The concept of the general force vector field

Transcription

1 OALib Journal, Vol. 3, P (16). The onept of the general fore vetor field Sergey G. Fedosin PO box 61488, Sviazeva str. -79, Perm, Russia A hypothesis is suggested that the fields assoiated with marosopi bodies, suh as lassial eletromagneti and gravitational fields, aeleration field, pressure field, dissipation field, strong interation field and weak interation field, are the manifestations of a single general field. Using the generalized fourveloity as the four-potential of the general field, with the help of the priniple of least ation it is shown that eah of these seven fields ontributes linearly to the formation of the total four-fore density. The general field equations, equation of the partiles motion in this field, equation for the metri and the system s energy are determined. It should be noted that the stress-energy tensor of the general field inludes not only the stressenergy tensors of these seven fields, but also the ross terms with the produts of various field strengths. As a result, the energy and momentum of the system with several fields an differ from the lassial values, not taking into aount suh ross terms in the general field energy and momentum. Keywords: general field, generalized veloity, aeleration field, pressure field, dissipation field. PACS Nos.: 3.5.x, 1.1. g. 1. Introdution Many sientists believe that there is generality between the fields known in physis whih is not fully understood so far. So in the Grand unified theory in order to desribe elementary partiles in unified quantumfield formalism an attempt is made to ombine strong, weak and eletromagneti interations [1]. In the Theory of everything gravitational interations are also taken into aount. However, so far there is inompatibility between the general theory of relativity, desribing gravitation at the marosopi level, and the quantum field theory, desribing interation of partiles at the mirosopi level []. One of the well-known models of unifiation of gravitation and eletromagnetism is the Kaluza-Klein theory [3-4]. This theory uses five-dimensional spaetime and some salar field, and the theory s onsequenes are the equations equivalent to Maxwell equations and the equations of general theory of relativity. Besides fundamental interations, there are other fields that influene diretly the matter partiles and transfer energy and momentum. These fields inlude the aeleration field and pressure field [5], as well as the field of energy dissipation due to visosity [6]. Under the influene of these fields almost uniform spatial and temporal distribution of veloities, pressure, energy dissipation, potentials and field strengths takes plae in bodies, whih arises from the wave equation of a standard form. Similarity of distribution of physial funtions indiates a single mehanism of their generation. In onnetion with this, we introdue a onept of a marosopi general fore vetor field, in whih we inlude the eletromagneti and gravitational fields, aeleration field, pressure field, dissipation field, strong interation field, weak interation field and other vetor fields. This general field is assumed to be the main 1

2 soure of ating fores, energy and momentum, as well as the basis for alulation of the system s metri from the standpoint of non-quantum lassial field theory. Inluding the marosopi fields of strong interation and weak interation in the general field is most neessary in those ases, when reations of radioative deay or nulear fusion take plae in massive bodies, as it happens in stars.. The struture of fields Table 1 and Table show the notation for the basi funtions of eah field that we use, inluding potentials, strengths, energy flux densities and field tensors. The last olumn of Table shows the notation for the funtions of the general field. In the following setions we will provide definitions of eah funtion of the general field, while the definitions of other fields were provided in [5] and [6]. Table 1. Field funtions Field Field funtion Eletromagneti field Gravitational field Aeleration field Pressure field 4-potential A D u Salar potential Vetor potential A D U Π Field strength E Γ S C Solenoidal vetor B Ω N I Field tensor F Φ u f Stress-energy tensor W U B P Energy-momentum flux vetor P H K F Table. Field funtions Field Dissipation Strong interation Weak interation General Field funtion field field field field 4-potential g w s Salar potential Vetor potential Θ G W Φ Field strength X L Q T Solenoidal vetor Y μ π χ Field tensor h w s

3 Stress-energy tensor Q L A T Energy-momentum flux vetor Z Σ V Ξ In Table 1 P is the Poynting vetor, H is the Heaviside vetor. The stress-energy tensor of the aeleration field B desribes the energy and momentum of direted motion of the large-sale substane fluxes, as well as the motion of bodies relative to an arbitrary referene frame or rotation of bodies around a fixed pole. The small-sale and random motion of the matter partiles are desribed by the stress-energy tensor of the dissipation field Q. We an assume that this tensor haraterizes the quantity and flux of internal energy in the form of heat and energy of phase transitions that our in the system as a result of visosity. Beause of visosity the direted substane fluxes are deelerated by the surrounding stationary medium and transfer part of their energy to this medium. The general field is haraterized by three three-dimensional vetors and one salar funtion: the field strength T and the solenoidal vetor χ are the omponents of the tensor s, and the salar potential and the vetor potential Φ are the omponents of the 4-potential s. Table 3 shows what field funtions and 4-urrents are inluded in these or those equations. It is assumed that the mass 4-urrent J and the harge 4-urrent j represent the matter properties, and the properties of fields are speified by the orresponding 4-potential. The field equations are usually divided into two fourdimensional equations one of them reflets the field s symmetry and does not ontain 4-urrents, and the other inludes the divergenes of field tensors and the 4-urrents as the soures that generate the fields. Field equations and relations Field equations Motion equation Energy, Lagrangian, Hamiltonian Equation for the metri Gauge of 4-potentials Continuity equations Table 3. Connetion between equations, field funtions and 4-urrents Field funtions, 4-urrents Divergenes of field tensors, 4-urrents Produts of field tensors and 4-urrents or divergenes of fields stress-energy tensors 4-potentials, field tensors and 4-urrents The Rii tensor, salar urvature, fields stressenergy tensors Divergenes of 4-potentials Divergenes of 4-urrents, field tensors, the Rii tensor We will note that aording to Table 3 the stress-energy tensors of fields are present only in the equation for the metri and the equation of the matter motion, but they do not allow us to alulate the system s energy. As it was shown in [7], the volume integral of the sum of stress-energy tensors of fields gives the integral 4-vetor of the system s field energy-momentum equal to zero. Therefore, the system s energy is alulated in another way 3

4 not as an invariant of the motion equation, but as an invariant onserved over time in the system, in whih the Lagrangian does not depend on time [8]. The gauge of 4-potentials allows us to simplify the field equations, espeially it is notieable in the flat spaetime of the speial theory of relativity. The ontinuity equations are obtained as a result of applying the divergene to the field equations with the soures in the form of 4-urrents. 3. The ation funtion and its variation Sine we are planning to replae all the fields existing in the matter with one general field, the ation funtion will inlude only the 4-potential of the general field, the tensor of this field and the mass 4-urrent: 1 S L dt k( R ) s J s s g d, 16 (1) where L is the Lagrange funtion or Lagrangian, R is the salar urvature, is the osmologial onstant, J u is the 4-vetor of mass (gravitational) urrent, is the mass density in the referene frame assoiated with the partile, dx u is the 4-veloity of a point partile, is the speed of light, ds s, Φ is the 4-potential of the general field, desribed with the salar potential and the vetor potential Φ of this field, s is the general field tensor, k and are assumed to be onstant oeffiients. The 4-potential of the general field is alulated as the sum of 4-potentials of the seven fields and at the same time as a generalized 4-veloity:. () q s A D u g w Here q is the harge density in the referene frame assoiated with the partile and we assume that the ratio of the harge density to the mass density is onstant. From () and the definition of s it follows that the salar and vetor Φ potentials of the general field are the sums of the respetive salar and vetor potentials of the fields under onsideration. 4

5 The general field tensor is defined as a 4-url of the 4-potential s : s s s s s. (3) Assuming that q onst, we substitute () into (3):. q q s A D u g w A D u g w q F Φ u f h w (4) In (4) the general field tensor is obtained as the sum of the seven field tensors. The ation funtion with the terms similar to the terms in (1) was varied in [5]. Using the results obtained there, we will make the appropriate onlusions regarding the general field. For the variation of the ation funtion we an write the following: S S S S, (5) 1 3 k S1 k R R g kg g g d, S s J s J g g J s g d, 1 S3 s s T g g d 4, where R is the Rii tensor, g is the metri tensor variation, 1 3 g d g dt dx dx dx is an invariant 4-volume, expressed in terms of the time oordinate differential dx dt, the produt 1 3 dx dx dx of the spae oordinate differentials, and the square root g of the determinant g of the metri tensor, taken with a negative sign, 5

6 is the variation of oordinates, due to whih the variation of the mass 4-urrent J takes plae, s is the variation of the 4-potential of the general field. The stress-energy tensor of the general field is given by expression: 1 T g s s g s s 4 4. (6) We present some harateristis of the general field in Appendix A. 4. The general field equations Substituting S1, S and S3 in (5) and summing up the terms with idential variations, we obtain the orresponding equations as a onsequene of the priniple of least ation. For example, for the variation we an write the following: s 1 s J s g d 4, 4 s J or s 4 J. (7) Sine the general field tensor is defined in (3) using a 4-url, this tensor is antisymmetri and the following relations hold for it: s s s or s. (8) Equation (8) is the equation of the general field without soures, and equation (7) is the general field equation with the soure in the form of mass 4-urrent. If we apply the ovariant derivative to (7) we obtain: 4 R s J. (9) In the flat spaetime the Rii tensor R beomes zero, the ovariant derivative beomes the partial derivative, and the ontinuity equation aquires its standard form in the speial theory of relativity: 6

7 . (1) J The gauge ondition of the 4-potential of the general field: s s. (11) We will substitute () into (11): q s A D u g w. (1) If we assume, as in [5-1], that all the fields appear and exist independently of eah other, then the gauges of 4-potentials of the fields ould also be independent of eah other: A A, D D, u u, (13),, g g, w w. Relations (13) are ompletely onsistent with (1), espeially if we assume that the ratio But the opposite statement is false in general, sine (13) does not follow diretly from (1). q is onstant. We an express (1) in terms of salar and vetor potentials, whih are part of the fields 4-potentials. In the flat spaetime an be used instead of, in whih ase the result is signifiantly simplified: q 1 q A D u g w t 1 q s A D U Π Θ G W t Φ. (14) The gauge of the general field (14) implies a onnetion between the time derivative of the sum of the salar potentials and the divergene of the sum of the vetor potentials of the seven fields. 5. The equation of motion The term with variation is present only in S in (5): 7

8 1 s J g d. Sine, then in order to onform to the priniple of least ation the equation must hold: s J. This an be written in more detail, if we take into aount (4): q F J Φ J u J f J h J J w J. (15) The harge 4-urrent an be defined with the mass 4-urrent as follows: j q J, and the tensor produt u J an be expressed in terms of the 4-aeleration a with the help of the operator of propertime-derivative: Du d u u J u u u u u u u a D d. With this in mind, (15) turns into the four-dimensional equation of motion of visous ompressible substane, whih was introdued and analyzed in [6], with addition of the density of 4-fores, arising due to strong and weak interations: a F j Φ J f J h J J w J. (16) Another way to define the equation of motion is to equate the divergene of the stress-energy tensor of the general field to zero, sine the following relation is valid: k s J T. (17) k To prove (17) we should expand the tensor derivative k to the tensor produts and then use equations (7) and (8). T k with the help of definition (6), apply the ovariant If we substitute s from (3) into the left side of (17), the equation of motion ould be expressed in terms of the 4-potential s of the general field: Ds d s u s u s s u D d. (18) 8

9 On the other hand, we have the relation: u s u s s u. Combining it with the previous equation, we find an equivalent definition of (18): ds u s u s d. 6. The equation for the metri After substituting S1, S and S3 in (5) we an distinguish the terms ontaining the metri tensor variation: k 1 1 kr R g kg s J g T g g d. Sine g, the equation for the metri is obtained by equating the expression in brakets inside the integral to zero : k 1 1 k R R g kg s J g T. (19) Let's ontrat equation (19) by multiplying by the metri tensor, given that g T, gr R, g g 4 : k R 4k s J. () In [5] we assumed the gauge of the osmologial onstant, whih aording to () orresponds to the following expression: q k s J A D u g w J. (1) 9

10 Gauge (1) means that the osmologial onstant is not an arbitrary quantity. For eah substane unit the value an be hosen so as to equal the total rest energy of all the partiles of the substane unit, inluding the energy of these partiles in the potentials of their own internal fields and exluding the energy of the partiles interation. The latter an be ahieved only when all the partiles are separated and sattered at infinity. With gauge (1), it follows from (): R. () Outside the matter J in (1), then, and the salar urvature is equal to zero: R. Let us substitute (1) and () into (19): 1 1 R R g T. (3) 4 k We will obtain the same if we multiply () by g and divide by 4 and then substitute in (19). The equation for the metri (3) oinides with the equivalent equation in [5] and [6], with the differene that in (3) the stress-energy tensor of the general field T, due to its definition (6) with regard to (4), ontains not only the stress-energy tensors of the seven fields, but also additional ross terms with the produts of strengths and solenoidal vetors of these fields. If we apply the ovariant derivative to (3), the right side beomes zero, as a onsequene of the equation of motion in the form of (17). We an apply in the left side of (3) the equality 1 R R g as the property of the Einstein tensor. We will obtain the equality 1 Rg or the equivalent equality R. If we take into aount (1-), this leads to the 4 following equation, whih must hold inside the matter: s J A j D J u J J J g J w J. The same expression will be obtained in ase when the ovariant derivative is applied diretly to (19). 7. The energy The energy of the system, onsisting of the matter and the fields, an be alulated by the same method as in [5]. If the Lagrangian does not depend on time, the system s energy will be equal to the Hamiltonian of this system. Taking into aount the gauge (1-), for the energy we obtain the following: 1

11 E s J s s g dx dx dx (4) The energy (4) depends on the time omponents of the 4-potential of the general field s and the mass 4- urrent J, and does not depend on the produt i sj i, where the index 1,,3 i speifies the spae E E E omponents of the 4-vetors. For the 4-momentum of the system we obtain: p,, p v, where p and v denote the system s momentum and the veloity of the enter of mass. 8. Conlusions Let us ompare our approah to unifying the eletromagneti and gravitational fields, aeleration field, pressure field, dissipation field, strong interation field and weak interation field with another attempt of unifying the eletromagneti, gravitational and other arbitrary vetor fields, whih was undertaken by Науменко [11]. His Unified theory of vetor fields (UTVF) is formulated in the framework of the speial theory of relativity. We present here a quote from [11]: Let us assume that there are n fields: X1, X,, X n, eah of whih has its orresponding harge: q q q. X 1, X,, X n It is suggested to onsider these fields as manifestations of a single field that onforms to the equations: div дl rot Y j, (5) дt Y YL L, YL L YL L L L where Y, L take values from a set of symbols X1, X,, X n, ( ) is a matrix of elrtri onstants, ( ) is a matrix of magneti onstants, ( ) is a matrix of eletrodynamial onstants, denotes harge densities, j denotes urrent densities. div j To these equations Науменко adds the onditions of harge onservation for eah field: Y Y t. As we an see, the equations of UTVF represent extended Maxwell equations. In these equations any field (for example, the eletri or magneti field) an influene the divergene or url of another field (for example, the gravitational field, torsion field or gravitomagneti field) or even influene this field s own divergene or url. Науменко also introdues a vetor of this unified field: ФY YL L or ФL YL Y, onsisting of the sum of strengths and solenoidal vetors of all the fields with the orresponding oeffiients. L Y 11

12 Multiplying equations (5) by the oeffiients equations: YL and summing over the index Y, he obtains additional div Ф. (6) rot L YL YL L YL L Y L L 1 1 Ф Ф j Y. (7) t L YL L L In (6) the soure of the unified field Ф L is the sum of produts of the fields harge densities and some oeffiients. In (7) the sum of the produts of urrents and some oeffiients gives the url and the time derivative of the unified field strength Ф L. It turns out that the unified field s divergene is formed of a multitude of available harge densities, and the urrents define the url of the unified field. The analysis of (5-7) shows that as the basis of the unified field equations of UTVF the idea is taken about the full symmetry of Maxwell-like equations relative to the ontribution of harges and urrents in the unified field, whih is oneived as linear ombination of strengths and solenoidal vetors of a set of vetor fields. Our approah differs by the fat that as a basis the 4-potential of the general field s is taken, onsisting of the sum of 4-potentials of the seven vetor fields. With the help of s, by means of antisymmetri ovariant differentiation we define the general field tensor s and its invariant s 1 s. These quantities are substituted into the Lagrangian, and the subsequent use of the priniple of least ation allows us to derive the neessary equations, inluding the general field equations, the equation of matter motion in the general field, the equation for alulation of the metri, the stress-energy tensor of the general field. The soure of the general field is the mass 4-urrent J, and the ontribution of the harge 4-urrent j revealed when the general field tensor s or the 4-potential s is multiplied by J. in the motion equation or in the energy is Aording to the method of onstrution of the 4-potential and the general field tensor, the salar (vetor) potential of the general field onsists of the sum of the salar (vetor) potentials of the seven fields. The same an be said about the strength and solenoidal vetor of the general field aording to (A1) they onsist of the sums of the orresponding vetors of the seven fields. As we an see in (4), the energy of the system of matter and seven fields in our approah appears to be dependent not only on the stress-energy tensors of these seven fields, but also on the sum of the ross terms with the produts of different strengths and solenoidal vetors of the fields. We remind that the Lorentz-invariant equations of the gravitational field, oiniding by their form with Maxwell equations for the eletromagneti field, first appeared in the works by Heaviside [1]. Subsequently, these equations were derived in a ovariant form and beame the basis of the ovariant theory of gravitation [13]. Later, based on the priniple of least ation the ovariant equations of the aeleration field, pressure field [5] and energy dissipation field [6] were derived. All these equations in the weak field limit have the form of Maxwell equations. Aording to [7-8], the potentials and strengths of these fields have the same dependene on

13 the oordinates and time, obeying the wave equation. Thus, there is every reason to aknowledge the existene of a single general field, for whih the above mentioned seven fields are the partiular forms. In our opinion, this situation is losely onneted with the theorem of equipartition of energy. Usually this theorem is interpreted as follows: when the system is in equilibrium, the kineti energy is distributed between all those degrees of freedom that appear in the energy as quadrati funtions. Apparently, this definition should be expanded so that the energy of the general field tends to be distributed also among the degrees of freedom in the form of strengths and solenoidal vetors of individual fields. Indeed, these field degrees of freedom are inluded in the expressions for the field energy as quadrati funtions. In turn, division of the general field into separate fields ours beause new degrees of freedom are released by means of physial analysis, whih are haraterized by their own fields. We an also say that the 4-potential of the general field an be divided to the 4-potentials of separate fields, and therefore it onsists of them. The tendeny to distributing the energy of interations between the fields and substane is a onsequene of the energy exhange between the fields and matter partiles, and the differene between the fields arises due to different types of interation. As it is shown in [14], the gravitational 4-potential of an arbitrary small partile an be presented as the produt of the partile s 4-veloity and the gravitational potential of this partile in its rest system, divided by the square of the speed of light. In this ase, the gravitational field of a system of moving partiles an be preisely alulated taking into aount the superposition priniple of potentials and field strengths of a multitude of partiles, taking into aount the propagation delay of the gravitational effet by using the method of retarded potentials and Lorentz transformations. Although the vetor potential of a single partile an be onsidered proportional to the salar potential, it is not so for a system of partiles, whih is the onsequene of different rules of summation of salars and vetors. The salar and vetor potentials of a system of partiles beome independent of eah other. Exatly the same applies to the eletromagneti field of a system of harged partiles. The aeleration field, pressure field and dissipation field were introdued by multiplying the 4-veloity of an arbitrary system s partile by the potential of the orresponding field at the loation of the partile, divided by the square of the speed of light [15]. This approah is suitable for desribing the strong interation field and weak interation field. In this ase the salar potentials of these fields are proportional to the density of the energy, aumulated by the matter during the reations of strong and weak interations per unit mass of the matter. This is why the 4-potential of the general field s is the sum of the 4-potentials of onstituent fields and at the same time it an haraterize the interation of all the fields with the matter. This interation is desribed by the produt s J in the ation funtion (1), while J denotes the mass 4-urrent. In [16], gravitation is seen as a onsequene of the pressure gradient of the quantum vauum, whih oupies the entire spae within and between the bodies. In this stati piture for the emergene of gravitation gravitons are not required. From the lassial point of view the universal harater of the equations of suh fundamental fields as eletromagneti and gravitational fields, is most naturally explained in the Fatio-Le Sage s theory of gravitation. This theory provides a lear physial mehanism of the gravitational fore origination [17-18], as a onsequene of the influene of ubiquitous fluxes of gravitons in the form of tiny partiles like neutrinos or photons on the 13

14 bodies. This mehanism also allows us to explain the eletromagneti interation [13], if we assume the presene of tiny harged partiles in graviton fluxes. These graviton fluxes penetrate all bodies and perform eletromagneti and gravitational interation by means of the field even between distant partiles. The partiles an also exert diret mehanial ation on eah other, whih an be represented by the pressure field. An inevitable onsequene of the ation of these fields is deeleration of fast partiles in the surrounding medium, whih is desribed by the dissipation field. Finally, the aeleration field is introdued for kinematial desription of the motion of partiles, the fores ating on them, the energy and momentum. As a result, the general field an be represented as a field, in whih neutral and harged partiles in the fluxes of neutral and harged gravitons exhange energy and momentum with eah other and with gravitons. The energy and momentum of the general field an be assoiated with the energy and momentum, aquired by the fluxes of gravitons during interation with the matter; and in order to take into aount the system s energy and momentum we need to add the matter s energy and momentum from its interation with gravitons. We should add to the above-mentioned, that the strong interation in our opinion an be redued to strong gravitation, ating at the level of atoms and elementary partiles [13], [19-], with replaement of the gravitational onstant by the strong gravitational onstant. As for the weak interation, from the standpoint of the theory of infinite nesting of matter, it is redued to the proesses of matter transformation under the ation of fundamental fields, taking into aount the ation of strong gravitation. Similarly, the pressure field and dissipation field ould be redued to fundamental fields, if we would know all the details of interatomi and intermoleular interations. Due to the diffiulties with suh detailed information, we assume the existene of own 4-potentials in the pressure field, energy dissipation field, strong interation field and weak interation field, and approximate the ation of these fields in the matter using these 4-potentials. On the other hand, Abdus Salam, Sheldon Glashow and Steven Weinberg have ombined with one formalism the weak and eletromagneti interations in the quantum field theory. This implies that suh ombination is also possible in the lassial desription of fields and their ation in massive bodies, and we make it based on the same proedure that was used in [5-6], [15]. As for the reations of strong and weak interations, we should take into aount that they hange the energy of massive objets in the marosopi gravitational and eletromagneti fields. These reations take plae due to emission or absorption of the energy of strong mirosopi fields, ating on the atomi level, lead to thermonulear reations and are the main soure of stellar radiation. The existene of additional thermonulear energy soures inside the stars shifts signifiantly the standard spatial distribution of physial quantities. For example, the estimate of the temperature in the enter of the Sun in [7] in general orresponds to the formula of temperature derease proportionally to the square of the radius, as it follows from the wave equation for the potential of the aeleration field. However, the pressure in the enter of the Sun is 58 times less than in the standard Sun model. This deviation ourred beause we did not take into aount the pressure effet from the energy and momentum aquired by the partiles in nulear reations due to strong and weak interations. If we assume that eah of the seven fields under onsideration is a speial manifestation of the general field, then in ase of equilibrium and steady distribution of parameters for the fields of strong and weak interations we an expet the field equations, similar in the form to the equations for other fields. These equations an be obtained from (7-8) and from equations (A1-A11) in Appendix, with replaement of the potentials and strengths 14

15 of the general field by similar quantities from Table for the strong interation field and weak interation field, respetively. In this ase the oeffiient in (6), in formulas (A4) and further on must be replaed by other onstant oeffiients to be determined for eah field. In partiular, for the salar potential of the strong interation field in the framework of the speial theory of relativity we expet the wave equation similar to equation (A1): 1 4, (8) t where is a ertain oeffiient. In stationary ase, the potential does not depend on time, and the solution, that follows from (8), is similar to the solution for the pressure field in [7] for a spherial massive body: 3 r r sin 4 r 4 3, (9) where is the salar potential of the strong interation field in the enter of the body, is the Lorentz fator for the partiles in the enter, is the oeffiient of the aeleration field. 1 1 v We an express the salar potential by a formula p s, where p s denotes the volume energy density or the pressure, arising from reations in the matter inluding strong interation. Nulear reations our mainly in the stellar ore, on the ore surfae the rate of reations is low, and at r R we an assume that. Then, from (9) we an estimate p s in the enter of the stellar ore: p R 3. M s R Assuming for simpliity that the solar energy is produed mainly in reations involving strong interation, equating p s to the pressure in the enter of the Sun equal to substituting the ore mass obtain the estimate of the onstant: M equal to.34 Solar masses and the ore radius Pa in the standard model [1], and R equal to. Solar radii, we m 3 /(kg s ). For omparison, in the formula for the salar potential of the pressure field, the same as in (9), a similar oeffiient in the absene of the strong interation 15

16 field equals oeffiient also equals 1 3G 1 m 3 /(kg s ). Aording to [7], for the aeleration field the orresponding 3G, where G is the gravitational onstant. Referenes [1] Georgi, H. and Glashow, S.L. (1974) Unity of All Elementary Partile Fores. Physial Review Letters, 3, [] Carlip, S. (1) Quantum Gravity: a Progress Report. Reports on Progress in Physis, 64, doi:1.188/ /64/8/31. [3] Kaluza, T. (191) Zum Unitätsproblem in der Physik. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): [4] Klein, O. (196) Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitshrift für Physik A, 37, [5] Fedosin, S.G. (16) About the osmologial onstant, aeleration field, pressure field and energy. Aepted by Jordan Journal of Physis. [6] Fedosin, S.G. (15) Four-Dimensional Equation of Motion for Visous Compressible and Charged Fluid with Regard to the Aeleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamis, 18, doi:1.5541/ijot [7] Fedosin, S.G. (14) The Integral Energy-Momentum 4-Vetor and Analysis of 4/3 Problem Based on the Pressure Field and Aeleration Field. Amerian Journal of Modern Physis, 3, doi: /j.ajmp [8] Fedosin, S.G. (15) Relativisti Energy and Mass in the Weak Field Limit. Jordan Journal of Physis, 8, [9] Fedosin, S.G. (1) The Priniple of Least Ation in Covariant Theory of Gravitation. Hadroni Journal, 35, [1] Fedosin, S.G. (1) The Hamiltonian in Covariant Theory of Gravitation. Advanes in Natural Siene, 5, doi:1.3968%fj.ans [11] Науменко Ю. В. (6) Единая теория векторных полей (от электродинамики Максвелла к единой теории поля). Армавирское полиграф-предприятие, Армавир. [1] Heaviside, O. (1893) A Gravitational and Eletromagneti Analogy, Part I. The Eletriian, 31, [13] Fedosin, S. (15) The physial theories and infinite hierarhial nesting of matter, Vol.. LAP LAMBERT Aademi Publishing, Saarbrüken. [14] Fedosin, S.G. (13) 4/3 Problem for the Gravitational Field. Advanes in Physis Theories and Appliations, 3, [15] Fedosin, S.G. (14) The proedure of finding the stress-energy tensor and vetor field equations of any form. Advaned Studies in Theoretial Physis, 8, doi:1.1988/astp [16] Caligiuri, L.M. and Sorli A. (14) Gravity Originates from Variable Energy Density of Quantum Vauum. Amerian Journal of Modern Physis, 3, doi: /j.ajmp [17] Fedosin, S.G. (9) Model of Gravitational Interation in the Conept of Gravitons. Journal of Vetorial Relativity, 4,

17 [18] Mihelini, M. (7) A flux of Miro-quanta explains Relativisti Mehanis and the Gravitational Interation. Apeiron Journal, 14, [19] Fedosin, S. G. (1999) Fizika i filosofiia podobiia ot preonov do metagalaktik. Style-Mg, Perm. [] Fedosin, S.G. (1) The radius of the proton in the self-onsistent model. Hadroni Journal, 35, [1] Christensen-Dalsgaard et al. (1996) The urrent state of solar modeling. Siene, 7, doi:1.116/siene Appendix. The harateristis of the general field The antisymmetri tensor omponents of the general field are obtained from relation (3). Let us introdue the following notations: 1 si si i s Ti, si j i s j j si k, (A1) where the indies i, j, k form triplets of non-reurrent numbers of the form 1,,3, or 3,1,, or,3,1; the 3- vetors T and χ an be written by omponents: T Ti ( T1, T, T3 ) ( Tx, Ty, Tz ) ; χ (,, ) (,, ). i 1 3 x y z Using these notations the tensor s an be represented as follows: s Tx Ty Tz T T x y T z z y z x y x. (A) The same tensor with ontravariant indies equals: s g g s. In Minkowski spae the metri tensor does not depend on the oordinates, and in this ase for the general tensor field we have the following: 17

18 s T T x y T z Tx z y s. (A3) Ty z x Tz y x The general field equation (7) an be expressed in Minkowski spae in terms of the vetors T and χ using the 4-vetor of mass urrent: J u ovariant derivatives v, where, with the partial derivatives, we find: 1 1 v. Substituting in (7) the T 4, 1 T 4 v t χ, χ, χ T. (A4) t If we multiply salarly the seond equation in (A4) by T, and multiply salarly the fourth equation by χ and sum up the results, we will obtain the following: 1 ( T ) 4 vt [ Tχ ]. (A5) t Equation (A5) ontains the Poynting theorem applied to the general field, it is written in a ovariant form as the time omponent of equation (17):. T s J If we substitute (A) in (17), we an obtain one salar and one vetor relation: s J T v, s J i T[ vχ ]. (A6) The first relation in (A6) is the time omponent of the motion equation (16) and the seond relation is the spae omponent of (16). The vetor T has the dimension of an ordinary 3-aeleration, and the dimension of the vetor χ is the same as that of the frequeny. 18

19 Let us substitute the 4-potential of the general field s, Φ in the definition (A1): Φ T, χ Φ. (A7) t The vetor T is the general field strength and it is expressed in terms of salar and vetor potentials of the seven fields. The vetor χ is the solenoidal vetor of the general field, depending on the vetor potentials of fields. We an substitute the tensors (A) and (A3) in (6) and express the stress-energy tensor of the general field T in terms of the vetors T and χ. Let us write here the expressions for the tensor invariant s s and the time omponents of the tensor T : ( ) s s T, 1 ( T T ), 8 T i [ Tχ ]. (A8) 4 Ξ The omponent i T T defines the energy density of the general field in the given volume, and the vetor [ T χ ] defines the energy flux density of the general field. 4 If we substitute T from (A7) in the first equation in (A4), and take into aount the gauge of the 4-potential (14) as follows: 1 s t Φ, (A9) we will obtain the wave equation for the salar potential: 1 4. (A1) t From (A7), (A9) and the seond equation in (A4) the wave equation follows for the vetor potential of the general field: 1 4 v t Φ Φ. (A11) 19

20 Let us now substitute in (A7) the general field potentials and Φ, expressed in terms of the potentials of the seven fields, aording to (14), provided q onst : T A D U Π Θ G W q q t E Γ S C X L Q. q q q χ A D U Π Θ G W B Ω N I Y μ π. (A1) In (A1) we used definitions of the field strengths, suh as A E, B A for the t eletromagneti field, and similar definitions for other fields. Aording to (A1), the strength T and the solenoidal vetor χ of the general field are expressed in terms of the sums of the orresponding strengths and solenoidal vetors of the seven fields. If we substitute (A1) in (A), we will obtain the relation, whih oinides with (4) for the general field tensor:. q s F Φ u f h w The vetors T and χ in (A1) are represented as the sums of the orresponding vetors of the seven fields. Therefore, after substituting (A1) in the general field equations (A4), these equations ould be divided into seven sets with four equations in eah set, separately for eah field. As a result, we ould assume that the fields and the equations for these fields are relatively independent of eah other. But in general ase, suh division of the general field equations to separate equations for eah field is not always possible. Probably division of equations and independene of fields an take plae when energy and momenta distribution between all the fields is ompleted in the system. As we an see from (A8), the stress-energy tensor of the general field T inludes the vetors produts of the vetors T and χ, as well as the squares of these vetors. If we take into aount (A1), then we an see that in the tensor T ross terms appear, ontaining the produts of strengths and solenoidal vetors of all the seven fields. This means that the fields tend to interat with eah other, introduing additional ross terms into the energy and momentum of the general field. This does not apply to the fore ation of the fields on the matter, sine there are no ross terms in the equation of motion, aording to (16) and (A6).