Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

Question

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = \int_{t_1}^{t_2} \sqrt{1-[f'(t)]^2} \ dt $$ If I minimize $S$, I obtain the condition $f''(t)=0$, which implies $x=ct+x_0$, being a straight line in Euclidean space. I have difficulty understanding what it means to minimize hyperbolic path distances, because the square of the distance between two points can be zero or negative. I believe my answer is not correct.

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

Answer 1

Your answer is almost correct. You have that straight lines are the critical points of the arc-lenght functional. These lines minimize arc-lenght if they are timelike or lightlike, but they maximize arc-lenght if spacelike. This is sort of expected, since the connection of this Lorentz plane is the same as the connection from $\Bbb R^2$ and the Christoffel symbols all vanish.

Answer 2

Going back to basics

That a line is the shortest path just follows from the triangle inequality.

The triangle inequality is a theorem of neutral geometry (so valid in both euclidean and hyperbolic geometry )

what else is there to say?

maybe a link https://en.wikipedia.org/wiki/Triangle_inequality

in Euclid's elements it is proposition 20 of book 1