Gravitational Collapse in Painlevé-Gullstrand Coordinates

We construct an exact solution for the spherical gravitational collapse in asingle coordinate patch. To describe the dynamics of collapse, we use ageneralized form of the Painleve-Gullstrand coordinates in the Schwarzschildspacetime. The time coordinate of the form is the proper time of a free-fallingobserver so that we can describe the collapsing star not only outside but alsoinside the event horizon in a single coordinate patch. We show the bothsolutions corresponding to the gravitational collapse from infinity and from afinite radius.