Archive for December, 2017


Structure and evolution of a dust star in the Oppenheimer-Snyder model

 Zahid Zakir [1]


In 1939 Oppenheimer and Snyder (OS) have found an exact solution of the Einstein equations for a collapsing homogeneous dust star at the parabolic velocity of dust particles by transforming the Tolman solution in the comoving coordinates to a solution in the Schwarzschild coordinates r,t and matching on the surface of the star with the exterior Schwarzschild solution. However, despite the regularly citation of the OS paper, the meaning and significance of their solution have so far remained unappreciated and poorly understood, in addition their method has been forgotten. In the present paper it is shown that the OS method allows one to describe correctly from the astrophysical point of view the structure and evolution of the dust star as a whole on hypersurfaces of simultaneity t=const. A detailed derivation of the parabolic OS solution and solutions for hyperbolic and elliptic velocities is given. The plots of the proper time rate and particle trajectories r(t,R) in different layers are presented, visualizing the structure of the dust star. At large t, not only the surface quickly freezes outside the gravitational radius, asymptotically approaching it, but the particles in the internal layers also freeze at certain distances from the center, and their worldlines approach their own asymptotes, rapidly becoming almost parallel to the worldlines of particles at the center and on the surface. This shows that in the OS model the frozen star picture refers not only to the surface, but also to the structure of the collapsed dust star as a whole. Thus, at any finite moment of cosmological time the collapsed OS dust star appears as not a black hole, but as a frozar, an object by practically totally frozen internal structure.

   PACS: 04.20.Dg;  04.70.-s;  97.60.-s,  98.54.-h

   Keywords: relativistic stars, gravitational collapse, black holes

Vol. 12, No 2, p. 17 – 40, v1,       December 14, 2017
Electron.: TPAC: 6192-044 v1,  December 14, 2017;             DOI: 10.9751/TPAC.6192-044

[1] Centre for Theoretical Physics and Astrophyics, Tashkent, Uzbekistan