The Geodesic Line on the Poincaré Half Plane

A Physics stack exchange question asked about the calculation of geodesics on the Poincaré Half Plane. Naturally, as Physics SE is a physics site, the poster was looking for help in deriving the geodesic equation with the wonted pseudo-Riemannian geometry techniques.

Riemannian geometry is powerful to be sure, but it kind of “clobbers” the geometry of the Poincaré Half Plane: there are more first principles and intuitive ways of thinking about hyperbolic geometry, so I couldn’t resist posting an analysis of the Poincaré Half Plane wielding Poincaré’s theorem that $PSL(2,\,\mathbb{R}) \subset PSL^+(2,\,\mathbb{C})$ is precisely the group of isometries of the half plane. (Here $PSL(2,\,\mathbb{C})$, the projective unimodular $2\times2$ complex matrix group, is isomorphic to the group of Möbius transformations of the one-point-compactified complex plane (Riemann sphere)).

So here is my answer …..

Although the following doesn’t directly answer your question (which of course is a request for a review of the “GR” method of calculating geodesics and which Stan Liou did perfectly), I can’t resist writing down the following elegant little characterisation of geodesics in the hyperbolic plane. I believe thoughts along the following lines, although very simple and specialised, help grow an intuition for hyperbolic geometry and for some of the basic behaviours of negatively curved spacetime. Naturally it does not replace the “GR” method as it is much more specialised.

I’ll simply sketch the proof as it is not quite as concise as I recalled when written out in full, but nonetheless it is a sequence of little jewels: readily grasped, simple and clear landmarks that one can keep in one’s head when thinking about this kind of thing. If you need me to fill details in, I’ll naturally add these to my answer.

The hyperbolic plane:

$$\mathbb{H}^2 = \{z\in\mathbb{C}: {\rm Im}(z)>0\}\qquad(1)$$

is kitted with the hyperbolic metric defined by:

$$ds^2 = \frac{dx^2 + dy^2}{y^2}\qquad(2)$$

which is the same as your metric, aside from a scaling constant. One wontedly studies $\mathbb{H}^2$ together with the Poincaré disk:

$$\mathbb{D}^2 = \{z\in\mathbb{C}: |z|<1\}\qquad(3)$$

that is the isometric image of $\mathbb{H}^2$ under the billinear transformation:

$$T:\mathbb{H}^2\to \mathbb{D}^2;\quad T(z) = \frac{1+i\,z}{z+i}\qquad(4)$$

and you can readily show that the metric in $\mathbb{D}^2$ is defined by:

$$ds^2 = \frac{4\,|d\omega|^2}{(1-|\omega|^2)^2}\qquad(5)$$

where $|d\omega|$ is the everyday Euclidean metric in $\mathbb{D}^2$. Witness that the following are clearly isometries of $\mathbb{H}^2$:

1. “Sideways translations”, i.e.

$$T_\lambda(z) = \lambda + z; \,\lambda \in \mathbb{R}\qquad(6)$$

2. “Dilations”, i.e.

$$D_\rho(z) = \rho\,z;\,\rho\in \mathbb{R},\,\rho>0\qquad(7)$$

and also

3. The transformation that corresponds to a rotation through any angle $\theta$ about the origin in the disk $\mathbb{D}^2$, i.e. $\omega\mapsto e^{i\theta} \omega$, which corresponds to the bilinear transformation:

$$R_\theta(z) = \frac{z \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right)}{-z \sin\left(\frac{\theta}{2}\right) + \cos\left(\frac{\theta}{2}\right)}\qquad(8)$$

because, from (5), it is clearly an isometry in $\mathbb{D}^2$ and $\mathbb{D}^2$ and $\mathbb{H}^2$ are isometricly equivalent.

So now we call on the following:

Theorem (Poincaré):

The group $PSL(2, \mathbb{R})$ of billinear transformations of the form:

$$f:\mathbb{H}^2\to\mathbb{H}^2;\;f(z) = \frac{\alpha\,z+\beta}{\gamma\,z+\delta};\;\alpha,\,\beta,\,\gamma,\,\delta \in \mathbb{R};\;\alpha\delta-\beta\gamma=1\qquad(9)$$

is precisely the group of isometries of $\mathbb{H}^2$; that is, every transformation of this kind is an isometry, and all isometries are of this kind.$\qquad\square$

Indeed, although not relevant here, $PSL(2, \mathbb{R})$ is also precisely the group of conformal transformations of $\mathbb{H}^2$, i.e. all maps of this kind are conformal, and any globally conformal bijection $\mathbb{H}^2\to \mathbb{H}^2$ is of this kind.

To prove the first part, one shows that all such billinear maps can be decomposed into the following composition of known distance-preserving maps above:

$$f = T_{\frac{\alpha}{\gamma}} \circ D_{\frac{\alpha\delta-\beta\gamma}{\gamma}} \circ R_\pi \circ D_\gamma\qquad(10)$$

To prove to converse, one shows that any distance preserving map is determined by the images of three non-collinear points $A, B, C\in\mathbb{H}^2$ because any other point $D\in \mathbb{H}^2$ is set by its distance from the reference points $A, B, C$ and their images.

So now, armed with this theorem, one considers line segments $PQ$ between any pair of points $P$ and $Q$ on the imaginary axis in $\mathbb{H}^2$ as in my drawing below:

Hyperbolic PlaneFigure 1: “Prototype” Geodesic compared with Perturbation in the Hyperbolic Half Plane

Clearly from (2) for any $C^0$ path $\Gamma$ between $P$ and $Q$, $ds^\prime \geq ds$ when we project the path onto the imaginary axis as shown. Therefore the unique geodesic linking two points on imaginary axis is simply a segment of the imagine axis between those two points.

It is then not hard to show that any two points $A,B\in\mathbb{H}^2$ are the images of two points $P$ and $Q$ on the imaginary axis under a mapping in the group $PSL(2,\mathbb{R})$ of isometries of $\mathbb{H}^2$ and this group member is uniquely defined by $A$ and $B$.

Therefore, by Poincaré’s theorem, the image of the line segment $PQ$ along the imaginary axis defined by this unique map is the unique geodesic between $A$ and $B$. Since bilinear transformations map circles to circles (i.e. “circle” as defined in the wonted Euclidean space), the geodesic between $A$ and $B$ is the arc of a Euclidean circle. Indeed it is the arc between the two points of the unique Euclidean circle passing through $A$ and $B$ with its centre on the real axis.

Thus you can find the geodesics you need, and, for some interesting background, if you want to knit your own hyperbolic plane over the holidays, try . Check out the Wikipedia articles:

  1. Hyperbolic Geometry,
  2. Poincaré Disk,
  3. Uniform Tilings in Hyperbolic Plane,
  4. Pseudosphere

and also look up the “circle limit” woodcuts by M. C. Escher, although these deal more directly with the Poincaré disk than the Poincaré half plane.