Gravitational Collapse in Painlevé-Gullstrand Coordinates

We construct a simple expression for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painlev-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both cases corresponding to the gravitational collapse from infinity and from a finite radius.