Computing the hyperbolic distance between two points

Question

In http://users.monash.edu/~jpurcell/papers/hyp-knot-theory.pdf, p. 22, it was left as an exercise that

The hyperbolic distance between the endpoints of the perpendiculars on the line from 0 to $\infty$ is $|\ln|z||$ (exercise).

However I was unable to show this result: what I tried was

$$ds^2=\dfrac{dx^2+dy^2+dt^2}{t^2} = \dfrac{\left|\dfrac{d(\frac{z+1}2 + \frac{z-1}2\cos u)}{du}\right|^2+\left|\dfrac{d(\frac{|z-1|}2\sin u)}{du}\right|^2}{\left(\frac{|z-1|}2\sin u\right)^2}=\dfrac{1}{\sin^2u}du^2$$ where $u$ ranges from $0$ to $\pi$. In this case $\sin u>0$, so we have $$\int ds = \int_0^\pi \csc udu = -[\ln|\csc u + \cot u|]_0^\pi $$ for which whatever the value is, is independent of $z$.

What is wrong?

Answer 1

The vertical line in question is the $t$-axis $\{(0,0,t) \mid 0 < t < +\infty\}$.

The two points in question on that line are:

• $P=(0,0,1)$, which is the endpoint of the geodesic from $1 \in \mathbb C$ to the $t$-axis, meeting the $t$-axis at a right angle.
• $Q=(0,0,|z|)$, which is the endpoint of the geodesic from $z \in \mathbb C$ to the $t$-axis, meeting the $t$-axis at a right angle.

Since the $t$-axis is a geodesic, the distance between $P$ and $Q$ is the absolute value of integral of $ds$ along the $z$-axis segment between $P$ to $Q$, which is $$\left|\int_1^{|z|} ds\right| = \left|\int_1^{|z|} \frac{dt}{t}\right| = \left|\ln|z|\right| $$