The Hilbert metric on Teichm\"uller space and Earthquake

Hamenstadt gave a parametrization of the Teichmuller space of punctured surfaces such that the image under this parametrization is the interior of a polytope. In this paper, we study the Hilbert metric on the Teichmuller space of punctured surfaces based on this parametrization. We prove that every earthquake ray is an almost geodesic under the Hilbert metric.