**Structure and evolution of a spherical dust star. **

**1. The modified Oppenheimer-Snyder solution**

**1. The modified Oppenheimer-Snyder solution**

**Abstract**

In the Oppenheimer-Snyder solution (OS) for the parabolic trajectory particle’s worldline r(t,R) in terms of world time t differs from its standard worldline in the Schwarzschild field outside and on the surface of the dust star. This is a consequence of the fact that the trajectory function r(t,R) were defined on the “homogeneity hypersurface”, when r = R at the zero initial moment of proper time in all layers and since these events are not simultaneous, the initial moments of the world time t(R) are nonzero. In view of the fact that the structure of the star at any moment means the determination of the positions of all particles on the hypersurface t=const., and the solution of the OS is used for checking more realistic models of stars, this incompleteness of the procedure for the transition to hypersurface t=const. leads to distortions of physical consequences other models too. A more consistent application of the OS method is proposed, where this problem does not arise. The modification consists in fixing the initial positions r=R for t(R)=0 and determining the shift of the proper time moments in different layers on the hypersurfaces t=const. from the condition of obtaining the standard trajectory function r(t,R). The pictures of particle trajectories of the dust star are presented, which clearly show the internal structure of the star at t=const. At large t, not only the surface asymptotically approaches the gravitational radius, but the world lines of particles in the inner layers also approach their asymptotes, rapidly becoming practically parallel to the world lines of particles at the center and on the surface. This shows that the frozen star picture refers not only to the surface, but also to the inner layers freezing at certain distances from the center.

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

* Keywords: relativistic stars, gravitational collapse, black holes, quark stars*

Vol. 12, No 1, p. 1 – 16, v1, June 5, 2017

Electron.: TPAC: 5200-041 v1, June 5, 2017; DOI: 10.9751/TPAC.6000-043

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[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan* zahidzakir@theor-phys.org

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