Gravitational Waves, Fields, and Particles in the Frame of (1 + 4)D Extended Space Model

1. Introduction

Since the late nineteenth century, there have been discussions about the issue of integrating electromagnetic and gravitational forces into a single field. These attempts have been made by constructing geometric models of physical interactions and interpretation of physics as geometry in the spaces of a larger number of dimensions. F. Klein [1] developed the Hamilton-Jacobi theory in the late nineteenth century as optics in the space with more dimensions. His thoughts did not, however, evolve at that time. By developing the general theory of relativity (GRT), interest in the issue of geometrization of physics has recently increased [2]. There have been attempts to use gravity as an analogy to define electromagnetic in geometric terms.

Instead of attempting to develop a new model, their authors tried to improve the GRT approach that has already existed. The most well known were the T. Kaluza [3] and O. Klein [4] models. The works of H. Mandel [5] and V. Fock [6] are also remarkable. The fact that they could only use a five-dimensional space is remarkable. The issue of the fifth coordinate’s physical interpretation has not yet been satisfactorily resolved. The development of these approaches was attempted by scientists and researchers, including Einstein [7], de Broglie, Gamow, and Rumer [8]; nevertheless, they were unsuccessful in generating any interesting results. We believe that the reason is because, without including novel physical concepts, their works were formal expansions of previous models. Another area of the geometrization of physical interactions that we should highlight is the theory of gauge fields [9]. According to this paradigm, all interactions: electromagnetism, gravity, and others are viewed geometrically [10].

Afterward, in arrangement to form the hypothesis of basic particles, another approach to combining gravity with other interactions was developed. K.P. Stanyukovich and M.A. Markov suggested attempts to account for the gravitational field in the description of the interaction of elementary particles 50 years ago [11, 12]. They proposed the idea of two types of heave particles-planckions and maximons.

Three essential constants in nature were supposed by the authors: the Planck constant ℏ, lights speed c, gravitational constant G. Above values can be used to build expressions with dimensions of time, length, and mass. They are called Planck time tPl, Planck length lPl_, and Planck mass mPl:

A particle with mass m corresponds to the Compton wavelength in the quantum theory

Particle size can be associated with this wavelength. In the case of the Planck mass mPl substituted in the formula (2), it turns out that the Compton wavelength coincides with the Planck length lPl.

But another linear parameter can be associated with mass m—the Schwarzschild gravitational radius

A spherically symmetric distribution of matter, according to the GRT, collapses into a black hole when it is squeezed to such a size. As a result, it is presently assumed that is the maximum value of an elementary particle’s mass mPl. Such particles were named maximons. Large mass particles might become black holes. The corresponding gravitational radius rgr can be considered as the minimum elementary particles’ possible size. If we substitute the Planck mass mPl in the formula (4), we will take the result

Thus, the gravitational radius of maximon coincides in order of magnitude with Planck length. In Landau’s work [13], estimates for the value of the “radius” of elementary particles were obtained, based on the limit of applicability of electrodynamics representations in quantum mechanics. Interestingly, the “radius” of the electron at the same time was equal to zero. Such relations were discussed in an attempt to take into account the gravitational forces in the processes of interaction of elementary particles. This approach assumes the initial existence of particles with a large rest mass, and since we do not observe such objects, it is not clear how it can be used to describe the processes occurring in the laboratory.

The name extended space model (ESM) is due to the fact that it is formulated in a flat five-dimensional space G(1,4) with the metric (+− − − −). ESM approach [14, 15, 16, 17] is fundamentally different from all these and similar theories. ESM is based on the physical hypothesis that mass (rest mass) and its conjugation (interval) are dynamic variables, and its values are determined by the field-particle interactions. So, ESM is a direct SRT generalization. Interval and rest mass are invariants but can be changed in ESM. In particular, photon mass can be positive and negative. This mass can vary because of electromagnetic interactions and generates gravity interactions. This situation allows us to consider gravity and electromagnetism as unit fields. The equations for the plane-wave potentials describing the process of its localization are presented. The exact solution of these equations is found.

In GRT, the definition of the momenta of material and light particles moving in curvilinear space-time, and the forces acting on them, aims to find relativistic corrections to Newton’s theory of gravitation for a weak gravitational field. In [18, 19, 20], the second derivatives of the coordinates along the path are considered as components of the 4-vector of the force acting on a material particle of a unit mass. However, in the gravitational field, not only the 4-momentum of matter alone, but the 4-momentum of matter together with the gravitational field should be preserved [18]. In the equation of particle motion containing force in this form, there is no energy and momentum transferred to the gravitational field.

Another approach is the choice of the Lagrangian of the particle, the definition of generalized forces as its partial derivatives with respect to the coordinate in accordance with Lagrange mechanics [21, 22, 23, 24]. In GRT, the physical velocities of particles are associated with the components of the contravariant 4-velocity vector. Therefore, the physical force is aligned with the upper index vector associated with the generalized force vector. The energy and momenta of particles are considered to be the components of the contravariant 4-vector of energy-momentum, as is done in [18] for a particle moving in the Minkovsky space-time. The equations of motion will contain an additional term, which express the rate of change of the energy and momentum acquired by the gravitational field when a particle moves in it.

In the Fock proof [25] of the light motion along geodesics, the time component of the covariant 4-velocity vector is taken as the Hamiltonian. Application of the variational principle of the energy stationary integral to the motion of a light-like [21, 22, 23, 24] particle in a gravitational field does not lead to a violation of the isotropy of the light path. In the generalized Fermat’s principle [10], a variation of the integral of the time component of the 4-velocity vector is used and gives the trajectory of light movement that coincides with the geodesic.

For weak gravity, the analogy of the particle dynamics in Schwarzschild space-time with Newtonian gravitational theory permits to determine the passive gravitational mass of a photon. It is equal to twice the material particle mass of the same energy corresponded to non-gravitational interactions. This agrees with the results of Tolman, Ehrenfest, and Podolsky for the photon effective gravitational mass in the interaction between light packets or beams and matter particles [26, 27]. Observance of conservation of energy as a gravitation source suggests that at annihilation of an electron and positron in addition to gamma quanta, the particles g− are released [22, 23, 24]. The birth of gamma-ray electron-positron pair leads to the appearance of a particle, which is identified [28] as a graviton.

2. ESM short description

According to the ESM, several physical values that are assumed to be constant under the conventional method are really not constant and can now change depending on the circumstances. We are discussing the zero mass of the photon as well as the rest mass of heavy particles. A model similar to the ESM model was developed by Wesson and his coauthors [29, 30, 31]. Wesson proposed to use “mass” as the fifth coordinate, in addition to the time and three spatial coordinates: “we … view mass as on the same footing as time and space…” [20] and “This means that the role of the 4D uncharged mass is played in 5D geometry by the extra coordinate.” We find this approach to be irrational. In this context, it poses a difficulty to generalize the energy-momentum-mass tensor in four-dimensional space to the tensor in five dimensions. The fifth coordinate, mass, can be utilized, but not in the coordinate space. The mass of the particle should be viewed as an additional value to the energy and other three momentum components. The fifth coordinate in coordinate space should not have a value related to the mass. It was hard to draw connections with actual experiments as a result of the assignment of mass as a fifth coordinate in addition to time and space.

The physical meaning of the fifth coordinate is action. This value is constant under the usual Lorentz transformations in M, but it changes when the transformations in the extended space G(T, X, Y, Z, S) are used. From a physical point of view, our expansion means that processes in which the rest mass of the particles changes are acceptable now. Lorentz transformations in the 4D Minkowski space M(T; X, Y, Z) in the planes (T, X), (T, Y), (T, Z) allow changing the energy and momentum of a particle in the conjugate space of the expanded 4D energy-momentum space M∗(E;Px, Py; Pz). In the ESM, gravity and electromagnetism are combined in one field, and it is possible to construct a 5×5 energy-momentum-mass tensor. Recently, Overduin and Henry [32] proposed the same idea of considering the fifth coordinate.

In ESM, the motion of a particle in a 5-dimensional cone

is considered. The parameters t,x,y,z,s are coordinates of a point in extended space G(1,4). The Minkowski space M(1,3) enters it as a subspace. In the extended space G(1,4) the particle 4-vector of energy and momentum is padded to a five-dimensional vector to a 5-vector

where E is energy, and px,py,pz are momenta.

For a free particle, the components of this vector satisfy

The interval s in Minkowski space serves as the fifth coordinate in the extended space G(1,4). The variations of mass m correspond to variations of the interval s. Let us now explain the fifth coordinate physical meaning in the ESM. For this purpose, we use the expression for the action S of a free particle [20, 21].

Here L is the Lagrangian of a particle. The integral is taken along the world line between two given events—the position of the particle at the beginning and end points of the Minkowski space.

Since the mass of the photon in STR is m=0, massless fields in the extended space G(1,4) are mapped to a 5-vector

We assume that a photon corresponds to a plane wave, which moves in M(1,3) with a speed c in the direction given by the vector k→. We want to consider a broader class of processes that can change mass. The energy-momentum-mass 5-vector (11) characterizes a particle for which all the parameters, energy, momentum, and mass are variables. The corresponding changes of these values can be described using transformations of the extended space G(1,4), given by the hyperbolic rotations on an angle ϕTS in the plane (TS)

and in the plane (XS)

The extended space G(1,4) can be viewed as a set of Minkowski spaces with the parameter n, which we conditionally refer to as the refractive index. We made this parameterization option because the physical meaning of our model depends on the photon’s movement and its velocity variations. We believe that the refractive index n of any subspace M(1,3) of space G(1,4) defines these subspaces. From the point of view of ESM, the transition from a medium with one refractive index n1 to a medium with another refractive index n2 can be interpreted as the movement along the fifth coordinate of the expanded space. In contrary to the usual relativistic mechanics, we now suppose that the mass m is also a variable, and it can vary at motion of a particle on the cone (6). In the ESM model when a particle enters an area of space with a non-zero density of matter or field, its mass changes. In such areas, the speed of light is reduced, and these areas can be characterized by the refractive index of a medium n. This parameter relates the speeds of light in the vacuum and in the medium, which is v=c/n. For example, the refractive index of a gravitational field that is described by the Schwarzchild solution [33] reads

where r is a distance from gravity center. The values of the components of photon 5-vector (7) corresponding to this field are found using (TS) rotation (11) and will be transformed as follows:

Electromagnetic and gravitational fields combined in a single gravitational-electromagnetic field. This is because in the ESM, electromagnetic interaction causes the formation and changing of fields and particle mass. Since mass is assumed to be variable in the ESM from the beginning, it is possible to describe new processes that the STR framework is unable to explain properly. This arises from the fact that in ESM, gravitational interaction of elementary particles occurs naturally.

3. Plane wave localization and mass appearance in the ESM

In the extended space G(1,4), the potentials of the field combining electromagnetism and gravity are determined by the equation [15]:

with

and the charge of a particle e.

On the right-hand side of the equations in system (15) are the components of the five-dimensional current vector ρ¯. This vector is a generalization of the four-dimensional current vector ρ∼, which in traditional four-dimensional electrodynamics is

where e is the charge of a particle.

In order to get ρ¯, we assign an additional component to the vector (16):

This is an isotropic vector: ρ¯2=0. We will consider an additional component as an analog of the momentum of a charged sphere moving at low speed [34] in an extra dimension

where ε0 is the vacuum permittivity, a is the sphere radius, μ is an arbitrary affine parameter along the path, and κ is a constant having the dimension of the ratio of charge to mass. Assuming that e0 and a=re are the charge and radius of the electron and the constant κ is equal to the ratio of its charge to mass: κ=e0/me, we get

The electromagnetic field is defined by the system of the first four Eq. (15), while the fifth equation describes the scalar gravitational field. A single electrogravity field results from the combination of these two fields if their values depend on the variable s. For such fields, Lienard-Wichert potentials were found in [35]. When the variables included in potentials have no dependence on the variable s, it splits into two independent subsystems.

From electrostatics theory [36], the potential energy of a uniformly charged sphere is given by

When the fifth equation describes the gravitational field created by the electrons charge, their gravity density js, Eq. (19), will not be equivalent to, their potential energy and will depend on the speed of movement in the extra dimension ds/dμ. Accordingly, the radius of an electron may differ from its classical radius recl=2.8•10−15m, determined from Eq. (20). This is confirmed by observation of a single electron in a Penning trap [37], which suggests the upper limit of the particle’s radius to be 10−22m.

Now let us look how a charged particle and a plane electromagnetic wave interact. We take it as given that a plane wave is an object in empty space that fills this infinite space. The following equations are used to find the field potentials without sources in the extended space G(1,4):

Let us consider the equation

We are looking for its solutions in the form

We assume that the function usxyz varies slowly over the variable s, compared with the variables x,y,z, so that the second derivative ∂2/∂s2u can be neglected. Now we get the equation

The neglect of the second derivative and presentation of Eq. (22) in the form (24) are similar to the searching a solution for the optical wave propagating shape in a laser along the z-axis [38]. Eq. (24) solution has the form of a three-dimensional Gaussian beam

Here w0 is the radius of the “neck” of the beam, that is, its minimum width at the point s=0. The value of w2=w02[1+2s/kw022 is the diameter of the beam at the point z. The radius of curvature of the beam wavefront is Rz=z1+kw02/2s2.

For s→∞, the radius and the beam width also tend to infinity. Solution (25) corresponds to a plane wave. If s→0, the plane wave is localized in a volume that looks like a ball with radius r=w0. Process of localization takes place without changing the energy. It is described by orthogonal rotations in the planes (SX), (SY), (SZ). The square of the wave modulus (25) reads as

As we can see, as the wave’s localization (25) diminishes, it grows and achieves its maximum value at s=0.

This degree of localization is not, however, really reached. The presence of a charged particle in space causes the localization of a plane wave. The charged particle effects at the wave field, but the field also has an impact on the particle. Since a charged particle’s mass distribution is defined by δ function, we suppose that it is concentrated at a single point in empty space (refractive indexn=1), and that this delta function is the free particle wave function. The-function starts to change into a Gaussian function as soon as such a charged particle reaches the plane-wave field.

We see that as s decreases, the localization of the wave (25) increases and reaches its maximum value at s=0. However, this degree of localization is not really achieved. The fact is that the process of localization of a plane wave is generated by the presence of a charged particle in space. The wave field is affected by the charged particle, but the particle itself is affected by the field. We assume that in empty space (refractive index n=1), a charged particle is concentrated at a point, that is, its mass distribution described by δ is a function, and we consider this delta function as the free particle wave function. When such a charged particle enters the plane-wave field, the δ-function begins to transform into a Gaussian function

This expression is structurally similar to the solution of the differential heat equation describing the temperature distribution. For an infinite body with an instantaneous point source at the origin, the temperature distribution has the following form:

Here T is the temperature at time t in coordinates x,y,z; Q is the heat emitted at the time t=0 at the origin; t is the time elapsed since the introduction of heat; a is the thermal diffusivity, ρ is the density of the body, and ς is its specific heat. Eq. (28) is the fundamental solution of the heat equation under the action of an instantaneous point source in an infinite body.

The localization of a plane wave and the “swelling” of a massive particle are both consistent processes. Our assumption is that their localizations must match in order for the field and particle to interact. We get an expression for the value s0that defines the lowest value of the plane-wave localization and the maximum value of the point particle swelling by comparing Eqs. (26) and (27):

This imposes restrictions on the dynamics of interaction of the field with the particle.

4. Equations of Lagrange mechanics

In GRT, a four-dimensional pseudo-Riemannian space-time with coordinates xi and metric coefficients gij is considered, the interval in which is written in the form

The 4-velocity vector of the particle is denoted as ui=dxi/dμ, where μ is the variable parameter. We obtain the equations of its dynamics in general form.

The particle Lagrangian corresponds to the covariant generalized momenta

and generalized forces

The particle motion is determined by Hamilton’s principle of stationary action δS=0 at

where μ0, μ1 are the values of the parameter at the points that are connected by the desired trajectory of motion. The extremum condition leads to the Euler-Lagrange equations

With generalized momenta (2.2) and forces (2.3), these equations are rewritten in the form

The Lagrangian is chosen [21] so that contravariant momenta bind to the physical energy and momentum of the particle

and the gravitational force acting on it is mapped to associated with (32) vector

Passing to them in Eq. (35), we find

Multiplying these equations by gkλ and summing over the twice occurring index λ, we obtain.

The presence of the second term on the right side reflects that in the gravitational field not only the 4-momentum of matter, but the 4-momentum of matter together with the gravitational field is stored [18]. Its components express the rate of change of the energy and momentum acquired by the gravitational field when a particle moves in it

Integration of this quantity over gives the energy and momentum received by the gravitational field at a certain interval of its trajectory. As a result, Eq. (39) can be written in the form

It follows from the laws of conservation of energy and momentum that the force acting on a particle is equal in magnitude and opposite in sign to the force acting by the source of gravity from the side of the particle. This is equivalent to fulfilling Newton’s third law.

5. Variational principle of the energy stationary integral for the photon motion

To determine the dynamics of a photon in a gravitational field, we will use principle of the energy stationary integral [21, 22, 23, 24]. Interval in pseudo-Riemannian space-time with metric coefficients g˜ij:

after substitutions g˜11=ρ2g11, g˜1p=ρg1p, g˜pq=gpq at p,q=2,3,4 is rewritten as

The condition ds=0 corresponds to the motion of light. With g11≠0, the variable ρ is given by the expression

where σ takes the values ±1 and 4-velocities ui are determined provided that μ is an affine parameter. Further, we will consider variations near ρ=1, to which the equality g˜ij=gij corresponds. If g11=0 and condition g1p≠0 is satisfied for at least one ρ, then it turns out

where k takes on the values (2)–(4).

The Lagrangian of a freely moving particle is chosen as

For both values (44), (45), the covariant generalized momenta (31) and forces (32) take the form

The chosen Lagrangian corresponds to the ratio

being the integral of motion [39] and, accordingly, ρ will be the energy of the system combining the light-like particle and the gravitational field given by the metric (30).

The equations of motion are found from Hamilton’s principle of stationary action (33), which, in view of (46), can be written in the form

The energy ρ is non-zero, its variations leave the interval light-like. The equations of motion will be Euler-Lagrange Eq. (35). The principle of the energy stationary integral for the photon motion is consistent [23, 24] with the generalized Fermat’s principle [40], and the resulting curves are null geodesics.

The contravariant vector of generalized momenta is written as

Physical energy and momenta of photon with frequency ν in Minkowski space-time with affine parameter μ=ct form contravariant 4-vector of momenta πi=ℏν/cui. For arbitrary affine parameter, it is rewritten as

And in pseudo-Riemannian space-time, similar energy and momenta of the photon will be put in line with the components of the contravariant vector of momenta. The photon frequency in coordinate frame is given by

where ν0 is a certain fixed value of its frequency. Comparing Eqs. (51) and (52) we obtain

The Lagrangian (46) corresponds to a particle with unit energy. For a photon, it is

and the gravitational forces acting on photon

are assigned to the components of the associated vector of generalized forces

6. Photon dynamics in the Schwarzschild field

Let us consider the dynamics of a light-like particle in a static centrally symmetric gravitational field described in spherical coordinates trθϕby the Schwarzschild metric

In plane θ=π/2, the energy-momentum vector of a photon (54) moving along an open trajectory [21, 22, 23, 24] is as follows:

where С is constant, and ν0 is a photon frequency away from the center of gravity. In radial motion, the components of this vector coincide with the first components of the vector (14) obtained by (TS)-rotation according to the ESM.

The only non-zero component of the associated vector of generalized forces (57) is

With radial motion (С=0), it is equal to

coinciding with the doubled force acting on the particle in Newtonian gravity. In view of (56), it corresponds to the passive gravitational mass of the photon

This result is consistent with a thought experiment on “weighing” a photon [41], in which it performs periodic motion in the vertical direction between two horizontal reflecting surfaces.

Considering the non-radial motion, in order to avoid the appearance of a fictitious component of momenta and force due to the sphericity of the coordinate system, we use the Schwarzschild metric in rectangular coordinates. It can be accessed using the transformation

of metric (58), which yields

The motion in the plane z=0 is studied and the force acting on a light-like particle at a point сtx00.

The single non-zero component of the force vector [21, 22, 23, 24] is

which taking into account transformation (63) can be rewritten as

The generalized force acting on a photon does not depend on the direction of its motion. This expression differs from the formula (61) corresponding to radial motion in spherical coordinates, which is a consequence of the non-covariance of the vector Fl. However, in the limit of weak gravity (r≫rg), these expressions converge asymptotically and give Newton’s law of gravitation with a passive gravitational mass of a photon (62) equal to twice the mass of a material particle of equivalent energy.

The gravitational field of the electromagnetic radiation flux is determined from the solution of the Einstein equations

with Ricci tensor Rij and χ=8πGc4 for the electromagnetic field energy-momentum tensor

where Fij is the electromagnetic field tensor. In case of weak gravity, it follows from analysis of acceleration of material particle that active gravitational mass of light beam or light packet is twice as much as similar mass of a rod of equivalent energy [26, 27, 42]. The equality of the active and passive gravitational masses of a photon means the fulfillment of Newton’s third law in the gravitational interaction of light and material particles and the laws of conservation of energy and momentum.

7. Electron positron pair production, gravitons

The expansion of the Birkhoff theorem to a sphere with equally distributed electrons and positrons provided the justification for applying the energy conservation law to a gravitational field source at the annihilation reaction [26]. This was done under the assumption that the gravitational mass of the sphere would not change because of electron-positron annihilation before the particles left it. Energy conservation law for the source of the gravitational field applies to any pair of annihilating particles if this condition is fulfilled for the entire sphere. Furthermore, due to the double gravitational mass of the photon compared with the total mass of the electron and positron 2me−, the annihilation reaction results in the appearance of particles g− with a negative gravitational mass

dissipating the negative energy as a gravitational source.

Annihilation process in this case looks as follows.

There is no energy for particles g− specified by non-gravitational interactions. This is because the total electron and positron energy is equal to the gamma quantum energy produced. These particles have no kinetic momentum. As a result, it is impossible to detect them using conventional particle registration techniques (such as a bubble chamber). However, when a ray of light passes through the negative energy region, a focusing effect can occur, as opposed to focusing by gravitational lensing [43].

Conditions for the formation of electron-positron pairs are produced by high-energy gamma radiation interactions (>1022 MeV) with matter. A “gravitational charge” particle, designated g+, appears as a result of the inverse annihilation reaction

Appearing in addition to the electron and positron particles are opposite to g− in gravitational mass. It occurs with extracting pairs g+,g− from a vacuum. Particle g− is immediately absorbed, producing an electron and a positron, leaving g+ with positive gravitational mass. Particles g− are immediately absorbed, producing an electron and a positron and leaving g+ with a positive gravitational mass.

We consider a model in which the boson-like particles g− and g+ have a rest mass of 0. These particles are assumed to have a spin of 2 and an electric charge of 0. The graviton, a hypothetical quantum of gravitational radiation, holds the property of particle g+ [28]. Fermi Gamma-ray Space Telescope [44] detects photons with an energy, sufficient for reaction (72), in pulsar jets, such as Cygnus X-3 [45], gamma-ray bursts from blazars [46], and cosmic ray generation in supernova remnants [47].

Gravitational mass of body is less than the sum of individual gravitational masses its constituent elements [48]. The gravastar [49] with negative mass component with half the mass two times less than the mass obtained by integrating the spatial volume is considered in [17]. In this paper, the gravitational mass defect is inspected as a result of the negative binding energy presence. This case reflects to the ratio between the electron mass and gravitational mass of the particle g− released during annihilation. Such condition corresponds to the negative internal pressure and positive pressure on the shell. This model is a gravastar with inside approaching a de Sitter interior vacuum with constant density and pressure, having a singularity near the shell [49].

8. Gravastar with constant pressure

The general static, spherically symmetric line element in Schwarzschild coordinates is

with metric functions fr and hr. The stress-energy tensor of a static, spherically symmetric distribution of matter with density q and isotropic pressure p is described (in units in which c=1) by the diagonal matrix

Metric functions are sought from the Einstein Eq. (68), which reduced [49] to

These equations have a solution [50]:

with constant density and pressure obeying the relation

Metric functions f and h must match the exterior Schwarzschild solution (58) in vacuum

where R is radius of matter distribution. Thus, we have the boundary conditions

and since function f is constant inside the sphere (78), it will have the value

The space-time inside the sphere is the following:

Equation of state (79) characterizes the vacuum pressure that balances the pressure of the gravitational field in a uniform sphere [21, 50, 51] or, in case of electromagnetic field, pressure of it [27]. This gravastar model will have the same dependence between its proper and negative mass components on radius and density as the gravastar with a de Sitter interior vacuum [28, 49].

9. Conclusions

The generalization of Einstein’s special theory of relativity on 5-dimentional space is considered, in which as fifth coordinate additional coordinate is identified with the interval of a particle. We obtained a 5D generalization of the 4D current vector under the assumption that the mass of an electron is equivalent to a component of its momentum dependent on its velocity in an additional dimension. As follows from this approach, the radius of an electron can be substantially less than its classical radius.

A point charged particle’s field is entered by a plane electromagnetic wave. It was found that such a wave can be described by potentials in a five-dimensional extended space. It is considered on how such a wave could interact with a charged particle. We can calculate the field strengths and determine their energy-momentum-mass tensor using the explicit form of these potentials.

The dynamics of particles in curvilinear space-time is considered using Lagrange mechanics. A correspondence is established between the physical energy and momentum of a particle, determined from non-gravitational interactions, and the contravariant vector of generalized momenta. The obtained dynamic equations include the rate of change of the energy-momentum vector, the components of which express the energy and momentum acquired by the gravitational field when a particle moves in it. This vector is an analog of the pseudotensor used in conservation laws in tensor form when considering the dynamics of an individual particle.

By choosing the Lagrangian of a photon corresponding to the principle of the energy stationary integral, a vector of forces acting on it in the Schwarzschild field is obtained. Although these generalized forces are not covariant quantities, in the limit of weak gravity, they express the Newtonian law of gravity with a passive mass of particles corresponding to the active gravitational mass of moving point bodies and a light beam. The passive gravitational mass of a photon does not depend on the direction of its motion. Coinciding with its active gravitational mass when interacting with a material particle, it is equal to twice the mass of a material particle having an energy equivalent to a photon. When a particle moves in a gravitational field, the non-covariance of the generalized forces vector and the vector composed of the rates of energy and momentum transfer to it has the same nature as the non-covariance of the gravitational field energy-momentum pseudo-tensor, with the help of which experimentally confirmed changes in the circular orbits of two bodies moving around a common center were calculated as a result of energy loss caused by the radiation of gravitational waves.

With twice the gravitational photon mass compared with a material particle of equivalent energy, the application of Birkhoff’s theorem to a sphere full of annihilating electrons and positrons leads to the following conjecture: during electron-positron annihilation, particles g− with a negative gravitational mass are released in addition to gamma quanta. The opposite “gravitational charge” particle g+ will appear as a result of the photon conversion event into an electron-positron pair. With a spin of 2 and no electric charge, these hypothetical particles are categorized as bosons. They are identified as gravitons, being the localization of elementary gravitational waves. Gamma-ray blazar outbursts, the jets of pulsars like Cygnus X-3, and cosmic ray emissions in supernova remnants contain photons with enough energy for their appearance.

We proposed a gravastar model with constant pressure and no singularity. The gravitational mass defect is explained by the presence of negative binding energy. This model allows for a relation between the gravitational mass and the negative component of mass corresponding to the relation between the mass of electron and particle g−. The equation of state in the inner region is such that the pressure of the vacuum balances the pressure of the gravitational or electromagnetic fields.