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.

Let us fix the notations. Let be the -dimensional Minkowski space defined by

equipped with the Lorentzian metric .

Fix and define the hyperbolic plane by

equipped with the induced metric.

It can be shown that is a surface with constant curvature (analogous to the fact that the sphere of radius , , is a surface with constant curvature ). It is easy to see that can be parametrized by

the “polar coordinates” around , with being the (geodesic) distance from .

Finally, let the (infinite) cylinder of radius be defined by

again equipped with the induced metric.

There is a natural orthogonal projection from to defined by

In polar coordintes, this is given by

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

because the projection of is exactly the finite cylinder

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 .

Now, it is easy to see that in the coordinates , the area form of is given by

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

From (1) and (3), we see that (note that )

Comparing with (2), we conclude that , i.e. is area-preserving. As a corollary, we get

Corollary 1The area of is