4/3 Problem for the Gravitational Field

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1 4/3 Proble for the Gravitational Field Serey G. Fedosin PO bo 6488 Sviazeva str. -79 Per Russia E-ail: Abstrat The ravitational field potentials outside and inside a unifor assive ball were deterined usin the superposition priniple the ethod of retarded potentials and Lorentz transforations. The ravitational field strenth the torsion field the enery and the oentu of the field as well as the effetive asses assoiated with the field enery and its oentu were alulated. It was shown that 4/3 proble eisted for the ravitational field as well as in the ase of the eletroaneti field. Keywords: enery oentu theory of relativity ravitation field potentials. Introdution In field theory there are a nuber of unsolved probles whih need deeper analysis and loial understandin. An eaple is the proble of hoosin a universal for of the stress-enery tensor of the body whih would inlude the rest enery of the substane as well as the field enery and at the sae tie would provide an univoal onnetion with therodynai variables of the substane in the lanuae of four-vetors and tensors. Another interestin proble is 4/3 proble aordin to whih the effetive ass of the body field whih is alulated throuh the field oentu and the effetive ass of the field found throuh the field enery for soe reason do not oinide with eah other with the ratio of the asses approiately equal to 4/3. The proble of 4/3 is known for a lon tie for the ass of eletroaneti field of a ovin hare. Joseph John Thoson Geore Franis FitzGerald Oliver Heaviside Geore Frederik Charles Searle and any others write about it (Heaviside 888/894 (Searle 897 (Hajra 99. We also disuss this question with respet to the ravitational field of a ovin ball (Fedosin 8. Now we present a ore aurate desription of the proble not liited to the approiation of sall veloities.. Methods In the alulation of the enery and the oentu of ravitational field of a unifor assive ball we will use the superposition priniple by eans of suin up the field eneries and oenta fro all point partiles forin the ovin ball. This approah is reasonable in the ase of a weak field when the eneral theory of relativity hanes to ravitoanetis and the ovariant theory of ravitation to the Lorentz-invariant theory of ravitation (Fedosin 9a. The field equations then beoe linear allowin the use of the superposition priniple. We will note that the ravitational field an be onsidered weak if the spaetie etri differs insinifiantly fro the Minkowski spaetie etri (the spaetie etri of the speial theory of relativity. If the effets of ravitational tie dilation and sizes ontration are sinifiantly less than the siilar effets due to the otion veloity of the referene frae under onsideration then this ravitational field an be onsidered weak. 3. Results and Disussions 3. The Gravitational Field Outside a nifor Massive Ball We will first define the ravitational field potentials for a ball ovin at a onstant veloity V alon the ais OХ of the referene frae K. We will proeed fro the so-alled Liénard-Wiehert potentials P a e

2 (Liénard 898; Wiehert 9 for any point partiles that ake up the ball. Popular presentation of the proble (for the eletroaneti field an be found in Feynan s book (Feynan at all Siilarly to this the differential salar Liénard-Wiehert potential for the ravitational field fro a point partile with ass dm has the followin for: dm d ( r V r / Where is the ravitational onstant is the veloity of ravitation propaation vetor r is the vetor onnetin the early position of the point partile at tie t and the position r ( y z at whih the potential is deterined at tie t. In this ase the equation ust hold: r t t. ( The eanin of equation ( is that durin the tie period t t the ravitational effet of the ass dm ust over the distane r at veloity up to the position r ( y z so that at this position the potential d would appear. Suppose there is ontinuous distribution of point partiles and at t these partiles are desribed by the oordinates ( y z and the enter of distribution of point partiles oinides with the oriin of the referene frae. Then at tie t the distribution enter of the point partiles would ove alon the ais OX to the position Vt and the radius vetor of an arbitrary partile of distribution would equal r ( Vt y z. At the early tie t the position of this point partile is speified by the vetor r ( Vt y z. Sine r we an write down: r = r r and r ( t t aordin to ( then for the square r ( Vt ( y y ( z z ( t t. (3 The riht side of (3 is a quadrati equation for the tie t. After we find t fro (3 we an then find r fro (. If we onsider that in ( the produt of vetors is V r = V ( Vt then substitutin r also in ( we obtain the followin epression (Fedosin 9b: d dm ( V t V ( y y ( z z V. (4 Aordin to (4 the differential ravitational potential d of the point ass dm at the tie t P a e

3 durin its otion alon the ais OX depends on the initial position ( y z of this ass at t. If we use the etended Lorentz transforations for the spatial oordinates in (4: V t V y y y z z z (5 and then let the veloity V tend to zero we obtain the forula for the potential in the referene frae K the oriin of whih oinides with the point ass dm : d dm y z. (6 In (6 in the referene frae K the vetor r ( y z at the proper tie t speifies the sae point in spae as the vetor r ( y z in the referene frae K at the tie t. If we introdue the ravitational four-potential D с D inludin the salar potential and the vetor potential D (Fedosin 999 then the relation between the salar potential (6 in the referene frae K and the salar potential (4 in the referene frae K an be onsidered as the onsequene of etended Lorentz transforations in four-diensional foralis whih are applied to the differential four-potential of a sinle point partile. These transforations are arried out by ultiplyin the orrespondin transforation atri by the four-potential whih ives the four-potential in a different referene frae with its own oordinates and tie. Sine in the referene frae K the point ass is at rest its vetor potential is dd and the d four-potential has the for: dd. In order to ove to the referene frae K in whih the с referene frae K is ovin at the onstant veloity V alon the ais OX we ust use the atri of inverse partial Lorentz transforation (Fedosin 9a: L k V V / V / V V / V / 3 P a e

4 d V d d ddk Lk dd dd. (7 с V / V / с Fro (7 takin into aount (6 and (5 we obtain the followin relations: d d dm V / ( V t V ( y y ( z z V dd d V dd ddz. (8 The first equation in (8 oinides with (4 and the differential vetor potential of the point ass is direted alon its otion veloity. After interation of (8 over all point asses inside the ball on the basis of the priniple of superposition the standard forulas are obtained for the potentials of ravitational field around the ovin ball with retardation of the ravitational interation taken into aount: M V t V y z ( ( V D (9 Where the salar potential of the ovin ball M the ass of the ball ( y z the oordinates of the point at whih the potential is deterined at the tie t (on the ondition that the enter of the ball at t was in the oriin of oordinate syste D the vetor potential of the ball. V In (9 it is assued that the ball is ovin alon the ais OX at a onstant speed V so that D D D. With the help of the field potentials we an alulate the field strenths around the ball y z by the forulas (Fedosin 999: D G D t ( Where G is the ravitational field strenth the ravitational torsion in Lorentz-invariant theory of ravitation (ravitoaneti field in ravitoanetis. In view of (9 and ( we find: 4 P a e

5 G z G G y M Vt V ( 3 [ Vt ( V ( y z ] M y V ( 3 [ Vt ( V ( y z ] M z V ( 3 [ Vt ( V ( y z ] M z V ( V y Vt V y z 3 [ ( ( ] M yv ( V z Vt V y z 3 [ ( ( ] ( The enery density of the ravitational field is deterined by the forula (Fedosin 999: 3 V t V y z M ( V [ V t ( V ( y z ] u G. ( 8 8 [ ( ( ] The total enery of the field outside the ball at a onstant veloity should not depend on tie. So it is possible to interate the enery density of the field ( over the eternal spae volue at t. For this purpose we shall introdue new oordinates: V r y rsin os z rsin sin. (3 os. The volue eleent is deterined by the forula d J dr d d where J is deterinant of Jaobian atri: J r ( y z y y y. ( r r z z z r It follows that d r V dr d d sin. The interal over the spae of the enery density ( will equal: b M u d 8 V [ (sin os ] sin V dr d d. (4 r 5 P a e

6 We shall take into aount that due to the Lorentz ontration durin the otion alon the ais OX the ball ust be as Heaviside ellipsoid the surfae equation of whih at t is the followin: y z R V. (5 After substitutin (3 in (5 it beoes apparent that the radius r at the interation in (4 ust hane fro R to and the anles and hane the sae way as in spherial oordinates (fro to for the anle and fro to for the anle. For the enery of the ravitational field outside the ovin ball we find: b M ( V 3 b ( V 3 (6 R V V Where b M is the field enery around the stationary ball. R We an introdue the effetive relativisti ass of the field related to the enery of ovin ball: b V ( V 3. (7 b b We shall now onsider the oentu density of the ravitational field: H (8 Where H [ G ] is the vetor of enery flu density of the ravitational field (Heaviside 4 vetor (Fedosin 999. Substitutin in (8 the oponents of the field ( we find: M ( V ( y z V 4 [ V t ( V ( y z ] 3 (9 y M ( V ( V t y V 4 [ V t ( V ( y z ] 3 z M ( V ( V t z V 4 [ V t ( V ( y z ] 3 We an see that the oponents of the oentu density of ravitational field (9 look the sae as if a liquid flowed around the ball fro the ais OX arryin siilar density of the oentu liquid spreads. 6 P a e

7 out to the sides when eetin with the ball and eres one aain on the opposite side of the ball. Interatin the oponents of the oentu density of the ravitational field (9 by volue outside the ovin ball at t as in (4 we obtain: 3 M V sin dr d d M V P d 4 V r 3R V. ( Py yd Pz zd. In ( the total oentu of the field has only the oponent alon the ais OX. By analoy with the forula for relativisti oentu the oeffiient before the veloity V in ( an be interpreted as the effetive ass of the eternal ravitational field ovin with the ball: pb P V ( M 4 b V 3R 3 Where b M R is the enery of the eternal stati field of the ball at rest. Coparin ( and (7 ives: b 3( V 3 pb. ( 4 The disrepany between the asses b and pb in ( shows the eistene of the proble of 4/3 for ravitational field in the Lorentz-invariant theory of ravitation. 3. The Gravitational Field Inside a Movin Ball For a hooeneous ball with the density of substane (easured in the oovin frae whih is ovin alon the ais OX the potentials inside the ball (denoted by subsript are as follows (Fedosin 9b: i depend on tie and ( Vt i R y z V 3V V i i D. (3 In view of ( we an alulate the internal field strenth and torsion field: G i 4 V t 3 V G yi 4 y 3 V G zi 4 z 3 V i yi 4 zv 3 V zi 4 yv 3 V. (4 7 P a e

8 Siilarly to ( for the enery density of the field we find: [ V t ( V ( y z ] ui Gi i. (5 8 9( V Aordin to (5 the iniu enery density inside a ovin ball is ahieved on its surfae and in the enter at t it is zero. The interal of (5 by volue of the ball at t in oordinates (3 with the volue eleent d r V dr d d sin equals: u d [ V (sin os ] r sin dr d d 4 i i 9 V. (6 Aordin to the theory of relativity the ovin ball looks like Heaviside ellipsoid with equation of the surfae (5 at t and in the oordinates (3 the radius in the interation in (6 varies fro to R. With this in ind for the enery of the ravitational field inside the ovin ball we have: i M ( V 3 i( V 3 (7 R V V Where i M is the field enery inside a stationary ball with radius R. R The effetive ass of the field assoiated with enery (7 is: i V ( V 3. (8 i i Substitutin in (8 the oponents of the field strenths (4 we find the oponents of the vetor of oentu density of ravitational field: i 4 ( y z V 9 ( V yi 4 ( V t yv 9 ( V zi 4 ( Vt zv 9 ( V. (9 The vetor onnetin the oriin of oordinate syste and enter of the ball depends on the tie and has the oponents ( Vt. Fro this in the point oinidin with the enter of the ball the oponents of the vetor of the oentu density of the ravitational field are always zero. At t the enter of the ball 8 P a e

9 passes throuh the oriin of the oordinate syste and at the tie fro (9 it follows that the aiu density of the field oentu 4 RV M V 9 ( 4 ( a 4 V R V is ahieved on the surfae of the ball on the irle of radius R in the plane YOZ whih is perpendiular to the line OX of the ball s otion. The sae follows fro (9. We an interate the oponents of the oentu density of ravitational field (9 over the volue inside the ovin ball at t in the oordinates (3 siilar to (: 4 V 4 3 M V Pi i d r sin dr d d 9 V 5R V. (3 Pyi yi d Pzi zi d. As in ( the total oentu of the field (3 has only the oponent alon the ais OX. By analoy with ( the oeffiient before the veloity V in (3 is interpreted as the effetive ass of the ravitational field inside the ball: pi P V (3 i M 4 i V 5R 3 Where i M R is the field enery inside a stationary ball. Coparin (8 and (3 ives: i 3( V 3 pi. (3 4 Connetion (3 between the asses of the field inside the ball is the sae as in ( for the asses of the eternal field so the proble of 4/3 eists inside the ball too. 4. Conlusion A harateristi feature of the fundaental fields whih inlude the ravitational and eletroaneti fields is the siilarity of their equations for the potentials and the field strenths. As it was shown above the eternal potentials (9 of the ravitational (and siilarly the eletroaneti field of the ovin ball are siilar by their for to the potentials of the point ass (point hare (8 and an be obtained both usin the superposition priniple of potentials of the point asses inside the ball and usin the Lorentz transforation. We also presented the eat field potentials (3 inside the ovin ball for whih both the superposition priniple and the Lorentz transforation are satisfied. Fro the stated above we saw that the 4/3 proble was oon for both the eletroaneti and the ravitational field. It also followed fro this that onsiderin the ontribution of the enery and the oentu of both fields into the ass of the ovin body were to be done in the sae way takin into aount the neative values of the enery and the oentu of ravitational field and the positive values 9 P a e

10 of the enery and the oentu of eletroaneti field. Referenes Fedosin S.G. (999 Fizika i filosofiia podobiia: ot preonov do etaalaktik Per: Style-MG. Fedosin S.G. (8 Mass Moentu and Enery of Gravitational Field Journal of Vetorial Relativity 3 ( Fedosin S.G. (9a Fiziheskie teorii i beskonehnaia vlozhennost aterii Per. Fedosin S.G. (9b Coents to the book: Fiziheskie teorii i beskonehnaia vlozhennost aterii Per. Feynan R.P. Leihton R. & Sands M. (964 The Feynan letures on physis. Massahussets: Addison-Wesley. (Vol.. Hajra S. (99 Classial Eletrodynais Reeained Indian Journal of Theoretial Physis 4 ( 64. Heaviside O. (888/894 Eletroaneti waves the propaation of potential and the eletroaneti effets of a ovin hare Eletrial papers Liénard A.M. (898 Chap életrique et Manétique L élairae életrique 6 ( Searle G.F.C. (897 On the steady otion of an eletrified ellipsoid The Philosophial Maazine Series 5 44 ( Wiehert E. (9 Elektrodynaishe Eleentaresetze Arhives Néerlandaises P a e

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