Consider $\mathbb{R}^{1+2}$ with coordinates $\{t, x, y\}$ and Minkowski metric $g = diag(-1,1,1)$. Suppose we have a hyperboloid $x^2 + y^2 - t^2 = 1$ inside the previous Minkowski spacetime. My question is the following: is it always true that two arbitrary points on the hyperboloid are spacelike separated in $\mathbb{R}^3$ if and only if they are spacelike separated on the hyperboloid? In other words, given two arbitrary points $P$ and $Q$ on the hyperboloid, are the following two statements logically equivalent?
- $(x_P - x_Q)^2 + (y_P - y_Q)^2 - (t_P - t_Q)^2 > 0$
- the geodesic connecting $P$ and $Q$ on the hyperboloid has tangent vector which is spacelike everywhere with respect to the induced metric on the hyperboloid
If this is not the case, is there any computationally simple way to establish if two arbitrary points on the hyperboloid are causally connected (within the hyperboloid as a spacetime manifold) without computing the parameterization of the geodesic explicitly, but rather exploiting the fact that it is immersed in $\mathbb{R}^{1+2}$ Minkowski?