Consider the set $A=\{x^2+y^2<1\}$ with the metric $ds^2=\frac{4dx^2+4dy^2}{(1-x^2-y^2)^2}$.
Question: What's the length of the segment $y=x$? (with respect to the given metric)
My attempt: I'll parametrize the line by $f(t)=(x(t),y(t))=(t,t)$.
Using the metric I know that the arc length is calculated by $\int_a^b2\frac{\sqrt{(x')^{2}+(y')^{2}}}{1-x^{2}-y^{2}}dt=\int_a^b\frac{2\sqrt2}{1-2t^{2}}dt$. Now I must go on the search of $a$ and $b$ , however as $2t^{2}<1$ we know that:
$-\sqrt2/2<t<\sqrt2/2$
As I plug those values for $a$ and $b$ something terrible happens,the integral diverges. What am I doing wrong?
edit: I have reasons to believe this answer is correct because this segment goes from the boundary of the Poincaré Disk to another point in the boundary which makes sense to have infinit lenght (even if the disk itself is "finite" in some sense,surely not in this metric sense), but still its just intuition.
More generally, the length of any curve $\gamma\colon [a,b)\to A$ such that $\gamma(t)$ approaches the "ideal" boundary of the disk as $t\to b$ is infinite. This is a consequence of the triangle inequality and of the fact that the hyperbolic metric $\rho(0,P)$ tends to infinity as $P$ tends to the boundary.
All of this reflects on the completeness of the open disk with the hyperbolic metric: what appears to be the boundary of the disk is not in fact its boundary (hence "ideal"), as those points cannot be reached from within the space.