2019, no. 4

### PAINLEVE-LIKE COORDINATES AND MODELING OF STATIC GRAVITATIONAL BALL

#### Baranov A.M.

The problem of coordinates introduction for the description of internal static spherical solutions of gravitating objects similar to the Painleve coordinates for the Sñhwarzschild external solution is considered. It is shown how the space-time metric for the Sñhwarzschild external solution in coordinates of the curvature can be rewritten in Bondi’s coordinates and the Painleve coordinates. For the known the Sñhwarzschild internal solution in coordinates of the curvature an analytical transformation to Painleve-like coordinates is found. The metric for Sñhwarzschild’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.

Keywords: Schwarzschild’s external and interior solutions, the Painleve coordinates, coordinates of curvatures, Bondi’s coordinates, the Painleve-like coordinates, 4-metric of static gravitating ball, parabolic law of mass density distribution.

UDC: 530.12:531.51

PACS: 04.20.-q , 04.20.Cv , 04.20.Jb

DOI: 10.17238/issn2226-8812.2019.4.13-22