**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

**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

**Abstract**

In the diffusion of cold light gas in warm heavy gas, initially and far before the relaxation, the thermal velocities of light and heavy atoms are the same order and the light gas remains cold, and mean energies of its particles are approximately conserved. The description of such conservative diffusion is analogous to the formalism of quantum mechanics, and quantum mechanics appears as a description of such diffusion of a particle in a physical vacuum, where the diffusion coefficient is proportional to the Planck constant inverse proportional to the mass. The growth of the diffusion flux in the localization of particles leads to an increase in the osmotic pressure, which reveals the “microscopic mechanism” of the uncertainty relations and allows us to identify cases where the prohibitions imposed by them can be circumvented, in particular, to solve the mass paradox for composite particles in the preon models. If two light particles (atoms) with different masses began to diffuse at distances much greater than the mean free path, then the diffusion mechanism prevents them from further joining and as more energy must be expended to form a composite particle, as smaller the final volume of localization. However, if these particles were initially located at a distance less than the mean free path and during a time shorter than the free pass time they formed a bound state (the atoms joined in the molecule), then this composite particle (molecule) diffuses as other light particles (atoms), but with a mass slightly less than the total mass of the initial particles.

*PACS*: 12.60.Rc, 12.60.Nz, 03.65.Ta, 05.30.Ch, 05.40.Jc

*Keywords*:* composite models, technicolor, quantum mechanics, conservative diffusion*

Vol. 11, No 1, p. 1 – 11, v1, May 1, 2016

Electron.: TPAC: 5200-041 v1, May 1, 2016 DOI: 10.9751/TPAC.5600-042

**Abstract**

At quantization of harmonic rotator, a rotational mode of planar oscillator, energy spectrum is linear on frequency and equidistant, but zero-point energy in ground state can not arise. This is in agreement with generalization of uncertainty relations to non-Hermitean canonical pairs. Quantization of waves at collective rotations of a chain of harmonic rotators allows one to model the fields with charge-conjugation and gauge symmetries. In quantum field theory (QFT) at quantization of rotational modes as harmonic rotators the observables of fields are normal ordered and zero-point energy and zero-point charge of vacuum do not arise. In this case frequencies of quanta are angular speeds of rotation of field vectors in real or field spaces and two signs of helicity correspond to a particle and an antiparticle. Photons with circular polarization and complex fields are examples of such fields, spin and isospins (charges) of particles can be related by their frequencies as angular momenta and helicities of the rotating field vectors. At rotational quantization of strings there are no zero-point energy of modes and here a conformal anomaly is absent, so spacetime dimensionality and gauge group are not fixed. In QFT the fields should be averaged in small cells of space and time, where distribution and evolution of fields are described classically, and field functions on borders of cells should be sewed. Then loop integrals are finite and the renormalized theories are invariant under reduction of the size of cells (a renormgroup with the cell regularization). The Planck scale cell is smallest because of freezing of proper times in a strong external gravitational field of the loop diagram with redshifting of frequencies up to zero. In the Standard Model and quantum gravity the loop contributions of fields, with exception of scalars, are small and the perturbation theory is convergent.

*PACS: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90. + t, 03.65.Ge, 03.70. + k, 11.10.Gh, 11.10.Hi, 11.15.Ha, 11.10.Ly, 12.20. – m, 11.25. – w, 12.10. – g,*

*Keywords: quantization, charge conjugation symmetry, harmonic rotator, quantum fields, vacuum energy, renormalization, regularization, strings, anomalies*

Vol. 10, No 1, p. 1 – 40, v1, March 28, 2015

Electron.: TPAC: 5200-040 v1, March 28, 2015 DOI: 10.9751/TPAC.5200-040

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

**Abstract**

In general relativity (GR) the worldline of a particle is unique and invariant, proper time and world time *t* are two parametrization of the same events on it only, that leads to a constraint for the proper time moments relating them by *t*. So, at contraction of a dust shell the proper times at finite *t* asymptotically freeze by no reaching a moment when the surface could cross the gravitational radius. Processes in entire volume of a star freeze at first at the center, then at higher layers, and at last the surface freezes outside the gravitational radius. Therefore in GR contraction leads to formation not black holes, but frozars (from “frozen star”) with the gravitationally frozen state of matter in entire volume, where the worldlines of particles are time-like everywhere, parallel to the *t*-axis and each other. Frozar formation in GR is shown for a thin dust shell, a dust ball, a star of uniform density and stars with ultrarelativistic matter. In real stars local temperatures in layers grow faster than temperature on the surface, and the last one grows on *t* exponentially fast. As high star’s mass, as high probability of that freezing occurs faster than warming up and the frozar will has time to be formed. But at lower masses, when the freezing does not enough fast, the warming up can stop contraction and can lead to explosion. During contraction a significant part of matter appear near the surface where in GR the physical volume sufficiently grows and energy of contraction is transformed to heat with transition of matter to the radiation dominated phase. If the star did not has time to be frozen, the part of ultrarelativistic matter and radiation leaves the star quickly, which appears as relativistic explosion, and the object will observed as relativistic supernova or hypernova. The observed lack of frozars of 2-4 solar mass and flat character of mass spectrum of more massive candidates to frozars confirm these predictions of the theory. the Big Bang and some explosions in astrophysics with large energy release probably are the cases of the relativistic explosion. In the frozar theory it appears a new GR phenomenon, the gravitational crystallization, defining structure of the most compact and massive objects in particle physics, astrophysics and cosmology. Gravitational radius of the system of few frozars sufficiently exceeds the radius of each of them and, therefore, at closing up they will not be able to merge and becomes frozen at distances larger their radii, forming a new state of matter – the gravitational crystal. Frozons, particles of the Planck energy, quantum fluctuations of which are frozen in their self gravitational field, also can not merge, i.e. for frozons there will be no interaction vertexes and they form only clusters and gravitational microcrystals. In astrophysics the supermassive frozen objects in the centers of star clusters, galaxies and quasars are probable gravitational supercrystals from frozars and ordinary matter. Relic frozons and frozar crystals could be the centers of inhomogeneities and also could be appear as a dark matter. If there is the backward contraction, the Universe as whole can be frozen also in the state of a global gravitational crystal which would stop the contraction.

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

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

Vol. 10, No 1, p. 41 – 67, v1, March 28, 2015

Electron.: TPAC: 5200-041 v1, March 28, 2015 DOI: 10.9751/TPAC.5200-040

**Abstract**

Stretching of the Newtonian potential (NP) at early epochs is investigated and it is shown that observed effects, usually ascribed to a dark matter, can by explained by such stretching only. Increasing by time a radius of the gravitationally-bound region (GBR) and conservation of gravitational energy lead to a new scenario in which values of NP in expanding volume are maintained, while in physical volume are stretched. Really, the energy conservation in expanding volume requires for NP values to be comoving to the expanding shells. In addition, the radius of gravitationally-bound region increases by time due to decreasing of expansion velocity and different shells around galaxy cease expansion at different times. Thus, as far a shell placed from galaxy, as longer it was expanded and thickened, while potential difference on its boundaries remained unchanged. This shifts the values of NP around galaxy proportional to the distance r and, as the result, the gravitational acceleration, from NP’s 1/r^2 dependence, turned to 1/r dependence, as for centrifugal acceleration. This fact naturally explains the known empirical facts, such as flatness of rotation curves and velocity-mass relationships for galaxies and velocity dispersion in clusters.

PACS: 95.30.Sf, 95.35.+d, 98.65.Cw, 98.62.Ck

*Key words*: *gravitation, cosmological expasion, rotation curves, galaxy clusters*

Vol. **9**, No 2, p. 73 –77, v1, 31 July 2014

Online: TPAC: 4960-039 v1, 31 July 2014; DOI: 10.9751/TPAC.4960-039

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

**Abstract**

The theory of conservative diffusion and its main applications are reviewed. A basic model for the theory is diffusion of a cold light gas in a warm heavy gas before relaxation when light gas remains cold and mean energy of its particles conserves. Unlike the Lorentz gas, where thermal energies of light and heavy atoms are equal, here the same order are their thermal speeds. Such conservative diffusion is described by two equations – the Hamilton-Jacobi and continuity equations, nonlinear under the probability density. They can be linearized by introduction of a complex probability amplitude, transforming them to the Schrödinger equation where one must add not probabilities, but probability amplitudes of alternatives. Mean free path and the corresponding momentum determine an elementary phase volume and a diffusion coefficient. The theory predicts a number of quasiquantum effects in classical systems. The formalism of quantum mechanics thus describes a classical conservative diffusion and quantum mechanics is only a special case of such diffusion in the vacuum, when the elementary phase volume is equal to the Planck constant. A conservative thermodiffusion at nonzero temperature gradient is studied also. Its properties, such as decreasing of intensity of fluctuations of particles (including redshift of frequencies), drift of particles to colder region and their thermodiffusive acceleration, not depending on the mass of particles, are similar to properties of gravitation. This allows us to identify gravitation by thermodiffusion in the physical vacuum. In the diffusive picture fluctuations of energy-momentum of classical particles due to interaction with vacuum lead to increasing of their mean energy, which appears as quantum phenomena, while corresponding local decreasing of vacuum energy density reveals as gravitation. The diffusive treatment of quantum theory thus leads to the thermodiffusive treatment of gravitation too with natural synthesis of theories of both phenomena. Observable effects following from the new theory are discussed.

PACS: 03.65.Ta, 04.20.Cv, 02.50.Ey, 05.40.Jc

*Key words*:* quantum fluctuations, vacuum energy, thermodiffusion, metrics, curvature*

Vol. **9**, No 2, p. 54 –72, v1, 31 July 2014

Online: TPAC: 4960-038 v1, 31 July 2014; DOI: 10.9751/TPAC. 4960-038

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

**Abstract**

New effects of stasis of radiation due to switching out from expansion flow at crossing largest gravitationally-bound regions (GBR), such as galaxy clusters, are considered and their observational consequences for supernovae and cosmic microwave background (CMB) data are discussed. The stasis of frequency and intensity of radiation at crossing of large number of clusters appreciably decreases observing redshifts z and magnifies apparent luminosities. Only normal redshifts z’ of photons not crossed clusters are directly related with the cosmic scale factor and thus true distances exceed those which follow from z. The effects increase for distant objects because of smaller inter-cluster distances at early epochs. For the relic radiation crossed the clusters the effects lead to the stasis of its temperature and “heating” with respect to a normally expanded flow outside the cluster. As a result, instead of former paradigm about almost sterile propagation of relic radiation from the recombination epoch, there is an opposite picture. Mixing of relic radiation flows, many times isolated from the expansion flow in GBRs along path, leads to their isotropy and loss of earlier perturbations. The observing anisotropy follows from the stasis effects at crossing of multiple layers of clusters at our nearest environment. The stasis effects allow one to do more exact conclusions from data analysis and lead to revising of distances and properties of extragalactic objects.

*PACS*: *04.20.Cv, 98.80.-k, 98.80.Jk 95.30.Sf, 97.60.Lf, 98.35.Jk, 98.54.-h*

*Key words: cosmological models, redshift, extra dimension*

Vol. **9**, No 1, p. 5 – 17, v1, 6 May 2014

Online: TPAC: 4874-035 v1, 6 May 2014; DOI: 10.9751/TPAC.4874-035

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

**Abstract**

Models of the closed Universe as a thin 3-sphere in 4-space, gravitating along 3-sphere’s radius, are reformulated in a new form, in which at a local environment of an observer the non-relativistic dynamics of a ball is reproduced with a correct energy conservation condition. Corresponding evolution equations for dust matter and radiation in the 3-sphere are obtained and their observational consequences are studied. It is shown that the closed models in 4-space also lead to the “Miniverse” model with a highly oscillating curve for the “distance modulus – redshift” relation.

*PACS*: *04.20.Cv, 98.80.-k, 98.80.Jk 95.30.Sf, 97.60.Lf, 98.35.Jk, 98.54.-h*

*Key words: cosmological models, redshift, extra dimension*

Vol. **9**, No 1, p. 1 – 4, v1, 6 May 2014

Online: TPAC: 4874-034 v1, 6 May 2014; DOI: 10.9751/TPAС.4874-034

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

**Abstract**

An influence of matter on the vacuum energy density is considered and a treatment of gravitation as inhomogeneity of quantum diffusion is developed. A treatment of quantum theory as conservative diffusion [1] is briefly presented, where quantum fluctuations of energy and momentum of a classical particle occur because of interaction with physical vacuum. The increasing of particle’s mean energy at such fluctuations appears as quantum phenomena, while corresponding local decreasing of mean vacuum energy to the same value appears as gravitation. For one a particle the decreasing is extremaly small and in particle physics it can be neglected. However, when large number of particles are concentrated in a small volume, consequences of the vacuum energy decreasing become appreciable and they appear as gravitation. The diffusion treatment of quantum processes thus leads to the diffusion treatment of gravitation with natural synthesis of theories of both phenomena. New properties of inhomogeneous diffusion related by local decreasing of vacuum energy, such as slowering of fluctuations of particles with delay of intensity of processes (including redshift of frequencies), drift of particles toward slower fluctuations region and their diffusive acceleration, which is independent on mass of particles, are studied. Observable effects following from the new treatment are discussed.

PACS: 02.50.Ey, 03.65.Ta , 05.40.Jc * *

*Key words*:* quantum mechanics, diffusion, Brownian motion, kinetic theory of gases*

Vol. **9**, No 1, p. 33 –53, v1, 6 May 2014

*Online: TPAC: 4874-037 v1, 6 May 2014*; DOI: 10.9751/TPAC.4874-037

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