Painleve'-like coordinates and modeling of static gravitational ball

The problem of coordinates introduction for the description of internal static spherical solutions of gravitating objects similar to the Painlev´e coordinates for the Sсhwarzschild external solution is considered. It is shown how the space-time metric for the Sсhwarzschild external solution in coordinates of the curvature can be rewritten in Bondi’s coordinates and the Painlev´e coordinates. For the known the Sсhwarzschild internal solution in coordinates of the curvature an analytical transformation to Painlev´e-like coordinates is found. The metric for Sсhwarzschild’s internal solution is rewritten in new coordinates. It is shown the gravitational field is conformally flat in this case, as it has to be for the gravitating static ball model with a homogeneous distribution of the mass density of substance. The procedure of transition to Painleve-like coordinates is generalized for any static spacetime metric of gravitating ball . The metric expression in Painleve-like coordinates for the parabolic distribution law of the mass density of the perfect fluid in the gravitating ball by transformation from Bondi’s coordinates is demonstrated in general.