(1) For the static field a. = ), i = 0,1,3 ; g R ( R R ) 2 = (2) Here 3 A (3)

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1 Title: The ravitation enery or a ylindrially and spherially symmetrial system Authors: oald Sosnovskiy (Tehnial University, 9, St. Petersbur, ussia It has been shown that t omponent o the enery-momentum pseudotensor in the ase o ylindrially symmetrial stati ravitational ield annot be interpreted as enery density o the ravitation ield. An approah has been suested allows one to express the enery density o the ylindrially or spherially symmetrial stati ravitation ield in terms o the metri tensor omponents. The approah based on the onsideration o the proess o isothermal ompression o a ylinder onsisted o inoherent matter The ravitation enery or a ylindrially and spherially symmetrial system oald Sosnovskiy Tehnial University, 9, St. Petersbur, ussia rosov@yandex.ru It has been shown that t omponent o the enery-momentum pseudotensor in the ase o ylindrially symmetrial stati ravitational ield annot be interpreted as enery density o the ravitation ield. An approah has been suested allows one to express the enery density o the ylindrially or spherially symmetrial stati ravitation ield in terms o the metri tensor omponents. The approah based on the onsideration o the proess o isothermal ompression o a ylinder onsisted o inoherent matter..introdution The relativisti theory o ravitation has an intrinsi ontradition, whih arise rom the at that the enery-momentum tensor is a nonovariant value. This ontradition produes some diiulties in the onsequent approah to the enery onservation priniple. This issue is a matter o disussion up to now [], []. As it is shown below in this paper, the t omponent o the enery-momentum pseudotensor or a ylindrially symmetrial ravitation ield annot be interpreted as the enery density o the ravitation ield. In this paper the approah is proposed allows one to express the enery density o suh a ield throuh the omponents o a metri tensor. This approah based on the onsideration o the isothermal ompression o the ylinder onsisted o the inoherent matter It has been shown [] that the lenth element or the ylindrially symmetrial system an be represented as ds dt + d + dφ + dz ( For the stati ield a i ii ( Here x a, i,,3 ; ( T ; x A ; x ( A ( A + 3 ; a 3 Φ ; x 3 A 33 a ( Z ; a + 3 ( A and is the radius o the ylinder. Formulas (, (3 orrespond to the ero salar urvature in the whole exterior o the ravitatin ylinder. A + 3 ; a (3

2 The hoie o the Laranian o the external ravitational ields is a matter o some diiulties. It is obvious that the metrial Laranian L ~ is equal to ero in the exterior o the ylinder. It is also the ase or L ~ µν (Γ α µ ν Γβ αβ - Γα β µ Γβ αν and the ormal enerymomentum pseudotensor density tensor [] t ν µ αβ,µ L / αβ,ν - ν µ L is equal to ero. On the other hand, the enery o the test partiles movin in the external ravitational ield o ylinder is not onstant. As well as the only soure o the enery variation o the test partile is the ravitation ield enery, the value t annot be onsidered as a density w o this enery. oreover, the value t does not ulill the orrespondene priniple (relativisti [5], aordin to whih in the limitin ase o weak ield all equations should rade into equations o the Newton theory. In partiular the enery density o the ield o the ylinder should transorm into w N - G /πr ( is a linear mass density o a ylinder. Another orm o the t ν µ 6πG t b b + b b + lm np il km,n,p lm,l,m 8 b has been suested in [6] : b il km,l,m + b ln lm,p il km lm nr pq ( ( b b np b pm,n qr il kn mp kl in mp ( b b + b b pq mn nr,p, l,l, m mn,p,l + ( Here b and index ",i" means the simple x i derivative. It an by easily obtained rom ( and ( that or a ylinder t 6 G π a ( 3a + a (5 It an be seen that in this ase t t, but in the limitin ase (no ravitin mass at all and a. The density o ield enery in this limitin ase should be ero, but aordin to (5 it does not. Thereore, t rom ( also annot be interpreted as a true density o ravitational ield enery. Nevertheless the expression or the density o the ravitational ield enery is neessary or the onsideration o the set o problems, or example in the lobal strins problem [7]. In this paper some approah or obtainin o suh expression is proposed. This approah an also be applied to spherially symmetrial system.. The enery density o the stati ravitation ield. The approah proposed in this paper based on the onsideration o proess o the isothermal staewise ompression o the hollow ylinder onsisted o inoherent matter whih produes ravitational ield. The dereasin o the ylinder radius is onsidered to be result o the onseutive displaement o the symmetrial ininitesimal layers ormed by partiles. The sel-ield dependene o the layer movement an be neleted. The proess onsidered to be isothermal, what means that the enery related to the partiles movement dissipated. The expressions we are oin to derive should ulill ollowin requirement: (a in the absene o mass enery density should be equal to ero (b the orrespondene priniple should be ulilled inludin the enery part ( the loal enery onservation law should be ulilled.

3 The ( requirement in the limitin ase o small masses is a orollary o (b and we suppose that it is orret or lare mass i the ield enery is loalied. Let us onsider the displaement o the partiles layer rom position x x to the position x x +, <. The ree partiles motion equations are d L dτ x& µ L x µ where τ is the intrinsi time, x µ µ and dτ L σ σ µ ν ( x& x x& x&, µν (7 For the radial movement x T& &, x & 3 &, x & x&. It an be shown rom ( and (7 that L x& + & + x& & x x (9 where ii ii (x due to (. For µ and initial onditions x x and (6 - (9 lead to ( x ( x ( ( x ( x The enery o the partile with rest mass δm an be obtained rom the relation [] (6 (8 E δm u i i, u i i ( dτ For i in (, the hane in enery near ( as [] : de δm ( x x an be obtained rom ormulas ( and, ( ( x The whole mass o the ravitatin matter an be expressed in ield produed by it as n α Γ ndsα πg (3 For the ylindrial symmetry α, n, Γ, and usin omponents ( one an be obtain rom (3 or the ylindrial surae a G/ ( This parameter does not depend on the mass distribution over. Equations (8, (9, ( and ( lead to the equation or the enery o the ree-movin partile whih satisy the (b requirement

4 E (, G (, δm δm Here U is the Newton potential. ln U Let mass onsider to be uniormly distributed over the ylindrial surae with radius. In the area < there is no ield. The layers disussed above move one by one rom this surae. The linear mass density onsider to be d. When the layer passes the area d ield hanes only in ylindrial area between and -d beause parameter a does not depend on, but it depends on the linear mass density inside the urrent area with radius. The requirement ( results in de + de where de is the hane o ield enery over the unit lenth o the ylinder. When δm d it an be seen rom ( and (5 that d E (5 d (,, d (6 (, where is the urrent linear density o mass inside the area with radius. The enery de is loalied in the layer d. Sequentially movin mass rom to in this way we an et the ield enery in the ylindrial layer as E d, (, (, d or takin into aount ( and (, E G ln( From (6 the ield enery density w(, is w с, (, d S (, ln (, (, d (7 (, (8 where S is the lateral area o the surae Const per lenth unit [] 3 S (9 Formula (8 an be rewritten usin (, ( and (9 as w( G, ( π Thus w(, is equal to the Newton s ield enery density. This result an be onneted with at, that salar urvature and the spae-time is lat. Obviously, the requirement (b is ulilled. It should be mention that ( does not restrit the enerality o the result obtained beause any initial oordinate system in the system with ylindrial symmetry an be transormed so that ( is ulilled. There are ood reasons to take into aount the intrinsi enery o ravitational ield. Partiularly, in [8] the ravitatin mass o the entral body or the spherially symmetrial system has been expressed as the sum o the own mass o the body and its ravitational exess, i.e. the mass o the ield. Nevertheless in the exterior o the body the enery o the ield has not been onsidered as a sours o the ield. The onsistent implementation o the approah to the whole enery o both the matter and the ield enery as a soure o ield has been provided [9] or the Newton ravitation. In [] the enery-momentum tensor has been onsidered as the soure o metri. The approah

5 disussed above in the paper an also be applied or suh onsideration. In this ase (6 beome (, + (, ln d (, ( where (, E (, / is the ield mass in the area < < and is the rest mass o the unit lenth o the ylinder. The equation ( results in v e (, ; v ( G ln( v When - so that the ield enery is inite and the whole enery o both matter and ield tend to. However dierenes between + and within the bounds o the osmoloial horion < 8 m beome notieable only when > /m. By suh reat distanes the dependene o ravity rom distane may be haned [].Thereore the ondut o quantities by it is neessary to onsider with the prudene 3. Spherial symmetry For the spherial symmetry ( substituted by -G/ ; - (-G/ - ; - ; 33 - Sin θ ( where is the whole mass o spherially distributed matter inside the sphere with radius. The ravitation ield in the exterior o the ravitatin sphere does not depends o the -distribution o the mass inside the sphere. In this mainin a spherial ield is similar to the ylindrial one. Thas, the equations (8, (9, (, (5, (7, (8 an be applied to a spherially symmetrial ield i one substitute the masses o the unit lenth o ylinder, by the mass o the interior o a sphere,. Then (5, ( ives the ollowin expression or the enery o a ree partile E δm G G U G where U is the Newton potential. From (8 and ( the enery density o the spherially symmetrial ield an be represented as (3 G G G w ln 6πG + ( 8π For the whole exterior o the body ( mass o the ield is equal to ( [ ( G ln( G ], -G /. All ormulas above ulill the orrespondene priniple (b. The assumption that the whole enery o ield and matter as the soure o ield and (,(3 lead to

6 (, G[ ' + (, '] d 'ln G ' The ield mass is essential omparin with the entral body mass when is about the ravitational radius. éérenes. A. Lounov. Letures in relativity and ravitation. A modern look. Nauka, oskow, 99...B. ensky. Phys. Lett. A 38 ( E. Frehland. Communi. in ath. Phys. 6 ( P.A.. Dira. General Theory o relativin. N.Y., N.V. itkevith. Physial ields in eneral relativity. Nauka, oskow, L.D. Landau, E.. Lishit. Field theory. Nauka, oskow, Orti., ui- ui F. Global strins oupled to ravity. In book "The ormation a. evolution o osmi strins". Cambr.univ.press,99, I.D. Novov, V.P. Fedorov. Physis o blak holes. Nauka, oskow, D. Giulini. Phys.Lett. A3 ( A.I. Nishov.arXiv:r-q/37v(3.A.A.Kirillov.arXiv: hep-th/86(,astro-ph/6(.j.l.syne. elativity: The eneral theory. Amsterdam, 966