Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form
$$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1} y_{n+1}. $$
A model for the hyperbolic space is given by the set $$\mathbb{H}^n = \{ x \in \mathbb{L}^{n+1} : \langle x, x \rangle = -1, \; x_{n+1} > 0\} $$ with the metric induced by $\mathbb{L}^{n+1}$ (It can be shown that the restriction of this bilinear form to the tangent spaces of $\mathbb{H}^n$ is positive definite).
Let $p_0 = (0, \dots, 0, 1) \in \mathbb{H}^n$. My question is: if $v \in T_p \mathbb{H}^n \subseteq \mathbb{L}^{n+1}$ is a tangent vector at $p \in \mathbb{H}^n$ such that $\langle v, v\rangle = 1$, can we say that $\langle v, p_0 \rangle \leq 1$? I'm a bit confused because Cauchy-Schwarz inequality is no longer valid.