GRAVITATION IN UNIFIED SCALAR FIELD THEORY

GRAVITATION IN UNIFIED SCALAR FIELD THEORY Alexander A. Chernitskii 1,†,2,∗ 0000-0003-1966-1338 1 Department of Mathematics, St. Petersburg State Chemical Pharmaceutical University; alexander.chernitskii@pharminnotech.com 2 A. Friedmann Laboratory for Theoretical Physics; chernitskii@friedmannlab.org * Correspondence: chernitskii@friedmannlab.org † Current address: Prof. Popov str. 14, St. Petersburg, 197022, Russia Version December 9, 2020 submitted to Universe Abstract: The scalar field of space-time film is considered as unified fundamental field. The field 1 model under consideration is the space-time generalization of the model for a two-dimensional 2 thin film. The force and metrical interactions between solitons are considered. These interactions 3 correspond to the electromagnetic and gravitational interactions respectively. The metrical interaction 4 and its correspondence to the gravitational one are considered in detail. The practical applications of 5 this approach are discussed. 6

All these attempts were not completely successful. In particular, within the framework of these 30 theories, no solution has been obtained that corresponds with a sufficient degree of realism to any 31 elementary particle. It should be noted that this problem is extremely difficult mathematically. 32 1.3. The question as to the tensor rank of the field 33 It is evident that the question as to the tensor rank of an unified field is highly important. 34 In the above-mentioned attempts to construct an unified field theory, the choice of the tensor 35 rank of the field was based on certain correspondence considerations. 36 For example, the nonlinear electrodynamics models generalized the linear electrodynamics which 37 was successful in certain limits. 38 The variants of Einstain's unified theories generalized his own gravitational theory which was 39 successful in certain limits. 40 Heisenberg's theory of an unified spinor field was based on Dirac's linear theory which was 41 successful in certain limits. In this case, the electromagnetic interaction of particles-solitons is a consequence of the integral 49 conservation law for energy-momentum. This type of interaction is called the force one. 50 The description of gravity in the framework of nonlinear electrodynamics is based on the effect 51 of induced space-time curvature by a weak field of distant particles-solitons at the location of the test 52 particle. This type of interaction is called the metrical one.  Thus we can consider a simpler model of a scalar unified field, since it also allows us to describe 57 electromagnetism and gravity [7].

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In particular, the tensor character of the electromagnetic field is due to the determination of the 59 integral force through the integral over a closed surface surrounding the test particle under the force 60 interaction [8].

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It should be emphasized that the electromagnetic field of antisymmetric tensor of the second 62 rank appears at a certain point when a test particle is placed there. In the absence of such a particle, 63 only a configuration of the scalar field is present near this point, but it generates the electromagnetic 64 interaction with any charged particle. We consider the following generally covariant world volume action and the appropriate variational principle [9]: where M det(M µν ), (dx) 4 dx 0 dx 1 dx 2 dx 3 , V is space-time volume, dV |m| (dx) 4 is four-dimensional volume element, m det m µν , Customary method gives the following canonical energy-momentum density tensor of the model in Cartesian coordinates where − m µν is the constant diagonal metrical tensor for flat space-time with signature {+, −, −, −} or 75 {−, +, +, +}. As we see, the canonical tensor is symmetrical.

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To obtain finite integral characteristics of solutions in infinite space-time we introduce the regularized energy-momentum density tensor with the following formula: where ∞ → T µν is a regularizing symmetrical energy-momentum density tensor. Here we will use the constant regularizing tensor In general case we can take a symmetrical tensor satisfying the differential conservation law as This field equation can be written in the following remarkable form [7]: where − → T µν is the canonical energy-momentum density tensor (2). Here we introduce the effective 81 metric ∼ m µν which will be considered below.
Equation (6) transforms to ordinary linear wave equation with χ = 0: 2.4. Effective metric and curved space-time 83 The characteristic equation of the model has the following form which is obtained directly from (6a):  When we move away from the localization region of a soliton-particle, the field of space-time film 120 satisfies approximately the linear wave equation (7).

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Elementary oscillating solutions of the linear wave equation in the spherical coordinate system decrease in amplitude as r −1 . Let us take the following simple solution as example: where the point under symbols denotes the belonging to the intrinsic coordinate system.

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According to the definition given in the previous subsection, this solution is a weak soliton.

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The asymptotic form such as we have in (9a) leads to the divergence of energy integral at infinity.

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However, the investigation of toroidal configurations [10] shows that there can be solitons with 125 oscillating part without the asymptotic form of type (9a) whose energy is finite. But we can suppose 126 in this case that the wave mode of type (9a) will appear for interacting solitons.

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Thus we assume that a soliton-particle in the intrinsic coordinate system has both static and oscillating parts. The oscillating part is a standing wave having perhaps sufficiently complicated configuration. Using a space-time rotation we can obtain a moving soliton from the rest one. In this case the standing wave transforms to moving one. The moving soliton is obtained with the help of the following substitution for the intrinsic coordinates of the soliton: transformations, {x ν } is the coordinate system in which the soliton is moving. 129 We have the following dispersion relation for the wave four-vector components of this moving wave: where · ω is the angular frequency of the standing wave in intrinsic coordinate system of the soliton, Here we consider the weak soliton with a constant amplitude • a as some approximation for a 145 soliton-particle whose amplitude is defined by an exact solution. If there is an additional field of 146 remote solitonsΦ, the soliton solution is modified, and its amplitude could also be changed. However,

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it is natural to assume that the soliton-particle has a maximum amplitude which significantly greater 148 than the field of distant solitons-particles. Then the weak field of distant particles will not significantly 149 affect the amplitude of the considered soliton-particle. Thus if we consider the weak soliton instead of 150 the soliton-particle, we actually investigate the influence of the distant solitons fieldΦ to the part of 151 soliton-particle which is sufficiently far from its center and has a small amplitude. The movement of 152 this weak part should direct the entire soliton-particle, since it is a modified exact solution. 153 We substitute this sum (11) in the equation of space-time film (6). Here we suppose an averaging of the effective metric ∼ m µν (6b) with the background fieldΦ over a defined space-time localization region for the weak soliton • Φ. Then we obtain the following equation: where the averaging effective metric∼ m µν depends on the derivatives of the background field:∼ m µν = 154∼ m µν (Φ). Here below we use the designation ∼ m µν for the averaging effective metric∼ m µν in an effort to 155 simplify the designations. 156 We consider that the averaging background effective metric ∼ m µν is almost constant in the 157 space-time localization region of the weak soliton • Φ. In this area, let us find a coordinate transformation 158 which reduces the equation (12) to ordinary wave one (7).

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Let we have the following relations for the coordinate differentials: where the matrixX µ .ν andX µ .ν are mutually inverse and satisfy the relations Then the solutions of equation (12) in the limited area under consideration has the form where • Φ · x σ is a solution of the linear wave equation (7), in particular, the weak soliton (9a). Here the 160 functions · x σ (x δ ) are defined by the transformation (13).

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The averaged background effective metric ∼ m µν calculated in an expanded four-dimensional space 162 defines a Riemann space that is not generally flat. Also, in general, it is not possible to find the 163 coordinate transformation · x σ = · x σ (x δ ) satisfying (13) and (14) everywhere.
where · x j (x 0 ) is a position of energy center for the weak soliton at the time x 0 ,k is called a normalized wave vector such that Taking into account the transformation (16) we obtain that · ω · x 0 is a phase of the wave • Φ. Then we have the following its wave vector components: It should be emphasized that the substitution of type (16b) for · x 0 to the solution (9a) is the known  Using (16b) and (14), we have the following dispersion relation for the weak soliton:

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Now let us obtain a trajectory of energy center for the weak soliton · x j = · x j (x 0 ). We use the intrinsic time · x 0 (x µ ) (16b) at the point · x µ as a parameter of movement s. Taking also · x 0 = x 0 we have where · u µ is the four velocity. We have the last relation in (19b) in accordance to (13) and (19a).

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The definitions · u µ (19b) andk ν (16b) with (14) gives also the following relations at the point {· x µ }: Let us introduce the inverse tensor∼ m µν to the tensor ∼ m µν : We have from (20) and (21) the following relation at the point {· x µ }: This leads to the well-known expression in general relativity theory: We have also from (20) the following relation at the point · x μ One must note that although · u µ andk µ are defined as components of second-order matrix, they 171 are actually four-vectors.

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Indeed, if we consider the problem in another coordinate system {x µ } then we have This means that the intrinsic time · x 0 is not changed for such transformations or it behaves as scalar.

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Thusk µ is a four-vector according to the definition (16b).

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According to (16a), the three-dimensional coordinate system { · x i } is moving coupled with the localization region of the weak soliton. Thus we have · x i = 0 and d · Therefore · u µ are a four-vector according to definition (19b).

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Now let us obtain the trajectory equation for · x µ (s). Differentiation of dispersion relation (18) with respect to certain coordinate x ρ with consideration of relations (16c) and (20) gives the following equation at the point · x µ : Substituting (24) into (27) and using (21) we obtain the trajectory equation in the following form: As can be seen, this equation is the geodesic line one for the introduced effective Riemann space 176 with the metric ∼ m µν .

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Thus a weak soliton with rest frequency and small constant amplitude under the influence of 178 distant solitons behaves as massive particle in gravitational field. 179 We can assume that the obtained equation (28) describes also the movement of a soliton-particle 180 under the influence of distant solitons-particles.
where ∼ ϕ is the scalar potential of the gravitational field,Ẽ is an averaging energy density for the field 183 of distant solitons. Here the averaging is performed over a space-time volume including a localization 184 region of the soliton and a relevant time interval 185 In order to have the real gravitation in our consideration we must obtain the following expression for the gravitational potential: where ∼ γ is the gravitational constant,m is a mass for agglomeration of distant solitons-particles, R is a 186 distance from the energy center of this agglomeration.

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Thus we must obtain the following asymptotic form for the averaging energy density as R → ∞: To obtain the asymptotic r −1 for the potential ∼ ϕ we must also take into account a wave background 196 with almost constant amplitude in space. This wave background must undoubtedly exist in the space 197 where there is a bulk of oscillating solitons-particles.

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Then we must represent the fieldΦ generating the effective metric in the form of the following sum: whereΦ • is the field of distant solitons containing the fast oscillating part with amplitude decreasing as 199 r −1 (just as in the weak soliton (9a)),Φ ∼ is the wave background field with almost constant amplitude.

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The substitution of the field (32) to the energy density gives the terms which are proportional to r −1 .

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The following averaging of energy density can give the appropriate term in expression for the potential 202 ∼ ϕ which is proportional to r −1 . This expectation seems reasonable because the interaction between the 203 fieldsΦ • andΦ ∼ caused by nonlinearity of the model can give a defined phase synchronization for 204 these fast oscillating fields.

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Thus the gravitational constant is defined by two factors We assume that the amplitude of the wave background (the factor∼ γ in (33)) may vary slightly in 211 space. In this case we must assume that the gravitational constant is not really constant but it can also We have seen that in this approach, the existence of the long-range gravitational potential ∼ r −1 221 is is associated with the presence of the wave background.

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Then if the wave background is cut off or weakened with the help of some method, then the 223 gravitational interaction will also be weakened. In this case we can talk about gravitational screening.

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Estimations of possible frequency of the wave background show [6] that the gravitational 225 screening requires a metamaterial from which a gamma-ray mirror can be made.