4/3 Problem for the Gravitational Field

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1 4/3 Problem for the ravitational Field Sere. Fedosin PO bo 6488 Sviaeva str. -79 Perm Russia Abstract The ravitational field potentials outside and inside a uniform massive ball were determined usin the superposition principle the method of retarded potentials and Lorent transformations. The ravitational field strenth the torsion field the ener and the momentum of the field as well as the effective masses associated with the field ener and its momentum were calculated. It was shown that 4/3 problem eisted for the ravitational field as well as in the case of the electromanetic field. Kewords: ener momentum theor of relativit ravitation field potentials. Introduction In field theor there are a number of unsolved problems which need deeper analsis and loical understandin. An eample is the problem of choosin a universal form of the stress-ener tensor of the bod which would include the rest ener of the substance as well as the field ener and at the same time would provide an univocal connection with thermodnamic variables of the substance in the lanuae of four-vectors and tensors. Another interestin problem is 4/3 problem accordin to which the effective mass of the bod field which is calculated throuh the field momentum and the effective mass of the field found throuh the field ener for some reason do not coincide with each other with the ratio of the masses approimatel equal to 4/3. The problem of 4/3 is known for a lon time for the mass of electromanetic field of a movin chare. Joseph John Thomson eore Francis Fiterald Oliver Heaviside eore Frederick Charles Searle and man others write about it (Heaviside 888/894) (Searle 897) (Hajra 99). We also discuss this question with respect to the ravitational field of a movin ball (Fedosin 8). Now we present a more accurate description of the problem not limited to the approimation of small velocities.. Methods In the calculation of the ener and the momentum of ravitational field of a uniform massive ball we will use the superposition principle b means of summin up the field eneries and momenta from all point particles formin the movin ball. This approach is reasonable in the case of a weak field when the eneral theor of relativit chanes to ravitomanetism and the covariant theor of ravitation to the Lorent-invariant theor of ravitation (Fedosin 9a). The field equations then become linear allowin the use of the superposition principle. We will note that the ravitational field can be considered weak if the spacetime metric differs insinificantl from the Minkowski spacetime metric (the spacetime metric of the special theor of relativit). If the effects of ravitational time dilation and sies contraction are sinificantl less than the similar effects due to the motion velocit of the reference frame under consideration then this ravitational field can be considered weak. 3. Results and Discussions 3. The ravitational Field Outside a Uniform Massive Ball We will first define the ravitational field potentials for a ball movin at a constant velocit V alon the ais OХ of the reference frame K. We will proceed from the so-called Liénard-Wiechert potentials (Liénard 898; Wiechert 9) for an point particles that make up the ball. Popular presentation of the problem (for the electromanetic field) can be found in Fenman s book (Fenman at all. 964). Similarl to this the differential scalar Liénard-Wiechert potential for the ravitational field from a point particle with mass dm has the followin form: γ dm dψ () r V r c Where γ is the ravitational constant c is the velocit of ravitation propaation vector r is the vector connectin the earl position of the point particle at time t and the position r ( ) at which the potential is determined at time t. In this case the equation must hold: / 9

2 r t t. () c The meanin of equation () is that durin the time period t t the ravitational effect of the mass dm must cover the distance r at velocit c up to the position r ( ) so that at this position the potential dψ would appear. Suppose there is continuous distribution of point particles and at t these particles are described b the coordinates ( ) and the center of distribution of point particles coincides with the oriin of the reference frame. Then at time t the distribution center of the point particles would move alon the ais OX to the position Vt and the radius vector of an arbitrar particle of distribution would equal r ( + Vt ). At the earl time t the position of this point particle is specified b the vector r ( + Vt ). Since can write down: r r r and r c ( t t ) accordin to () then for the square r we r ( Vt ) + ( ) + ( ) c ( t t ). (3) The riht side of (3) is a quadratic equation for the time t. After we find t from (3) we can then find r from (). If we consider that in () the product of vectors is V r V( Vt ) then substitutin r also in () we obtain the followin epression (Fedosin 9b): γ dm dψ. (4) ( Vt) V c + ( ) + ( ) V c Accordin to (4) the differential ravitational potential dψ of the point mass dm at the time t durin its motion alon the ais OX depends on the initial position ( ) of this mass at t. If we use the etended Lorent transformations for the spatial coordinates in (4): Vt V c (5) and then let the velocit V tend to ero we obtain the formula for the potential in the reference frame K the oriin of which coincides with the point mass dm : γ dm dψ. (6) + + In (6) in the reference frame K the vector r ( ) at the proper time t specifies the same point in space as the vector r ( ) in the reference frame K at the time t. If we introduce the ravitational fourpotential D µ ψ с D includin the scalar potential ψ and the vector potential D (Fedosin 999) then the relation between the scalar potential (6) in the reference frame K and the scalar potential (4) in the reference frame K can be considered as the consequence of etended Lorent transformations in fourdimensional formalism which are applied to the differential four-potential of a sinle point particle. These transformations are carried out b multiplin the correspondin transformation matri b the four-potential which ives the four-potential in a different reference frame with its own coordinates and time. Since in the reference frame K the point mass is at rest its vector potential is dd and the four-potential dψ has the form: ddµ. In order to move to the reference frame K in which the reference frame с K is movin at the constant velocit V alon the ais OX we must use the matri of inverse partial Lorent transformation (Fedosin 9a):

3 V V / c c V / c V µ L k c V / c V / c µ dψ Vdψ dψ ddk Lk ddµ dd с V / c c V / c с From (7) takin into account (6) and (5) we obtain the followin relations: dψ γ dm dψ V / c ( Vt) V c + + V c dd. (7) dψ V dd dd. (8) c The first equation in (8) coincides with (4) and the differential vector potential of the point mass is directed alon its motion velocit. After interation of (8) over all point masses inside the ball on the basis of the principle of superposition the standard formulas are obtained for the potentials of ravitational field around the movin ball with retardation of the ravitational interaction taken into account: γ M ψ ψ D V (9) Vt + ( V c )( + ) c Where ψ the scalar potential of the movin ball M the mass of the ball ( ) the coordinates of the point at which the potential is determined at the time t (on the condition that the center of the ball at t was in the oriin of coordinate sstem) D the vector potential of the ball. ψv In (9) it is assumed that the ball is movin alon the ais OX at a constant speed V so that D c D D. With the help of the field potentials we can calculate the field strenths around the ball b the formulas (Fedosin 999): ψ D Ω D () t Where is the ravitational field strenth Ω the ravitational torsion in Lorent-invariant theor of ravitation (ravitomanetic field in ravitomanetism). In view of (9) and () we find: Vt ( V c) 3 [ Vt + ( V c )( + )] ( V c) 3 [ Vt + ( V c )( + )] ( V c) 3 [ Vt + ( V c )( + )] Ω ()

4 γ M V( V c) Ω c Vt V c 3 [ + ( )( + )] V( V c) Ω c Vt V c 3 [ + ( )( + )] The ener densit of the ravitational field is determined b the formula (Fedosin 999): π Vt + V c + ( V c )[ Vt ( V c )( )] u ( + cω ) 8πγ 8 [ ].. () The total ener of the field outside the ball at a constant velocit should not depend on time. So it is possible to interate the ener densit of the field () over the eternal space volume at t. For this purpose we shall introduce new coordinates: V c r θ rsinθcosϕ rsinθsinϕ. (3) cos The volume element is determined b the formula d matri: ϒ J drdθ dϕ r θ ϕ ( ) J ( r θ ϕ) r θ ϕ r θ ϕ where J is determinant of Jacobian It follows that dϒ r sinθ V c drdθ dϕ. The interal over the space of the ener densit () will equal: U b udϒ 8π c V c [ (sin cos )]sin. c + V θ θ θ drdθ dϕ. (4) We shall take into account that due to the Lorent contraction durin the motion alon the ais OX the ball must be as Heaviside ellipsoid the surface equation of which at t is the followin: R V c + + r. (5) After substitutin (3) in (5) it becomes apparent that the radius r at the interation in (4) must chane from R to and the anles θ and ϕ chane the same wa as in spherical coordinates (from to π for the anle θ and from to π for the anle ϕ). For the ener of the ravitational field outside the movin ball we find: U b γ M (+ V 3 c) Ub(+ V 3 c) R V c V c Where Ub is the field ener around the stationar ball. R We can introduce the effective relativistic mass of the field related to the ener of movin ball: We shall now consider the momentum densit of the ravitational field: m b b b c c (6) U V c U (+ V 3 c ). (7) (8) H c

5 Where c H [ Ω ] is the vector of ener flu densit of the ravitational field (Heaviside vector) 4πγ (Fedosin 999). Substitutin in (8) the components of the field () we find: ( V c ) ( + ) V 4 πc [ Vt ( V c )( )] ( ) ( V c ) ( Vt) V 4 πc [ Vt ( V c )( )] 3 ( ) + + ( V c ) ( Vt) V 4 πc [ Vt ( V c )( )] 3 ( ) + +. (9) We can see that the components of the momentum densit of ravitational field (9) look the same as if a liquid flowed around the ball from the ais OX carrin similar densit of the momentum liquid spreads out to the sides when meetin with the ball and meres once aain on the opposite side of the ball. Interatin the components of the momentum densit of the ravitational field (9) b volume outside the movin ball at t as in (4) we obtain: 3 γ M V sin θ drdθ dϕ γ M V P dϒ 4 c V c r. () 3Rc V c π P dϒ P dϒ. In () the total momentum of the field has onl the component alon the ais OX. B analo with the formula for relativistic momentum the coefficient before the velocit V in () can be interpreted as the effective mass of the eternal ravitational field movin with the ball: m pb P V c U () γ M 4 b V 3Rc 3c Where Ub is the ener of the eternal static field of the ball at rest. R Comparin () and (7) ives: 3(+ V 3 c) mpb mb. () 4 The discrepanc between the masses m b and m pb in () shows the eistence of the problem of 4/3 for ravitational field in the Lorent-invariant theor of ravitation. 3. The ravitational Field Inside a Movin Ball For a homoeneous ball with the densit of substance ρ (measured in the comovin frame) which is movin alon the ais OX the potentials inside the ball (denoted b subscript i ) depend on time and are as follows (Fedosin 9b): πγ ρ ( Vt) ψiv ψi R + + D i. (3) V c 3 V c c In view of () we can calculate the internal field strenth and torsion field: 3

6 i ( Vt) 4πγ ρ 3 V c Ω i i 4πγ ρ 3 V c 4πγ ρ V Ω i 3c V c Similarl to () for the ener densit of the field we find: i 4πγ ρ 3 V c 4πγ ρ V Ω i 3c V c π γ ρ [ Vt + ( + V c )( + )] ui ( i + cωi ) 8πγ 9 V c. (4). (5) Accordin to (5) the minimum ener densit inside a movin ball is achieved on its surface and in the center at t it is ero. The interal of (5) b volume of the ball at dϒ r sinθ V c drdθ dϕ equals: t in coordinates (3) with the volume element π γ ρ U u d [ c V (sin θ cos θ)] r sinθ drdθ dϕ 4 i i ϒ + 9c V c. (6) Accordin to the theor of relativit the movin ball looks like Heaviside ellipsoid with equation of the surface (5) at t and in the coordinates (3) the radius in the interation in (6) varies from tor. With this in mind for the ener of the ravitational field inside the movin ball we have: Where U i U i γ M (+ V 3 c) Ui(+ V 3 c) R V c V c is the field ener inside a stationar ball with radius R. R The effective mass of the field associated with ener (7) is: m i i i c c (7) U V c U (+ V 3 c ). (8) Substitutin in (8) the components of the field strenths (4) we find the components of the vector of momentum densit of ravitational field: i 4 πγ ρ ( + ) V 9 c V c i i 4 πγ ρ ( Vt) V 9 c V c 4 πγ ρ ( Vt) V 9 c V c. (9) The vector connectin the oriin of coordinate sstem and center of the ball depends on the time and has the components ( Vt ). From this in the point coincidin with the center of the ball the components of the vector of the momentum densit of the ravitational field are alwas ero. At t the center of the ball passes throuh the oriin of the coordinate sstem and at the time from (9) it follows that the maimum densit of the field momentum 4πγ ρ R V V 9 4 ma 4 c V c πr c V c is achieved on the surface of the ball on the circle of radius R in the plane YOZ which is perpendicular to the line OX of the ball s motion. The same follows from (9). We can interate the components of the momentum densit of ravitational field (9) over the volume inside the movin ball at t in the coordinates (3) similar to (): 4

7 4πγ ρv 4 3 γ M V Pi i dϒ r sin θ drdθ dϕ 9c V c 5Rc V c. (3) Pi i dϒ Pi i dϒ. As in () the total momentum of the field (3) has onl the component alon the ais OX. B analo with () the coefficient before the velocit V in (3) is interpreted as the effective mass of the ravitational field inside the ball: m pi P V c U (3) i γ M 4 i V 5Rc 3c Where Ui is the field ener inside a stationar ball. R Comparin (8) and (3) ives: 3(+ V 3 c) mpi mi. (3) 4 Connection (3) between the masses of the field inside the ball is the same as in () for the masses of the eternal field so the problem of 4/3 eists inside the ball too. 4. Conclusion A characteristic feature of the fundamental fields which include the ravitational and electromanetic fields is the similarit of their equations for the potentials and the field strenths. As it was shown above the eternal potentials (9) of the ravitational (and similarl the electromanetic) field of the movin ball are similar b their form to the potentials of the point mass (point chare) (8) and can be obtained both usin the superposition principle of potentials of the point masses inside the ball and usin the Lorent transformation. We also presented the eact field potentials (3) inside the movin ball for which both the superposition principle and the Lorent transformation are satisfied. From the stated above we saw that the 4/3 problem was common for both the electromanetic and the ravitational field. It also followed from this that considerin the contribution of the ener and the momentum of both fields into the mass of the movin bod were to be done in the same wa takin into account the neative values of the ener and the momentum of ravitational field and the positive values of the ener and the momentum of electromanetic field. References Fedosin S.. (999) Fiika i filosofiia podobiia: ot preonov do metaalaktik Perm: Stle-M. Fedosin S.. (8) Mass Momentum and Ener of ravitational Field Journal of Vectorial Relativit 3 (3) Fedosin S.. (9a) Fiicheskie teorii i beskonechnaia vlohennost materii Perm. Fedosin S.. (9b) Comments to the book: Fiicheskie teorii i beskonechnaia vlohennost materii Perm. Fenman R.P. Leihton R. & Sands M. (964) The Fenman lectures on phsics. Massachussets: Addison- Wesle. (Vol. ). Hajra S. (99) Classical Electrodnamics Reeamined Indian Journal of Theoretical Phsics 4 () 64. Heaviside O. (888/894) Electromanetic waves the propaation of potential and the electromanetic effects of a movin chare Electrical papers Liénard A.M. (898) Champ électrique et Manétique L éclairae électrique 6 (7-9) Searle.F.C. (897) On the stead motion of an electrified ellipsoid The Philosophical Maaine Series 5 44 (69) Wiechert E. (9) Elektrodnamische Elementaresete Archives Néerlandaises

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