The gravitational field energy density for symmetrical and asymmetrical systems

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1 The ravtatonal eld enery denty or yetrcal and ayetrcal yte Roald Sonovy Techncal Unverty 90 St. Peterbur Rua Abtract. The relatvtc theory o ravtaton ha the conderable dculte by decrpton o the ravtatonal eld enery. Peudotenor t 0 n the oe cae 0 cannot be nterpreted a enery denty o the ravtatonal eld. In [] the approach wa propoed whch allow to expre the enery denty o uch a eld throuh the coponent o a etrc tenor. Th approach baed on the conderaton o the otheral copreon o the layer conted o the ncoherent atter. It wa eploy to the cylndrcally and phercally yetrcal tatc ravtatonal eld. In preented paper the approach developed.. Introducton. The proble o the ravtatonal eld enery dcued a lon te [] [3]. However peudotenor t ν der ro author to author relectn the abuty n denn ravtatonal eld enery denty [3]. In [] the approach ha propoed allow one to expre the enery denty o uch a eld throuh the coponent o a etrc tenor. Th approach baed on the conderaton o the otheral copreon o a layer conted o the ncoherent atter n the eld o the nnteal thn ateral hell by ulllent o the requreent []: (a) the local enery conervaton law hould be ullled and (b) the correpondence prncple hould be ullled ncludn the enery part. In the preented paper proved that th approach can be ued or arbtrary yte. Here proved that the requreent o the nvarance o the ravtatonal eld enery denty [] ullled. For the cylndrcally and phercally yetrcal yte obtaned eld enery denty orula contaned only the etrc tenor coponent and h dervatve.. The derental o the ravtatonal eld enery In [] ha obtaned the orula o the ravtatonal eld enery or the pecal coordnate connected wth type o yetry. Here t condered the orula o the eld enery or the arbtrary tatc coordnate yte. The oluton analoou to one n []... The otheral copreon. Here t condered the oveent o the partcle layer when acqured enery o partcle ha eradated or dpated. The oveent condered a conted o dcrete nnteal tep when the partcle all ree and n end o tep enery o partcle ha dpated. Concrete way o dpaton no dcued. Sucently to uppoe that uch way can be on prncple approxately realzed. For exaple ree all o partcle n thn lay on the old urace wth ollown cooln o the old. The partcle condered a tet-partcle. However the chane o eld caued wth accuulaton o the atter on hell urace calculated ater every tep. Aue x the ntal coordnate yte n 3-pace. Let u conder the dplaceent o the partcle layer ro poton x = x to po- ton x = x + dx dx < 0. The ree partcle all equaton are [5]

2 d L L = 0 dτ x where τ the ntrnc te dx x & = and dτ L σ σ ν 3 ( x ) = ( x x x ) ν For tatc yte ()() lead to C ν () ν = () 0 0 x & = (3) where C 0 contant on all tep. Fro (3) and ro equaton c = 0 + we et o ar a all C 0 = c (5) Fro orula () () or =3 and (5) (6) reult 0 & x = (6) and ro (5)(6) = (7) c δτ where τ the ntrnc te o partcle oveent. Coponent u o the 3 axu velocty o ree partcle all near by pont x r ( x x x ) ay wrtten u = β (8) where ß nnteal coecent.. The tatc ravtatonal eld enery. The enery o the partcle by ree all can obtaned ro the relaton [6] E c u ν 0ν = δ () ν ν dx u = (9) cdτ where δ ret a o the partcle roup. Fro (3) and o ar a 0 = δ the chane o partcle enery on way d x 0ν ν δc = dx (0) Th enery ha dpated on way d x. I the local enery conervaton law ullled then the enery chane o partcle ut reult ro chane o eld enery on way d x. Thereore

3 δc = dx () Fro (8) ollow that the coponent o the o the partcle coordnate ean chane dx = λ () and calar dplaceent equal λ = (3) dl where dl λ = () I ubttute λ n () and then d x n () then we et δc dδ E = dl (5) Here δ and δl are calar by 0 =0 no depend on the pace coordnate tranoraton. The quantty nvarant by the pace coordnate tranoraton x = A x det A = Thereore dδe alo nvarant. 3. The eld ource ravtatonal eld nteral enery. 3.. The object. Condered the tatc eld o the ayetrc convex ooth nntealy thn ateral hell wth urace a denty σ c.the quantty σ c a 3 nle-valued uncton o the coordnate o hell pont x = c xc ( x x ) ( ) σ x x 3 c = σ c. Aued that t nown how the etrc o the urroundn pace throuh σ c to nd. Th poble at leat by en o the coputer ethod [7]. It condered the pace 3 between a hell and oe convex ooth external urace xe = xe ( x x ) wth urace a denty σ ( x x 3 e = σ e ). 3.. The calculaton order. I condered the oton o N j dcrete tet partcle layer ro the external urace to hell. The oton dcrete; the nuber o tep N q. For every layer poton the calculaton ade or N n pont. The every pont poton deterned n coordnate yte x.for every pont P(nq) and every layer j are calculated ro (5) the eld enery derental dδe and ae denty σ(nq+j) or pont P(nq+).Aterward the layer j arrve the poton q = N q etrc coponent calculated or all pont P(nq). The ethod o uch calculaton no condered becaue that doe not atter or the purpoe o th paper. For every pont P(nq) the volue eleent bult at the vector r r 3 d z = dz( dx d x d x ). One de o th eleent dpoed at the layer poton q (6)

4 and the oppote de at the layer poton q+. The vector dl r decrbe the all o partcle ro pont P(nq) up to pont F(nq+) at the layer poton q The ravtatonal eld nteral enery r and enery denty nvarance. r r dl d z d z where Let the volue eleent bult at the vector ( ) r d z dl = (7) Co r r ( dl d z ) Here r Co ( dl dz ) = dl dl dl d x d x d x 3 (8) nvarant by coordnate tranoraton (6). Scalar o the volue eleent dz 3 3 dz dz dz de ( dz ) n coordnate ( ) 3 3 ( n q) a( z z ) dz dz ds = (9) a 3 3 ( ) ( x x ) z z = 3 ( z z ) + ( x x ) 3 ( z z ) + 3 ( x x ) 3 ( z z ) The a o the partcle roup paed throuh th eleent equal ( n q j) = δσ ( n q j) ds( n q) δ (0) where δσ(nqj) the atter denty n the partcle layer j. The a δ o the area eleent o layer whch condered a n the one pont concentrated all ro pont P(nq) n pont F(nq+). By ean nterpolaton can be calculate the a δ(nq+j) and a denty δσ(nq+j).conder the ucceve pa o the layer throuh the area eleent ( dz dz 3 ) wth pont P(nq). By every tep j the eld enery chane n volue eleent equal N j c ds( n q) dl j n q j ( n q) = δσ () where [] depend on (nql). The quantte under the ybol Σ are the nvarant thereore (nq) nvarant. The u o enery n all pont o the eld alo nvarant. The enery denty n pont P(nq) ven by ( q) w n ( n q) = dv ( n q) dv 3 = dz dz dz () where dv(nq) the volue o the volue eleent bult at the vector r r r dl d z d z or tep j=n j ; deternant o the pace etrc coponent. The ( ) 3 quantty dv calar thereore w(nq) nvarant.. The tranoraton o the orula or eld enery and enery denty o the yetrcal yte. The orula or thee quantte n the paper [] antan bede the etrc tenor coponent the eld ource a M and the dtance to yetry centre R. A the etrc tenor coponent are the uncton o M and R t poble to except M and R ro thee orula.

5 .. The cylndrcal yetry. In [] there are the orula a0 R GM z = R a0 = (3) 0 c where R radu R 0 radu o the eld ource M z the lnear a denty. Fro (3) ollow GM z = () R Let the coordnate dx = R dx = Rdφ dx 3 = dz ue. Fro () the a o the cylndrcal area eleent dx dx 3 3 c dx dx δ = d (5) 8πG where d (B) the chane o B caued wth the chane o the eld ource a. I (5) ubttute n () then by = 3 d πg dδ E = (6) where δv = dx dx dx 3. Ater nteraton wth ntal condton a 0 (=0)=0 we obtan or the eld enery 3 δ E = (7) π G and enery denty δe w = δv c = 3πG.. The phercal yetry. In th cae (8) GM = ; dx =R; dx =Rdθ; dx =RSnθdφ (9) c R For the area eleent dx dx 3 by analoy wth (5) 3 c dx dx δ = d (30) 8πG Then or the eld enery dδ E = d ( ) (3) 3πG and ater nteraton δ E = ( ) (3) 6 π G Then the eld enery δe c w = = (33) δv 6πG

6 .3. The dcuon. Thu the obtaned orula or eld enery and eld enery denty antan only the etrc tenor coponent. A t clear ro (7)(8)(3) and (33) thee orula are analoou but have oe derence. Thee derence caued by the or derence o Enten equaton oluton or derence yetre..reerence.r.sonovy.r-qc K.S.Vrbhadra.A coent on the enery-oentu peudotenor o Landau and Lhtz. Phy. Lett.A 57(99) J.Katz. r-qc 059. N.V.Mtzevtch. Phycal eld n eneral relatvty. Naua Moow 5.J.L.Martn. General Relatvty. N.Y A.Lounov. Lecture n relatvty and ravtaton. A odern Loo. Naua Moow L.Lehner. r-qc 00607