The first fully integrated explicit exact solution of the Einstein field equations corresponding to the superposition of a counterrotating dust disk with a central black hole is presented. The obtained solution represents an infinite annular thin disk (a disk with an inner edge) around the Schwarszchild black hole. The mass of the disk is finite and the energy-momentum tensor agrees with all the energy conditions. Furthermore, the total mass of the disk when the black hole is present is less than the total mass of the disk alone. The solution can also be interpreted as describing a thin disk made of two counterrotanting dust fluids that are also in agreement with all the energy conditions. Additionally, as we will show shortly in a subsequent paper, the above solution is the first one of an infinite family of solutions.
A detailed study is presented of the counterrotating model (CRM) for generic electrovacuum static axially symmetric relativistic thin disks without radial pressure. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We also find explicit expressions for the energy densities, charge densities and velocities of the counterrotating fluids. We then show that this constraint can be satisfied if we take the two counterrotating streams as circulating …
Disks of collisionless particles are important models for certain galaxies and accretion disks in astrophysics. We present here a solution to the stationary axisymmetric Einstein equations which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocity in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk previously given by Neugebauer and Meinel, in the limit …
A detailed study is presented of the counterrotating model (CRM) for electrovacuum stationary axially symmetric relativistic thin disks of infinite extension without radial stress, in the case when the eigenvalues of the energy-momentum tensor of the disk are real quantities, so that there is not heat flow. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We then show that, in some cases, this constraint can be satisfied if we take the two co…
We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the $\ell=2$ axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge-Wheeler-Zerilli approach. This formulation is useful, for instance, for the treatment in the ``close limit approximation'' of the collision of counterrotating black holes.
A family of models of counterrotating and rotating relativistic thin discs of infinite extension based on a charged and magnetized Kerr-NUT metric are constructed using the well-known "displace, cut and reflect" method extended to solutions of vacuum Einstein-Maxwell equations. The metric considered has as limiting cases a charged and magnetized Taub-NUT solution and the well known Kerr-Newman solutions. We show that for Kerr-Newman fields the eigenvalues of the energy-momentum tensor of the disc are for all the values of the parameters real quantities so that these discs do not present heat f…
We investigate gravitational waves from a dust disk around a Schwarzschild black hole to focus on whether we can extract any of its physical properties from a direct detection of gravitational waves. We adopt a black hole perturbation approach in a time domain, which is a satisfactory approximation to illustrate a dust disk in a supermassive black hole. We find that we can determine the radius of the disk by using the power spectrum of gravitational waves and that our method to extract the radius works for a disk of arbitrary density distribution. Therefore we believe a possibility exists for …
A "temporal analogue" of the standard Poynting-Robertson effect is analyzed as induced by a dust of particles (instead of a gas of photons) surrounding a Schwarzschild black hole. Test particles inside this cloud undergo acceleration effects due to the presence of a friction force, so that the fate of their evolution can be completely different from the corresponding geodesic motion. Typical situations are discussed of hyperbolic motion of particles scattered by the black hole in the presence of a dust filling the whole spacetime region outside the horizon as well as particles which free fall …
We consider the problem of gravitational collapse of a fluid under the effect of a Schwarzschild black hole (e.g. a primordial one) that suddenly forms inside the fluid. We assume the fluid initially be a uniform dust. Starting from this configuration we obtain a class of metrics under some assumptions. We find that the metric we obtain includes the dust collapse as a subcase. After discussing some basic properties of the solution, we discuss the case of dust collapse in more detail. We find that the radial and tangential pressures outside the horizon may take positive or negative values depen…
A physics-first derivation of the Schwarzschild metric is given. Gravitation is described in terms of the effects of tidal forces (or of spacetime curvature) on the volume of a small ball of test particles (a dust ball), freely falling after all particles were at rest with respect to each other initially. The possibility to express Einstein's equation this way and some of its ramifications have been enjoyably discussed by Baez and Bunn [Am. J. Phys. 73, 644 (2005)]. Since the formulation avoids the use of tensors, neither advanced tensor calculus nor sophisticated differential geometry are nee…
The equation of motion announced in the title was already deduced for the cases the inner metric being flat and the shell being negligibly small (test matter), using surface layers and geodesic trajectories resp. Here we derive the general equation of motion and solve it in closed form for the case of parabolic motion. Especially the motion near the horizon and near the singularity are examined.
An exact solution of the Lemaître--Tolman--Bondi class is investigated as a possible model of the Schwarzschild-like black hole embedded in a non-static dust-filled universe for the three types of spatial curvature. The solution is obtained in comoving coordinates by means of the mass function method. It is shown that the central part of space contains a Schwarzschild-like black hole. The R-T-structure of the resulting spacetime is built. It is shown that the solution includes both the Schwarzschild and Friedmann solutions as its natural limits. The geodesic equations for test particles are an…
A concentric charged thin-shell encircling a Reissner-Nordström black hole screens the clectric / magnetic charge completely to match with an external Schwarzschild black hole. The negative mass thin-shell is shown to be stable against radial perturbations. It is shown further that by reversing the roles of inside Reissner-Nordström and outside Schwarzschild geometries the mass of the appropriate shell becomes positive.
The dynamics of the dust-shell model of universe is exactly solved for the modified Schwarzschild solution. This solution is used to derive the cosmology corresponding to the modified gravity.
We present a time dependent isotropic fluid solution around a Schwarzschild black hole. We offer the solutions and discuss the effects on the field equations and the horizon. We derive the energy density, pressure and the equation of state parameter. In the second part, we generalize the rotating black hole solution to an expanding universe. We derive from the proposed metric the special solutions of the field equations for the dust approximation and the dark energy solution. We show that the presence of a rotating black hole does not modify the scale factor $b(t)=t^{2/3}$ law for dust, nor $b…
We derive the linearly perturbed matching conditions between a Schwarzschild spacetime region with stationary and axially symmetric perturbations and a FLRW spacetime with arbitrary perturbations. The matching hypersurface is also perturbed arbitrarily and, in all cases, the perturbations are decomposed into scalars using the Hodge operator on the sphere. This allows us to write down the matching conditions in a compact way. In particular, we find that the existence of a perturbed (rotating, stationary and vacuum) Schwarzschild cavity in a perturbed FLRW universe forces the cosmological pertur…
The explicit form of perturbation equation for the $Ψ_4$ Weyl scalar, containing the matter source terms, is derived for general type D spacetimes. It is described in detail the particular case of the Schwarzschild spacetime using in-going penetrating coordinates. As a practical application, we focused on the emission of gravitational waves when a black hole is perturbed by a surrounding dust-like fluid matter. The symmetries of the spacetime and the simplicity of the matter source allow, by means of a spherical harmonic decomposition, to study the problem by means of a one dimensional numeric…
We derive a class of non-static inhomogeneous dust solutions in f(R) gravity described by the Lemaitre-Tolman-Bondi (LTB) metric. The field equations are fully integrated for all parameter subcases and compared with analogous subcases of LTB dust solutions of GR. Since the solutions do not admit regular symmetry centres, we have two possibilities: (i) a spherical dust cloud with angle deficit acting as the source of a vacuum Schwarzschild-like solution associated with a global monopole, or (ii) fully regular dust wormholes without angle deficit, whose rest frames are homeomorphic to the Schwar…
By combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a Schwarzschild black hole. The matter models considered include quadrupolar dust shells and thick accretion disks, permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger. We focus on the interference pattern appearing in the…
We investigate quantum dynamics of self-gravitating spherical dust shell. The wave functions of discrete spectrum are not localized inside the Schwarzschild radius. We argue that such shells can transform into white holes (in another space). It is plausible that shells with bare masses larger than the Planck mass loose their mass emitting lighter shells.
Marginally trapped surfaces (MTSs) are commonly used in numerical relativity to locate black holes. For dynamical black holes, it is not known generally if this procedure is sufficiently reliable. Even for Schwarzschild black holes, Wald and Iyer constructed foliations which come arbitrarily close to the singularity but do not contain any MTSs. In this paper, we review the Wald-Iyer construction, discuss some implications for numerical relativity, and generalize to the well known Vaidya spacetime describing spherically symmetric collapse of null dust. In the Vaidya spacetime, we numerically lo…
The standard cosmological model posits a spatially flat universe of infinite extent. However, no observation, even in principle, could verify that the matter extends to infinity. In this work we model the universe as a finite spherical ball of dust and dark energy, and obtain a lower limit estimate of its mass and present size: the mass is at least 5 x 10^23 solar masses and the present radius is at least 50 Gly. If we are not too far from the dust-ball edge we might expect to see a cold spot in the cosmic microwave background, and there might be suppression of the low multipoles in the angula…
It is shown that the description of collapse given by the classic model of Oppenheimer and Snyder fails to satisfy a crucial matching condition at the surface of the ball. After correcting the model so that the interior and exterior metrics match correctly, it is established that the contraction process stops at the Schwarzschild radius, that there is an accumulation of particles at the surface of the ball, and that in the limit of infinite time lapse the density of particles at the surface becomes infinite. A black hole cannot form. This result confirms the judgements of both Einstein and Edd…
The dynamics of the gravitational collapse is examined in the realm of string based formalism of D-branes that encompass General Relativity as a low energy limit. A complete analytical solution is given to the spherically symmetric collapse of a pure dust star, including its matching with a corrected Schwarzschild exterior spacetime. The collapse forms a black hole (an exterior event horizon) enclosing not a singularity but perpetually bouncing matter in the infinite chain of spacetime maximal analytical extensions inside the outer event horizon. This chain of analytical extensions has a struc…
We construct a one-loop effective metric describing the evaporation phase of a Schwarzschild black hole in a spherically symmetric null-dust model. This is achieved by quantising the Vaidya solution and by chosing a time dependent quantum state. This state describes a black hole which is initially in thermal equilibrium and then the equilibrium is switched off, so that the black hole starts to evaporate, shrinking to a zero radius in a finite proper time. The naked singularity appears, and the Hawking flux diverges at the end-point. However, a static metric can be imposed in the future of the …
Semi-classical corrections at large curvature are employed in toy models of spherically symmetric gravitational collapse in order to avoid the formation of singularities. The resulting spacetimes may produce bounces, compact remnants or regular black holes in place of the usual Schwarzschild black hole. Within these models, a whole class of collapse scenarios leading to the formation of regular black holes may be obtained from General Relativity coupled to some theory of non-linear electrodynamics. In the present chapter we provide a thorough exposition of semi-classical dust collapse with par…
We study quantum corrections for the Schwarzshild black hole by considering it as a vacuum solution of a 2D dilaton gravity theory obtained by spherical reduction of 4D gravity coupled with matter. We find perturbatively the vacuum solution for the standard one-loop effective action in the case of null-dust matter and in the case of minimally coupled scalar field. The corresponding state is in both cases 2D Hartle-Hawking vacuum, and we evaluate the corresponding quantum corrections for the thermodynamical parameters of the black hole. We also find that the standard effective action does not a…
Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole …
We show that the mass function method makes it much easier to obtain the exact solutions of Einstein equations. The known solutions for empty space and for the universe filled with dust-like matter are considered from the point of view of the method and new exact solutions are obtained. Generalized solutions for the Schwarzschild type black holes are considered. It is shown that such black holes always contain non-baryonic matter. New exact solution for the black hole embedded into dust matter cosmological background is obtained.
Using a static massive spherically symmetric scalar field coupled to gravity in the Schwarzschild-de Sitter (SdS) background, first we consider some asymptotic solutions near horizon and their local equations of state(E.O.S) on them. We show that near cosmological and event horizons our scalar field behaves as a dust. At the next step near two pure de-Sitter or Schwarzschild horizons we obtain a coupling dependent pressure to energy density ratio. In the case of a minimally couplling this ratio is -1 which springs to the mind thermodynamical behavior of dark energy. If having a negative pressu…
We present a new type of gravitational mass defect in which an infinite amount of matter may be bounded in a zero ADM mass. This interpolates between effects typical of closed worlds and T-spheres. We consider the Tolman model of dust distribution and show that this phenomenon reveals itself for a solution that has no origin on one side but is closed on the other side. The second class of examples corresponds to smooth gluing T-spheres to the portion of the Friedmann-Robertson-Walker solution. The procedure is generalized to combinations of smoothly connected T-spheres, FRW and Schwarzschild m…
We consider the gravastar model where the vacuum phase transition between the de Sitter interior and the Schwarzschild or Schwarzschild-de Sitter exterior geometries takes place at a single spherical delta-shell. We derive sharp analytic bounds on the surface compactness (2m/r) that follow from the requirement that the dominant energy condition (DEC) holds at the shell. In the case of Schwarzschild exterior, the highest surface compactness is achieved with the stiff shell in the limit of vanishing (dark) energy density in the interior. In the case of Schwarzschild-de Sitter exterior, in additi…
An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their …
Will (1974) treated the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to the perturbation by a thin disc (which is more relevant astrophysically), but, due to a rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green's functions represented by the Will's result can be expressed i…
In general relativity, the endpoint of spherically symmetric gravitational collapse is a Schwarzschild--[(A)dS] black hole. In bimetric gravity, it has been speculated that a static end state must also be Schwarzschild--[(A)dS]. To this end, we present a set of exact solutions, including collapsing massless dust particles. For these, the speculation is confirmed.
We propose and analyze a new metric that has two conformal factors a(t) and b(t) that combine the expansion of the universe and its effects on the spatial and temporal part of the Schwarzschild metric in isotropic coordinates. We present the solutions, their descriptions and we comment on their shortcomings. In the spatially flat case of an expanding universe, we derive from the proposed metric the special solutions of the field equations for the dust approximation and the McVittie metric. We show that the presence of a black hole does not modify the a(t) proportional to t^{2/3} law for dust a…
The work is devoted to the construction of explicit embeddings for the metrics of the black holes, formed by nonsingular matter distribution. One of the possible examples of such type of solutions is regular black hole. Using the existing classification of minimal symmetric embeddings of the Schwarzschild metric as a base, we construct embeddings for regular black holes with de Sitter interior. Another simple example is black hole, formed by collapsing homogeneous spherically symmetric cloud of dustlike matter. We discuss embeddings for two variants of such black holes - the one with the etern…
Quantum gravity suggests that the paradox recently put forward by Almheiri et. al. (AMPS) can be resolved if matter does not undergo continuous collapse to a singularity but condenses on the apparent horizon. One can then expect a quasi-static object to form even after the gravitational field has overcome any degeneracy pressure of the matter fields. We consider dust collapse. If the collapse terminates on the apparent horizon, the Misner-Sharp mass function of the dust ball is predicted and we construct static solutions with no tangential pressure that would represent such a compact object. T…
The collapse of marginally bound, inhomogeneous, pressureless (dust) matter, in spherical symmetry, is considered. The starting point is not, in this case, the integration of the Einstein equations from some suitable initial conditions. Instead, starting from the corresponding general exact solution of these equations, depending on two arbitrary functions of the radial coordinate, the fulfillment of the Lichnerowicz matching conditions of the interior collapsing metric and the exterior Schwarzschild one is tentatively assumed (the continuity of the metric and its first derivatives on the time-…
We perform a Hamiltonian reduction of spherically symmetric Einstein gravity with a thin dust shell of positive rest mass. Three spatial topologies are considered: Euclidean (R^3), Kruskal (S^2 x R), and the spatial topology of a diametrically identified Kruskal (RP^3 - {a point at infinity}). For the Kruskal and RP^3 topologies the reduced phase space is four-dimensional, with one canonical pair associated with the shell and the other with the geometry; the latter pair disappears if one prescribes the value of the Schwarzschild mass at an asymptopia or at a throat. For the Euclidean topology …
We consider the quantum vacuum effects of the massless scalar fields that are non-minimally coupled to the background geometry of a collapsing homogeneous ball of dust. It is shown that for a definite range of coupling constants, there are repulsive quantum vacuum effects, capable of stopping the collapse process inside the black hole and precluding the formation of singularity. The final fate of the collapse will be a black hole with no singularity, inside which the matter stays balanced. The density of the final static matter will be close to the Planck density. We show that the largest poss…
The bizarre behaviour of the apparent (black hole and cosmological) horizons of the McVittie spacetime is discussed using, as an analogy, the Schwarzschild-de Sitter-Kottler spacetime (which is a special case of McVittie anyway). For a dust-dominated "background" universe, a black hole cannot exist at early times because its (apparent) horizon would be larger than the cosmological(apparent) horizon. A phantom-dominated "background" universe causes this situation, and the horizon behaviour, to be time-reversed.
The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by Háj{\'ı}ček and Kuchař. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in…
We point out that although the pioneering work of Oppenheimer and Snyder (OS), technically, indicated the formation of an event horizon for a collapsing homogeneous dust ball of mass $M_b$ and radius $R_b$, the Eqs. (32) and (36) of their paper actually demand that no trapped surface is formed $ 2GM_b/ R_b c^2 \le 1$, and $M_b \equiv 0$! Anybody may verify it. Further, by analyzing the general inhomogeneous dust solutions of Tolman in a proper physical perspective, we show that, all dust solutions are characterized by $M_b=0$. Next, for a collapsing fluid endowed with radiation pressure, we di…
The Schwarzschild black hole can be viewed as the special case of the marginally bound Lema\^ıtre-Tolman-Bondi models of dust collapse which corresponds to a constant mass function. We have presented a midi-superspace quantization of this model for an arbitrary mass-function in a separate publication. In this communication we show that our solution leads both to Bekenstein's area spectrum for black holes as well as to the black hole entropy, which, in this context, is naturally interpreted as the loss of information of the original matter distribution within the collapsing dust cloud.
We consider the generalized Tolman solution of general relativity, describing the evolution of a spherical dust cloud in the presence of an external electric or magnetic field. The solution contains three arbitrary functions $f(R)$, $F(R)$ and $τ_0(R)$, where $R$ is a radial coordinate in the comoving reference frame. The solution splits into three branches corresponding to hyperbolic ($f >0$), parabolic ($f=0$) and elliptic ($f < 0$) types of motion. In such models, we study the possible existence of wormhole throats defined as spheres of minimum radius at a fixed time instant, and prove the …
In a previous paper we studied the collapse of a spherically symmetric dust distribution (marginally bound LTB) in d-dimensional AdS spacetime and obtained the condition for the formation of trapped surfaces. Here we extend the analysis by giving the canonical theory for the same and subsequently quantize the system by solving the Wheeler-DeWitt equation. We show that for the case of small dust perturbations around a black hole the wave functionals so obtained describe an AdS-Schwarzschild black hole in equilibrium with a thermal bath at Hawking temperature and show the non-trivial dependence …
We investigate the thermodynamics and phase transition for Kiselev black hole and dilaton black hole. Specifically we consider Reissner Nordstrom black hole surrounded by radiation and dust, and Schwarzschild black hole surrounded by quintessence, as special cases of Kiselev solution. We have calculated the products relating the surface gravities, surface temperatures, Komar energies, areas, entropies, horizon radii and the irreducible masses at the Cauchy and the event horizons. It is observed that the product of surface gravities, surface temperature product and product of Komar energies at…
In the former part, we study the gravitational collapse of pressureless dust and find special solutions, where, in both the physical and fiducial sectors, the exterior and interior spacetime geometries are given by the Schwarzschild spacetimes and the Friedmann-Lemaître-Robertson-Walker universes dominated by pressureless dust, respectively, with specific time slicings. In the case where the Lagrange multipliers are trivial and have no jump across the matter interfaces in both the physical and fiducial sectors, the junction conditions across them remain the same as those in general relativity …
The interpretation of so-called cosmic string in black hole spacetimes has settled down to an unsatisfactory state. In this article we try to provide a different model for these cosmic strings by explicit construction of these spacetimes from the Bonnor rocket solution. It is shown that the correct stress-energy tensor is that of null dust with a rather strange energy density --- first derivative of Dirac delta distribution. We will discuss the Schwarzschild solution and the C--metric. In the latter case we will show that there is a momentum flux through the cosmic string, causing the accelera…
For an extension of the previous work on gravitational lensing by massless braneworld black holes, we investigate their microlensing phenomena and shadows and discuss how to distinguish them from standard Schwarzschild black holes and Ellis wormholes. Microlensing is known as the phenomenon in which luminosity amplification appears when a bright object passes behind a black hole or another massive object. We find that, for the braneworld black hole as well as for the Ellis wormhole, there appears luminosity reduction just before and after the amplification. This means that observation of such …
We examine in this paper the possibility of finding exact solutions for Teleparallel Gravity (TG) of the type of spherically symmetric Lema\^ıtre-Tolman-Bondi (LTB) dust models. We apply to the LTB metric, as obtained from the Schwarzschild solution in General Relativity, the formalism of Teleparallel Gravity in its extension to $f(T,B)$ models. An exact LTB solution is obtained that is compatible with a specific $f(T,B)$ model that seems to be appropriate to fit observations when applied to standard spatially flat Robertson-Walker geometry.
We study the covariant entropy bound in the context of gravitational collapse. First, we discuss critically the heuristic arguments advanced by Bousso. Then we solve the problem through an exact model: a Tolman-Bondi dust shell collapsing into a Schwarzschild black hole. After the collapse, a new black hole with a larger mass is formed. The horizon, $L$, of the old black hole then terminates at the singularity. We show that the entropy crossing $L$ does not exceed a quarter of the area of the old horizon. Therefore, the covariant entropy bound is satisfied in this process.
We examine the gravitational collapse and black hole formation of multiple non--spherical configurations constructed from Szekeres dust models with positive spatial curvature that smoothly match to a Schwarzschild exterior. These configurations are made of an almost spherical central core region surrounded by a network of "pancake-like" overdensities and voids with spatial positions prescribed through standard initial conditions. We show that a full collapse into a focusing singularity, without shell crossings appearing before the formation of an apparent horizon, is not possible unless the fu…
Probing black holes surrounded by Dust field in Quantum Fluctuation Modified Gravity with twin-peak QPOs
National University of Uzbekistan（乌兹别克斯坦国立大学）；Samarqand State Pedagogical Institute（撒马尔罕国立师范学院）；Institute of Fundamental and Applied Research, National Research University TIIAME（塔什干信息技术与管理系统国立研究大学基础与应用研究所）；Tashkent State Technical University（塔什干国立技术大学）；School of Physics, Harbin Institute of Technology（哈尔滨工业大学物理学院）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Institute of Theoretical Physics, National University of Uzbekistan（乌兹别克斯坦国立大学理论物理研究所）；New Uzbekistan University（新乌兹别克斯坦大学）；Kimyo International University in Tashkent（塔什干基莫国际大学）；University of Economics and Pedagogy（经济与教育学院）；Tashkent International University（塔什干国际大学）
