  Hamenst\"adt gave a parametrization of the Teichm\"uller 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 Teichm\"uller space
of punctured surfaces based on this parametrization. We prove that every
earthquake ray is an almost geodesic under the Hilbert metric.
