Vladimir Belayev - Academia.edu (original) (raw)

Papers by Vladimir Belayev

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected when determining their motion. The gravitational mass of the cloud is obtained by applying Lorentz transformations to the isotropic Schwarzschild metric, which defines the field of each particle, and using the superposition principle. It is found for the region where the cloud can be considered as a point body. The dependence of this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established. This result is used to determine the upper limit of the gravitational mass of a neutrino sufficient for its density to be close to the density of dark energy.

D Yu Tsipenyuk , W B Belayev. Gravitational Waves, Fields, and Particles in the Frame of (1 + 4)D Extended Space Model. Chapter in "Gravitational Waves - Theory and Observations". IntechOpen, 2023

The generalization of Einstein's special theory of relativity (SRT) is proposed. Interval s is us... more The generalization of Einstein's special theory of relativity (SRT) is proposed. Interval s is used as a fifth additional coordinate. We employ (1+4)-dimensional space G, an extension of the (1+3)-dimensional Minkowski space M. Interval changes under the transformations in the extended space G. From a physical point of view our expansion means that processes in which the rest mass of the particle's changes are acceptable now. In the Extended space model (ESM) a photon can have a nonzero variable mass. The equations for the plane-wave potentials describing the process of its localization are presented and their solution is found. The dynamics of particles in a gravitational field is investigated using Lagrange mechanics, a photon gravitational mass is determined for weak gravitation. Conservation of energy as a source of gravity in case of sphericity consisting of electrons and positrons, provided a twice gravitational mass of photons in comparison with a material particle of equivalent energy, leads to the presence of particles with negative gravitational mass and zero kinematic momentum in the results of the annihilation. Reaction of the conversion of a photon into an electron-positron pair will lead to the appearance particles, which are identified as graviton.

AIP Conf. Proc., 2023

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected. The gravitational mass of the cloud is determined for the region where it can be considered as a point body. The dependence of this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established.

Preprint in Cambrige Open Engage, 2022

The pressure of the vacuum is detected as per the geometry of the space around the local gravity ... more The pressure of the vacuum is detected as per the geometry of the space around the local gravity system according to Sakharov's idea of a "metrical elasticity" of space, i.e., of the emergence of a generalized force, preventing distortion of space. It is assumed that energy equivalent to the gravitational defect of masses is spent on vacuum deformation. We determine the gravitational impact of matter on the vacuum in case of weakly gravitating static centrally symmetric distribution of matter. Opposite in the sign pressure of vacuum corresponds to appropriate solution of Einstein's equations. It can be used to determine the course of time below the Earth's surface.

Prespacetime Journal, Vol 13, No 1 (2022), 32, 2022

The dynamics of particles in a gravitational field is investigated using Lagrange mechanics. Dyna... more The dynamics of particles in a gravitational field is investigated using Lagrange mechanics. Dynamic equations are obtained, including the rate of energy and momentum transfer to the gravitational field. The motion of particles in the Schwarzschild field is considered, and in the case of weak gravitation, it is determined the passive gravitational mass of a photon and a material particle under condition of the relationship between the gravitational potential and its velocity 22 / << / r V c . The active gravitational mass is found for the system special case of two identical bodies moving in opposite directions.

D. Yu. Tsipenyuk and W. B. Belayev, Rom. Rep. Phys. 72, 113 (2020), 2020

Based on Sakharov's idea of a 'metrical elasticity' of space, i.e., of the emergence of a general... more Based on Sakharov's idea of a 'metrical elasticity' of space, i.e., of the emergence of a generalized force, preventing distortion of space, we detect the pressure of the vacuum as per the geometry of the space around the local gravity system. The gravitational defect of mass is interpreted as the transfer of energy to the vacuum, which becomes apparent from its deformation. We determine the gravitational impact of matter on the vacuum and opposite in the sign pressure of it in case of weakly gravitating static centrally symmetric distribution of matter using appropriate solution of Einstein's equations. A possibility to extend the obtained results to arbitrary gravitational systems is evaluated. A non-conservation of energy in gravitational systems is interpreted by the Extended Space Model (ESM) as the rotation of the energy-momentum vector in 5-dimensional space. A proposed approach to determining pressure as a source of gravity leads to a revision of the dependence of the deceleration parameter of the Universe on the density parameters. Under this condition we examine the ratio between the density parameters of dark energy and cosmological constant depending on the deceleration parameter.

Dmitry Yu. Tsipenyuk, Vladimir B. Belyaev, RENSIT, 11(3):249-260, 2019

Abstract. It is considered (1+4)-dimensional Extended Space Model (ESM) as a generalization of th... more Abstract. It is considered (1+4)-dimensional Extended Space Model (ESM) as a generalization of the special theory of relativity at a 5-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. Within the framework of ESM the photons have a nonzero mass in a gravitational field. We study how a rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The force vector acting on the photon in Schwarzschild space-time is found for the weak gravity and corresponds to photon's gravitational mass equal to the twice mass of a material particle of the same energy. Compliance with the law of conservation of energy as a source of gravity leads to the presence of particles with negative gravitational mass and zero kinematic momentum in the results of the annihilation reaction. Bubble gravitational structures of a type gravastar are considered. Accretion of matter onto compact stars results in their absorption of positive energy from the vacuum and the release negative energy in a free deep space. The particles with negative gravitational mass create there antigravitating vacuum with negative pressure. In the present work a non-conservation of energy in gravitational systems is interpreted by the ESM as the rotation of the energy-momentum vector in 5-dimensional space. The comparison bubble cosmic structures with similar possible bubble structures in microphysical objects is made.

: Dmitry Yu. Tsipenyuk, Vladimir B. Belayev, Rom. Rep. in Phys., 71 (4), 2019

Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensi... more Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensional space, and more specifically at (1 + 4)-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. The possibility of a transition from the components of the 5-momentum of a particle in extended flat space to the components of a 4-momentum in an arbitrary 4-dimensional space by means of rotations is considered. Variational principle of the stationary energy integral of photon allows us to determine its dynamics. We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. We study how (TS)-rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The produced equations are solved for the metrics of Schwarzschild and Goedel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.

D. Yu. Tsipenyuk and W. B. Belayev 2019 J. Phys.: Conf. Ser. 1251 012048, 2019

Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensi... more Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensional space, and more specifically at (1 + 4)-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. The possibility of a transition from the components of the 5-momentum of a particle in extended flat space to the components of a 4-momentum in an arbitrary4-dimensional space by means of rotations is considered. Variational principle of the stationary energy integral of photon allows us to determine its dynamics. We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. We study how (TS)-rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle for the stationary gravity field and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The produced equations are solved for the metrics of Schwarzschild and Gödel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.

The existence of negative pressure of vacuum follows from the cosmological models, based on the r... more The existence of negative pressure of vacuum follows from the cosmological models, based on the results of observations. But, is it possible to detect the pressure of the vacuum as per the geometry of the space around the local gravity system? The gravitational defect of mass is interpreted as the transfer of energy to the vacuum, which becomes apparent from its deformation. We determine the gravitational impact of matter on the vacuum and opposite in the sign pressure of it in case of weakly gravitating static centrally symmetric distribution of matter. We have evaluated a possibility to extend the obtained results to arbitrary gravitational systems. The equation of state gives the deceleration parameter of the universe consistent with its constant expansion rate.

Arxiv: 0911.0614, Jul 13, 2018

We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find... more We consider variation of energy of
the light-like particle in the pseudo-Riemann space-time, find
lagrangian, canonical momenta and forces. Equations of the
critical curve are obtained by the nonzero energy integral
variation in accordance with principles of the calculus of
variations in mechanics. This method is compared with the Fermat's
principle for the stationary gravity field and geodesics
principle. Energy and momentum of the particle transferred to the
gravity field is defined. The produced equations are solved for
the metrics of Schwarzschild, FLRW model for the flat space and
Goedel. The gravitation mass of the photon is found in central
gravity field in the Newtonian limit.

A time-like geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory a... more A time-like geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory and astrophysical applications are proposed. For example of such space-time in 4D spherical coordinates with the space-like fifth dimension a transition to the cylindrical frame is implemented. After this transformation a space is divided into two parts that is interpreted physically as pair of the universe and anti-universe. The rotation in 5D space-time exhibits itself in 4D as action of extra force, which reveals itself with observed radial motion of the material particle. Also it is considered similar (3+2)D space-time in which additional dimension is time-like and a motion of space is hyperbolic. For these space-times it is found geodesics with constant radius in 4D spherical coordinates and studied their small deviations in cylindrical frames. It is shown that model with space-like fifth coordinate conforms to the basic properties of the Pioneer-effect, namely, a) constant additional acceleration of apparatus on distance from 20 to 50 a.e., b) its increase from 5 to 20 a.e., c) observed absence of one in motion of planets.

Books by Vladimir Belayev

The book is mainly used an algebraic approach to the presentation of the theory of relativity, mu... more The book is mainly used an algebraic approach to the presentation of the theory of relativity, much attention is paid to the variational methods and mechanics of motion of particles in the gravitational field, including the dynamics. The introduction of metrical energy of the light-like particle allows to determine its Lagrangian and use not only the kinematics, but also the dynamics for the analysis of its free movement. The problem of energy and momentum transferred to the particle or system of the gravitational field, for different types of energy and momentum. An analogy with Newtonian gravity and the gravitational mass of the photon is obtained in a weak gravitational field.
Differentiated approach to the conservation laws allows to determine the pressure of gravity field and vacuum. This is done on the basis of solutions of Einstein's field equations for a static spherically symmetric source. The speed of gravity in curved space obtained by using analogies with the gaseous medium.
The bases of the theory of Kaluza-Klein for a five-dimensional space-time, the theory of induced gravity is considered. Models of the rotating hyperspace with space- and time-like fifth coordinate, dynamics of the material particle in them are investigated..
The book is intended for scientists, graduate students, undergraduate students majoring in physics and astronomy. As in the first part contains the basis of the special theory of relativity, it is also of interest for those who begin to study this branch of physics.

Conference Presentations by Vladimir Belayev

Report at IC-MSQUARE 11, Mathematical Modeling in Physical Sciences, September 5-8, 2022, Virtual, on-line Conference, Publ. in: AIP Conf. Proc., 2023

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected.
The gravitational mass of the cloud is determined for the region where it can be considered as a point body. The dependence of
this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established.

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected when determining their motion. The gravitational mass of the cloud is obtained by applying Lorentz transformations to the isotropic Schwarzschild metric, which defines the field of each particle, and using the superposition principle. It is found for the region where the cloud can be considered as a point body. The dependence of this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established. This result is used to determine the upper limit of the gravitational mass of a neutrino sufficient for its density to be close to the density of dark energy.

D Yu Tsipenyuk , W B Belayev. Gravitational Waves, Fields, and Particles in the Frame of (1 + 4)D Extended Space Model. Chapter in "Gravitational Waves - Theory and Observations". IntechOpen, 2023

The generalization of Einstein's special theory of relativity (SRT) is proposed. Interval s is us... more The generalization of Einstein's special theory of relativity (SRT) is proposed. Interval s is used as a fifth additional coordinate. We employ (1+4)-dimensional space G, an extension of the (1+3)-dimensional Minkowski space M. Interval changes under the transformations in the extended space G. From a physical point of view our expansion means that processes in which the rest mass of the particle's changes are acceptable now. In the Extended space model (ESM) a photon can have a nonzero variable mass. The equations for the plane-wave potentials describing the process of its localization are presented and their solution is found. The dynamics of particles in a gravitational field is investigated using Lagrange mechanics, a photon gravitational mass is determined for weak gravitation. Conservation of energy as a source of gravity in case of sphericity consisting of electrons and positrons, provided a twice gravitational mass of photons in comparison with a material particle of equivalent energy, leads to the presence of particles with negative gravitational mass and zero kinematic momentum in the results of the annihilation. Reaction of the conversion of a photon into an electron-positron pair will lead to the appearance particles, which are identified as graviton.

AIP Conf. Proc., 2023

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected. The gravitational mass of the cloud is determined for the region where it can be considered as a point body. The dependence of this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established.

Preprint in Cambrige Open Engage, 2022

The pressure of the vacuum is detected as per the geometry of the space around the local gravity ... more The pressure of the vacuum is detected as per the geometry of the space around the local gravity system according to Sakharov's idea of a "metrical elasticity" of space, i.e., of the emergence of a generalized force, preventing distortion of space. It is assumed that energy equivalent to the gravitational defect of masses is spent on vacuum deformation. We determine the gravitational impact of matter on the vacuum in case of weakly gravitating static centrally symmetric distribution of matter. Opposite in the sign pressure of vacuum corresponds to appropriate solution of Einstein's equations. It can be used to determine the course of time below the Earth's surface.

Prespacetime Journal, Vol 13, No 1 (2022), 32, 2022

The dynamics of particles in a gravitational field is investigated using Lagrange mechanics. Dyna... more The dynamics of particles in a gravitational field is investigated using Lagrange mechanics. Dynamic equations are obtained, including the rate of energy and momentum transfer to the gravitational field. The motion of particles in the Schwarzschild field is considered, and in the case of weak gravitation, it is determined the passive gravitational mass of a photon and a material particle under condition of the relationship between the gravitational potential and its velocity 22 / << / r V c . The active gravitational mass is found for the system special case of two identical bodies moving in opposite directions.

D. Yu. Tsipenyuk and W. B. Belayev, Rom. Rep. Phys. 72, 113 (2020), 2020

Based on Sakharov's idea of a 'metrical elasticity' of space, i.e., of the emergence of a general... more Based on Sakharov's idea of a 'metrical elasticity' of space, i.e., of the emergence of a generalized force, preventing distortion of space, we detect the pressure of the vacuum as per the geometry of the space around the local gravity system. The gravitational defect of mass is interpreted as the transfer of energy to the vacuum, which becomes apparent from its deformation. We determine the gravitational impact of matter on the vacuum and opposite in the sign pressure of it in case of weakly gravitating static centrally symmetric distribution of matter using appropriate solution of Einstein's equations. A possibility to extend the obtained results to arbitrary gravitational systems is evaluated. A non-conservation of energy in gravitational systems is interpreted by the Extended Space Model (ESM) as the rotation of the energy-momentum vector in 5-dimensional space. A proposed approach to determining pressure as a source of gravity leads to a revision of the dependence of the deceleration parameter of the Universe on the density parameters. Under this condition we examine the ratio between the density parameters of dark energy and cosmological constant depending on the deceleration parameter.

Dmitry Yu. Tsipenyuk, Vladimir B. Belyaev, RENSIT, 11(3):249-260, 2019

Abstract. It is considered (1+4)-dimensional Extended Space Model (ESM) as a generalization of th... more Abstract. It is considered (1+4)-dimensional Extended Space Model (ESM) as a generalization of the special theory of relativity at a 5-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. Within the framework of ESM the photons have a nonzero mass in a gravitational field. We study how a rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The force vector acting on the photon in Schwarzschild space-time is found for the weak gravity and corresponds to photon's gravitational mass equal to the twice mass of a material particle of the same energy. Compliance with the law of conservation of energy as a source of gravity leads to the presence of particles with negative gravitational mass and zero kinematic momentum in the results of the annihilation reaction. Bubble gravitational structures of a type gravastar are considered. Accretion of matter onto compact stars results in their absorption of positive energy from the vacuum and the release negative energy in a free deep space. The particles with negative gravitational mass create there antigravitating vacuum with negative pressure. In the present work a non-conservation of energy in gravitational systems is interpreted by the ESM as the rotation of the energy-momentum vector in 5-dimensional space. The comparison bubble cosmic structures with similar possible bubble structures in microphysical objects is made.

: Dmitry Yu. Tsipenyuk, Vladimir B. Belayev, Rom. Rep. in Phys., 71 (4), 2019

Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensi... more Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensional space, and more specifically at (1 + 4)-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. The possibility of a transition from the components of the 5-momentum of a particle in extended flat space to the components of a 4-momentum in an arbitrary 4-dimensional space by means of rotations is considered. Variational principle of the stationary energy integral of photon allows us to determine its dynamics. We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. We study how (TS)-rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The produced equations are solved for the metrics of Schwarzschild and Goedel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.

D. Yu. Tsipenyuk and W. B. Belayev 2019 J. Phys.: Conf. Ser. 1251 012048, 2019

Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensi... more Extended space model (ESM) is a generalization of the special theory of relativity at a 5-dimensional space, and more specifically at (1 + 4)-dimensional space. Rotations in extended space correspond to the motion of a particle in gravity field in the embedded four-dimensional space-time. The possibility of a transition from the components of the 5-momentum of a particle in extended flat space to the components of a 4-momentum in an arbitrary4-dimensional space by means of rotations is considered. Variational principle of the stationary energy integral of photon allows us to determine its dynamics. We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. We study how (TS)-rotation in ESM agrees with photon dynamics in the Schwarzschild field. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's principle for the stationary gravity field and geodesics principle. Energy and momentum of the particle transferred to the gravity field is defined. The produced equations are solved for the metrics of Schwarzschild and Gödel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.

The existence of negative pressure of vacuum follows from the cosmological models, based on the r... more The existence of negative pressure of vacuum follows from the cosmological models, based on the results of observations. But, is it possible to detect the pressure of the vacuum as per the geometry of the space around the local gravity system? The gravitational defect of mass is interpreted as the transfer of energy to the vacuum, which becomes apparent from its deformation. We determine the gravitational impact of matter on the vacuum and opposite in the sign pressure of it in case of weakly gravitating static centrally symmetric distribution of matter. We have evaluated a possibility to extend the obtained results to arbitrary gravitational systems. The equation of state gives the deceleration parameter of the universe consistent with its constant expansion rate.

Arxiv: 0911.0614, Jul 13, 2018

We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find... more We consider variation of energy of
the light-like particle in the pseudo-Riemann space-time, find
lagrangian, canonical momenta and forces. Equations of the
critical curve are obtained by the nonzero energy integral
variation in accordance with principles of the calculus of
variations in mechanics. This method is compared with the Fermat's
principle for the stationary gravity field and geodesics
principle. Energy and momentum of the particle transferred to the
gravity field is defined. The produced equations are solved for
the metrics of Schwarzschild, FLRW model for the flat space and
Goedel. The gravitation mass of the photon is found in central
gravity field in the Newtonian limit.

A time-like geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory a... more A time-like geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory and astrophysical applications are proposed. For example of such space-time in 4D spherical coordinates with the space-like fifth dimension a transition to the cylindrical frame is implemented. After this transformation a space is divided into two parts that is interpreted physically as pair of the universe and anti-universe. The rotation in 5D space-time exhibits itself in 4D as action of extra force, which reveals itself with observed radial motion of the material particle. Also it is considered similar (3+2)D space-time in which additional dimension is time-like and a motion of space is hyperbolic. For these space-times it is found geodesics with constant radius in 4D spherical coordinates and studied their small deviations in cylindrical frames. It is shown that model with space-like fifth coordinate conforms to the basic properties of the Pioneer-effect, namely, a) constant additional acceleration of apparatus on distance from 20 to 50 a.e., b) its increase from 5 to 20 a.e., c) observed absence of one in motion of planets.

The book is mainly used an algebraic approach to the presentation of the theory of relativity, mu... more The book is mainly used an algebraic approach to the presentation of the theory of relativity, much attention is paid to the variational methods and mechanics of motion of particles in the gravitational field, including the dynamics. The introduction of metrical energy of the light-like particle allows to determine its Lagrangian and use not only the kinematics, but also the dynamics for the analysis of its free movement. The problem of energy and momentum transferred to the particle or system of the gravitational field, for different types of energy and momentum. An analogy with Newtonian gravity and the gravitational mass of the photon is obtained in a weak gravitational field.
Differentiated approach to the conservation laws allows to determine the pressure of gravity field and vacuum. This is done on the basis of solutions of Einstein's field equations for a static spherically symmetric source. The speed of gravity in curved space obtained by using analogies with the gaseous medium.
The bases of the theory of Kaluza-Klein for a five-dimensional space-time, the theory of induced gravity is considered. Models of the rotating hyperspace with space- and time-like fifth coordinate, dynamics of the material particle in them are investigated..
The book is intended for scientists, graduate students, undergraduate students majoring in physics and astronomy. As in the first part contains the basis of the special theory of relativity, it is also of interest for those who begin to study this branch of physics.

Report at IC-MSQUARE 11, Mathematical Modeling in Physical Sciences, September 5-8, 2022, Virtual, on-line Conference, Publ. in: AIP Conf. Proc., 2023

A cloud of relativistic material particles is considered, the gravitational interaction between w... more A cloud of relativistic material particles is considered, the gravitational interaction between which can be neglected.
The gravitational mass of the cloud is determined for the region where it can be considered as a point body. The dependence of
this mass on complete elliptic integral of the 2nd kind on the ratio of the particle velocity to the speed of light is established.