ is the intersection of a $2$-dimensional subspace $P \subset \mathbb{R}^{n+1}$ containing $x$ and which is of type $(1,1)$ (i.e. it has an orthornomal basis consisting of a unit timelike vector and a unit spacelike vector). WLOG, up to using a Lorentz transformation, one may assume that $P$ is the span of $e_1$ and $e_{n+1}$ and that $x$ is some positive constant times $e_1 + e_{n+1}$. In order to fix the notation, let us assume that $x = e_1 + e_{n+1}$. The interested reader can then modify the proof in order to accomodate for $x$ being a positive constant times $e_1 + e_{n+1}$. Then:
$$H_x = \{ y \in \mathbb{R}^{n+1}; y_1 - y_{n+1} = -1 \}.$$
Then the intersection of $H_x$ with $P$ and $\mathbb{H}^n$ consists of just the point: $q = (0,0,\ldots,0,1)$. The tangent line $T_q\gamma$ to $\gamma$ at $q$ is the span of $e_1$. On the other hand, the tangent space $T_q(H_x \cap \mathbb{H}^n)$ to $H_x \cap \mathbb{H}^n$ at $q$ is the span of $e_2, \ldots, e_n$. We thus see that
$$T_q\gamma \perp T_q(H_x \cap \mathbb{H}^n)$$
as required to prove. It remains to check that the proof also holds if $x$ is a positive constant times $e_1 + e_{n+1}$, but this part should be ok (I think), and I leave it to the interested reader.
