## Archimedes’ principle for hyperbolic plane

After writing the previous post, I realized that there is an exact analogue of the Archimedes’ principle for the surface area of the hyperbolic disk inside the hyperbolic plane.

Archimedes (287 –  212 BC)

Let us fix the notations. Let ${\mathbb R^{2,1}}$ be the ${3}$-dimensional Minkowski space defined by

$\displaystyle \begin{array}{rl} \displaystyle \mathbb R^{2,1}:=\{(x, y, t)\in \mathbb R^{3}\}, \end{array}$

equipped with the Lorentzian metric ${-dt^2+dx^2+dy^2}$.

Fix ${r >0}$ and define the hyperbolic plane ${\mathbb H^2(r)}$ by

$\displaystyle \begin{array}{rl} \displaystyle \mathbb H^2(r):=\{(x, y, t)\in \mathbb R^{2,1}: \quad t>0,\quad t^2-x^2-y^2=r^2\}, \end{array}$

equipped with the induced metric.

Hyperbolic plane

It can be shown that ${\mathbb H^2(r)}$ is a surface with constant curvature ${-\frac{1}{r^2}}$ (analogous to the fact that the sphere of radius ${r}$, ${\mathbb S^2(r)}$, is a surface with constant curvature ${\frac{1}{r^2}}$). It is easy to see that ${\mathbb H^2(r)}$ can be parametrized by

$\displaystyle \begin{array}{rl} \displaystyle X(\theta, \phi)= (r\sinh \theta \cos \phi,r \sinh \theta \sin \phi, r\cosh \theta), \quad \theta\ge 0, \quad \phi\in[0, 2\pi],\end{array}$

the “polar coordinates” around ${P=(0, 0, r)}$, with ${r\theta}$ being the (geodesic) distance from ${P}$.

Finally, let the (infinite) cylinder ${C}$ of radius ${r}$ be defined by

$\displaystyle \begin{array}{rl} \displaystyle C= \{(x, y, t)\in \mathbb R^{2,1}: x^2+y^2=r^2\}, \end{array}$

again equipped with the induced metric.

Cylinder

There is a natural orthogonal projection ${\Pi}$ from ${\mathbb H^2(r)\setminus \{P\}}$ to ${C}$ defined by

$\displaystyle \begin{array}{rl} \displaystyle \Pi(x, y, t)=\left(\frac{r x}{\sqrt{x^2+y^2}}, \frac{r y}{\sqrt{x^2+y^2}}, t\right). \end{array}$

The natural projection

In polar coordintes, this is given by

$\displaystyle \begin{array}{rl} \displaystyle \Pi(X(\theta, \phi ))=(r\cos \phi, r\sin \phi, r\cosh \theta). \ \ \ \ \ (1)\end{array}$

We claim that the map ${\Pi}$ is area-preserving. If this is true, then we can easily calculate the area of the hyperbolic geodesic disk (with radius ${rR}$)

$\displaystyle \begin{array}{rl} \displaystyle B_R:=\{X(\theta, \phi): 0\le \theta\le R\}\end{array}$

because the projection of ${B_R\setminus \{P\}}$ is exactly the finite cylinder

$\displaystyle \begin{array}{rl} \displaystyle \Pi(B_R\setminus \{P\})=\{(x, y, t)\in C: r

whose area is easy to be calculated. Indeed, the area of the cylinder is exactly the same as the ordinary cylinder in the Euclidean space ${\mathbb R^3}$.

A geodesic disk inside the hyperbolic plane

Now, it is easy to see that in the coordinates ${(\theta, \phi)}$, the area form of ${\mathbb H^2(r)}$ is given by

$\displaystyle \begin{array}{rl} \displaystyle \omega_{\mathbb H^2(r)}= r^2\sinh \theta d\theta\wedge d\phi. \ \ \ \ \ (2)\end{array}$

On the other hand, the area form of the cylinder is given by

$\displaystyle \begin{array}{rl} \displaystyle \omega_{C}=rdt\wedge d\phi \ \ \ \ \ (3)\end{array}$

if ${C}$ is parametrized by ${(r\cos \phi, r\sin \phi, t)}$.

From (1) and (3), we see that (note that ${d\phi(-r \sin \phi, r\cos \phi, 0)=1}$)

$\displaystyle \begin{array}{rl} \displaystyle \omega_C(d \Pi(X_\theta), d\Pi(X_\phi))=r^2\sinh\theta. \end{array}$

Comparing with (2), we conclude that ${\Pi^*\omega_C=\omega_{\mathbb H^2(r)}}$, i.e. ${\Pi}$ is area-preserving. As a corollary, we get

 Corollary 1 The area of ${B_R}$ is $\displaystyle \begin{array}{rl} \displaystyle \mathrm{Area}(B_R)=2\pi r^2(\cosh R-1). \end{array}$