The Procedure of Finding the Stress-Energy. Tensor and Equations of Vector Field of Any Form

Transcription

1 Advaned Studies in Theoretial Phsis Vol. 8, 14, no. 18, HIKARI Ltd, htt://d.doi.org/1.1988/ast The Proedure of Finding the Stress-Energ Tensor and Equations of Vetor Field of An Form Serge G. Fedosin Sviaeva Str. -79, Perm, Perm region, Russian Federation Coright 14 Serge G. Fedosin. This is an oen aess artile distributed under the Creative Commons Attribution Liense, whih ermits unrestrited use, distribution, and rerodution in an medium, rovided the original work is roerl ited. Abstrat A method allowing us to introdue into the Lagrangian the terms, whih haraterie an arbitrar vetor field of a sstem, is desribed. As a result of aling the rinile of least ation it beomes ossible to find all the main harateristis of this field, inluding its energ and momentum, field equations, fore of interation with the matter. Kewords: Four-otential; ressure field; aeleration field; field equations. 1. Introdution The field onet is widel used not onl in the gravitation theor but also in other hsial theories. Further we will onsider the roerties of vetor fields in four-dimensional sae. The main harateristi of the eletromagneti vetor field is the 4-otential A, A, where is the seed of light, and A denote the salar and vetor otentials, resetivel. If the sstem ontains a set of artiles, eah of whih generates its own otential, then the otentials and A of the sstem of artiles deend mainl on the general sstem arameters the dimensions of the sstem, the total harge, et. Besides the sstem s otentials orresond to the suerosition rinile the otentials of all the artiles. We an determine all the

2 77 Serge G. Fedosin main harateristis of the sstem s eletromagneti field with the hel of the 4- otential. Thus before we find the 4- otential of the sstem, we need to determine the 4- otential of a single artile. This an be done as follows: the invariant salar otential of the artile should be divided b the square of the seed of light, so that it beomes a dimensionless quantit, and then multil it b the ovariant 4-veloit: d d 1 d d 3 A u,,,, A 1, A, A3. (1) d d d d The otential referene frame is onsidered to be invariant, if it is determined in the K, whih is rigidl assoiated with the artile. We an see in (1), that is assoiated with the salar otential and the three omonents of the vetor otential A of a artile in an arbitrar referene frame K, in whih d the artile has a 4-veloit u (where d is a 4-dislaement with the d ovariant inde, is the roer time of the artile). In order to find the 4- otential of the sstem, it is neessar to integrate (1) over all of the sstem s artiles.. Pressure field We will turn now to the ressure field, the roerties of whih must be taken into aount in alulating the metri inside the material bodies, as well as in determining the equation of motion and of the state of matter. The eisting definitions of the ressure field and its energ-momentum are derived b means of generaliation of the formulas of lassial mehanis. For eamle, in the general theor of relativit (GTR) the following ressure tensor [1] is used for ideal liquid: P u u g. () In () the ressure reresents a salar field. The tensor uu is onsidered to be the matter harateristi in GTR, and the total stress-energ tensor of the matter with ressure and densit is equal to: T P.

3 Proedure of finding the stress-energ tensor 773 Now we will answer the question whether we an onsider the ressure field not just salar but a four-dimensional vetor field? B definition, a vetor field at eah oint is desribed b a ertain vetor. In ontinuousl distributed matter the artiles are so lose to eah other, that the onstantl interat with eah other. In this ase, we an assume that the diretion of the vetor of one artile s ressure on another is arallel to the vetor of artile veloit. If a vetor field of veloit is seified for the artiles, then the vetor field of ressure an be onsidered as the onsequene of the veloit field. On the other hand, the ressure, even in the absene of artiles motion when it looks like a salar field, makes its ontribution to the mass-energ of the artiles. Sine the ressure has meaning both of a salar and of a threedimensional vetor, there must be a 4-vetor, where the ressure is art of the salar and vetor omonents. It is natural to all suh a 4-vetor a 4-otential of the ressure field. We will determine the 4-otential of the ressure field similarl to (1):, Π, (3) u where and denote the ressure and densit in the referene frame K of the artile, the dimensionless ratio is roortional to the ressure energ of the artile er artile s unit mass, and Π are the salar and vetor otentials of the ressure field. Then aling the 4-rotor we find the antismmetri ressure tensor f, onsisting of si omonents, belonging to two vetors C ( C, C, C ), I ( I, I, I ): f. (4) f C C C C C C I I I I I I. (5)

4 774 Serge G. Fedosin Now, using (4) we an onstrut a tensor invariant f f, where 16 should be determined. The ressure field equations are obtained from the rinile of least ation, while the sum should be substituted into the Lagrangian: 1 J f f, where J is the mass 4-urrent. For omarison, all 16 the roerties of the eletromagneti field are obtained b varing a similar sum: 1 A j F F, where j is the eletromagneti urrent, is the 4 vauum ermittivit, F is the eletromagneti tensor. One of the results of the Lagrangian variation is the stress-energ tensor of the ressure field []: 1 P g f f g f f 4 4. (6) This tensor with other fields tensors is art of the right side of the equation for determining the metri, and the left side of this equation ontains the Rii tensor and salar urvature. With the hel of tensor (4) or tensor (6) we an determine the densit of 4-fore in the equation of matter motion that arises due to the ressure: k f J P k. We also find the ressure field equations: where 4 f J, f, (7) is the Levi-Civita smbol. Equations (7) in the limit of the seial theor of relativit with regard to (5) look like Mawell equations: 1 C 4 v C 4, I, I, t I C. (8) t Here of matter. 1 1 v is the Lorent fator, v is the veloit of a oint artile

5 Proedure of finding the stress-energ tensor 775 If we substitute (4) in the first equation in (7) in the form:, we obtain: f 4 J. In ase if 4-otential gauge in the left side of the equation we have s R s, where R s denotes the Rii tensor with mied indies. On the other hand, 4-d'Alembertian ating on the 4-vetor is determined in the general ase as follows: g R s s s s s s g ( ). s s s s As a result the terms with the Rii tensor are aneled and we have the following: s s s s 4 g g ( s s s s ) J. (9) Equation (9) reresents the wave equation for the 4-otential of the ressure field, whih allows us to find the ressure distribution inside the massive bodies. In artiular, for sherial bodies with aroimatel onstant densit the ressure dereases from the bod enter to its surfae, due to the resene of the negative term in the formula for the ressure, whih is roortional to the square of the urrent radius. From (9) we an estimate the ressure at the enter of the massive bod: 3 M, (1) 4 8 R where 3G, G is the gravitational onstant, M and R denote the bod mass and radius. 3. Aeleration field The foregoing desribes the roedure of obtaining the stress-energ tensor and the vetor field equations of an kind. In artiular, the above-mentioned roedure was also alied in [] in order to find the stress-energ tensor of matter

6 776 Serge G. Fedosin in a ovariant wa. As the 4-otential of the aeleration field of a artile the ovariant 4- veloit was used without additional fators: u, U, where and U denote the salar and vetor otentials, resetivel. The tensor of the aeleration field is given b: u u u u u. (11) u S S S S S S N N N N N N, (1) where the vetors S ( S, S, S ) and N ( N, N, N ) define the artile s aelerations. The ontribution of the aeleration field into the Lagrangian is given b the 1 sum: u J uu, where is to be determined. The stress-energ 16 tensor of the aeleration field aears as a result of variation of ation: 1 B g u u g u u 4 4. (13) The 4-aeleration in the equation of motion of a small artile of ontinuousl distributed matter is found either with the hel of tensor (11) or tensor (13): Du k u J B k. D Like an vetor field, the aeleration field is given b the orresonding equations:

7 Proedure of finding the stress-energ tensor u J, u. These equations in the seial theor of relativit are the equations for the vetors S and N from (1): 1 S 4 v S 4, N, N, t N S. (14) t The stress-energ tensors of the ressure field P (6) and the aeleration field B (13) are onstruted in a ovariant wa using the 4-otentials and in the ovariant theor of gravitation the substitute the tensor P in () and the tensor uu, resetivel. Similarl to (9) we obtain the wave equation for the veloit field inside the bodies: s s s s g u g ( u u u u ) 4 J s s s s 4 u. (15) The solution of this equation allows us to alulate the veloit of the artiles motion inside the sherial bod as a funtion of the urrent radius. The kineti energ of the artiles deends on their veloit and seifies the kineti temerature. Consequentl, it beomes ossible to find the equilibrium temerature distribution inside the massive bodies. In artiular, for the temerature at the enter we an aroimatel write the following: T M M. (16) 3kR where 3G, M denote the mass of a tial bod artile, usuall it is the mass of a hdrogen atom, k is the Boltmann onstant. 4. Conlusion Desite the fat that the formulas (1) and (16) were found in the assumtion of uniform densit, the are well satisfied for gas louds, lanets and stars. Good agreement is observed for Bok globules, the Earth and neutron star, as well as for

8 778 Serge G. Fedosin the temerature inside the Sun [3]. The differene ours onl for the ressure inside the Sun, where it is 58 times less than in the standard model. This is robabl due to the fat that thermonulear reations take lae inside the Sun, whih inrease the ressure. We believe that the massive bodies, held in equilibrium b gravitation fore, ontain radial gradients of the otentials of gravitation, ressure, artiles kineti energ and other quantities. These gradients are the essential omonents that ensure the sstem s stabilit at a given matter state. If we assume the validit of the gravitation mehanism in Le Sage s theor [4], then at equilibrium the temerature of the interior of a massive osmi bod annot fall below the value that is obtained from the virial theorem. Desite the onstant emission from the surfae of the bod, the neessar energ inflow is ensured b gravitons falling on the bod. In Le Sage s model the graviton flues enetrating the matter not onl reate the gravitational fore, but also leave some art of their energ inside the bod, warming it. Thus, we have introdued a roedure, aording to whih it is neessar first to determine the salar otential of an arbitrar vetor field, inherent in a single artile. After that b means of standard methods all the harateristis of this field are derived, inluding the field equations, its stress-energ tensor and the te of the fore, eerted b the field on the artiles. We must note that the 4-otential of the field is eressed as a ovariant 4- vetor, and the matter energ in this field deends on the rodut of the 4- otential and the mass (eletromagneti) 4-urrent, taken with the ontravariant inde. The field tensor has doubl ovariant indies as the onsequene of the 4- url ating on the 4-otential. In order to find this tensor with ontravariant indies the metri tensor is required. Contration of the field tensor with itself gives the tensor invariant, whih is required in the Lagrangian to arr out variation and to eress relationshi between the matter, metri and field in the aroriate equations. Another euliarit of this aroah is that the field equations (8) and (14) are similar in form to Mawell equations. The reviousl desribed aroah was used in [5], [6], [7] to find also the 4- otential of the gravitational field, its stress-energ tensor, gravitational 4-fore and field equations in the framework of the Covariant Theor of Gravitation (CTG). In CTG the gravitational field is divided from the metri field, gravitation beomes an indeendent field with its own energ, momentum and ation in the form of gravitational fore. As a result, the metri is onl neessar to desribe deviations of the results of gravitational eeriments from their form in the seial theor of relativit. The essential art of the Lagrangian in CTG is the onstant, whih is alled osmologial onstant. With the hel of this onstant the Hamiltonian gauge is erformed so that the sstem s energ ould be determined unambiguousl.

9 Proedure of finding the stress-energ tensor 779 REFERENCES [1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, San Franiso, CA, [] S.G. Fedosin, About the osmologial onstant, aeleration field, ressure field and energ. vira.org, 5 Mar 14. htt://vira.org/abs/143.3 [3] S.G. Fedosin, The integral energ-momentum 4-vetor and analsis of 4/3 roblem based on the ressure field and aeleration field. Amerian Journal of Modern Phsis, 3 (14), [4] S.G. Fedosin, Model of gravitational interation in the onet of gravitons. Journal of Vetorial Relativit, 4 (9), 1-4. [5] S.G. Fedosin, The rinile of least ation in ovariant theor of gravitation. Hadroni Journal, 35 (1), [6] S.G. Fedosin, Fiiheskie Teorii i Beskonehnaia Vlohennost Materii, Perm, 9. ISBN [7] S.G. Fedosin, Fiika i Filosofiia Podobiia ot Preonov do Metagalaktik, Perm, 1999, ISBN Reeived: Jul 3, 14