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

**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

**Download pdf**1.14 Mb

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

If they would just tell me that everything is frozen, I would have accepted it without proof. It has been said many times that a distant observer will never see the fall of a particle on the event horizon, no matter whether it is a single particle or infinitely many, whether the horizon is at rest or expanding.

The value of this work, in my opinion, is that the problem of collapse – freezing is solved analytically.

At the time of Oppenheimer and Tolman, they tried to solve it analytically, but with the advent of the computer, analysts were completely superseded by the numerals.

They declared an ideology that asserted that all problems that could be solved analytically had long been resolved and the future of physics is numericals. This ideology was necessary only to them, since justified the priority in physics to those who can solve problems by numerical simulations only.

They most likely conceal the fact that 100 times the calculation gives 100 contradictory results, they choose the one that they think is the most believable and pretend that their results should always be believed.

The analytical result obtained in this article is a worthy answer to all of them. If they now compare their results with this, they will know which one of them happened to be right accidentally.

OS could solve the problem analytically (i.e. exactly), confining itself to the diagonal internal metric, which gave the diagonality condition, strongly limiting the class of solutions.

This condition is also important in that it allows us to introduce a hypersurface of simultaneity with the world time t inside the star too (the clocks can be synchronized uniquely only for a diagonal metric).

By introducing the auxiliary function y(t,r) and finding it from the diagonality condition, the OCs were able to find exact solutions for both the metric and the particle trajectories inside the dust star in the Schwarzschild coordinates r,t. In the OS article only the final result is given for y(t,r), but I was able to restore their derivation in details and presented it in the Appendix.

In the literature (in textbooks too), a misleading tradition has settled, that as the OS solution, the same general Tolman solution in the comoving coordinates, but for a homogeneous star is given.

If this were the case, then it would only be renaming the homogeneous Friedman solution, well known even before Tolman, to the OS solution, which is strange in the first place, and secondly it is humiliating for the OS reputation, since it would simply be plagiarism.

But what about in reality?

In fact, for such an extended object as a star, an exact solution must be given on the hypersurface of simultaneity, so that

a) the positions of all particles were given simultaneously and

b) on the surface the solution was smoothly matched with the external Schwarzschild solution.

OS is well understood that and the internal Friedman-Tolman solution transformed into the Schwarzschild coordinates, which allowed them to carefully sew it with an external static solution.

But for an explicit solution, one must introduce and calculate the function y(t,r) from the diagonality condition, which is the highlight of the OS method. Without this function, there is no exact solution in these coordinates, and hence it is impossible to determine the structure of the star as a whole.

Thus, the OS solution represents the Friedmann-Tolman solution, transformed from the comoving coordinates into the Schwarzschild coordinates for the diagonal internal metric and the matching with the external static solution by means of the function y(t,r) following from all that.

Once only analytical (i.e. exact) solutions are trustworthy, the result of this article is, perhaps the first in the history of the topic, and everything else is nothing more than walking around the bush.

I am glad that freezing is strictly proved and let all those who believe that they see the collapse into the horizon, or show on the world lines of the exact solution where and when this happens or will be silent.

I’ll explain what’s new in my paper compared to OS:

1. OS thought, and this was accepted by the BH’s supporters, that the picture in terms of the world time t is not complete, but in terms of the proper time is complete, since the latter covers that part of the particle’s evolution that goes “after” t=infinity and therefore “is not visible” to the external obs.

The paper shows that the description in terms of t is complete, since “the worldlines of the star’s particles in the exact solutions of the Einstein equations obtained by the OS method cover every moment of the existence of these particles in the real world and therefore these solutions give a complete picture of the evolution of the star. The irreversible dilation of the proper times due to relativistic and gravitational time dilations is the objective physical phenomenon that stops all processes in the star, including the process of collapse itself. This specific and fundamental physical phenomenon basically distinguishes the collapse scenario of the Einstein gravity from the Newtonian one, where there is no such stopping mechanism. “(from the Conclusion)

2. The OS has found a particular internal solution (i.e., y(t,r), metric and world lines) only for the parabolic velocity (velocity=0 at r=infinity). I was able to find a complete exact solution (including y(t,r)) for all velocities that preserve the homogeneity of the density, i.e. including elliptic (velocity=0 at r=R) and hyperbolic (the velocity is finite at r=infinity).

3. In addition to the solution, a physical interpretation is also needed, which includes a physical picture of the process (including visualization), a clear separation of the physical and non-physical regions of variables, finding expressions for asymptotes for the inner layers, the moments of freezing of the proper times. All this is done for the first time for all three velocity regimes.

4. It is shown that the BH picture refers to the non-physical region of the variables (t>infinity) and is therefore forbidden by general relativity, since in stars embedded in the real universe all worldlines of particles remain timelike during of real cosmological time (t<infinity) (for non-zero mass particles) and the singularity at the center and the event horizon are never formed.