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Archive for the ‘2017 Volume 12’ Category

Dec
14

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

 Zahid Zakir [1]

Abstract

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  zahidzakir@theor-phys.org

Jun
05

Structure and evolution of a spherical dust star.

1. The modified Oppenheimer-Snyder solution

 Zahid Zakir [1]

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


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