Let us fix the notations. Let be the -dimensional Minkowski space defined by
equipped with the Lorentzian metric .
equipped with the induced metric.
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
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
Comparing with (2), we conclude that , i.e. is area-preserving. As a corollary, we get
Corollary 1 The area of is