The gravitational field energy density for symmetrical and asymmetrical systems

Transcription

1 Th ravtatonal ld nry dnsty or symmtral and asymmtral systms Roald Sosnovsy Thnal Unvrsty 1941 St. Ptrsbur Russa Abstrat. Th rlatvst thory o ravtaton has th onsdrabl dults by dsrpton o th ravtatonal ld nry. Psudotnsor t n th som ass annot b ntrprtd as nry dnsty o th ravtatonal ld. In [1] th approah was proposd whh allow to xprss th nry dnsty o suh a ld throuh th omponnts o a mtr tnsor. Ths approah basd on th onsdraton o th sothrmal omprsson o th layr onsstd o th nohrnt mattr. It was mploy to th ylndrally and sphrally symmtral stat ravtatonal ld. In prsntd papr th approah s dvlopd. 1. Introduton. Th problm o th ravtatonal ld nry dsussd a lon tm [] [3]. Howvr psudotnsor t ν drs rom author to author rltn th ambuty n dnn ravtatonal ld nry dnsty [3]. In [1] th approah has proposd allows on to xprss th nry dnsty o suh a ld throuh th omponnts o a mtr tnsor. Ths approah basd on th onsdraton o th sothrmal omprsson o a layr onsstd o th nohrnt mattr n th ld o th nntsmal thn matral shll by ulllmnt o th rqurmnts [4]: (a) th loal nry onsrvaton law should b ullld and (b) th orrspondn prnpl should b ullld nludn th nry part. In th prsntd papr provd that ths approah an b usd or asymmtral systms. Hr provd that th rqurmnt o th nvaran o th ravtatonal ld nry dnsty [4] ullld. For th ylndrally and sphrally symmtral systms s obtand ld nry dnsty ormulas ontand only th mtr tnsor omponnt and hs drvatv.. Th drntal o th ravtatonal ld nry In [1] has obtand th ormulas o th ravtatonal ld nry or th spal oordnats onntd wth typ o symmtry. Hr t onsdrd th ormula o th ld nry or th arbtrary stat oordnats systms. Th soluton s analoous to on n [1]..1. Th sothrmal omprsson. Hr t onsdrd th movmnt o th partls layrs whn aqurd nry o partls has radatd or dsspatd. Th movmnt onsdrd as onsstd o dsrt nntsmal stps whn th partls all r and n nd o stp nry o partls has dsspatd. Conrt ways o dsspaton no dsussd. Suntly to suppos that suh way an b on prnpl approxmatly rald. For xampl r all o partls n thn lay on th sold sura wth ollown ooln o th sold. Th partls onsdrd as tst-partls. Howvr th han o ld ausd wth aumulaton o th mattr on sold sura alulatd atr vry stp. α Assum x s th ntal oordnat systm admssbl or th systm onuraton wth mtr =. Lt us onsdr th dsplamnt o th partls ν layr rom poston x = x1 to poston x = x1 + dx. Th r partls all quatons ar [5] d L L = (1) dτ x whr τ s th ntrns tm dx x & = and dτ

2 σ σ 1 ν 1 3 ( x ) = ( x x x ) L ν For stat systm (1)() lad to C ν ν = () x & = (3) whr C s onstant on all stp. From (3) and rom quaton = + w t so ar as s small (4) C = (5) From ormulas (1) () or =13 rsult 1 & x = (6) and rom (5)(6) 1 = (7) δτ whr τ s th ntrns tm o partls movmnt. Componnts u o th 1 3 maxmum vloty o r partls all nar by pont x r ( x x x ) may wrttn u = β (8) whr ß s nntsmal ont... Th stat ravtatonal ld nry. Gnral ormulas. Th nry o th partls by r all an obtand rom th rlaton [6] E m u ν ν = δ ν ν dx u = (9) dτ whr δm s rst mass o th partls roup. From (3) and so ar as ν = δν th han o partls nry on way d x s = dx (1) Ths nry has dsspatd on way d x. I th loal nry onsrvaton law ullld thn th nry han o partls must rsult rom han o ld nry on way d x. Thror = dx (11) From (8) ollow that th omponnts o th o th partl oordnats man han s dx = λ (1) and salar dsplamnt s qual

3 λ = (13) dl whr dl λ = (14) I substtut λ n (1) and thn dδe d x n (11) thn w t = dl (15) Hr δm and δl ar salars by = no dpnd on th spa oordnats transormaton. Th quantty s nvarant by th spa oordnats transormaton x = A x (16) Thror dδe s also nvarant. 3. Enry o th asymmtral ravtatonal ld Th objt. Consdrd th stat ld o th asymmtr onvx smooth nntsmally thn matral shll wth sura mass dnsty σ.th quantty σ s a snl-valud unton o th oordnats o shll pont x = x ( x x ) σ ( x x 3 ) = σ. It onsdrd th spa btwn a shll and som onvx smooth xtrnal sura x x x x σ = x x 3. Assumd that t nown how = ( ) wth sura mass dnsty σ ( ) th mtr o th spa ron btwn x and x to nd. Ths s possbl at last by mns o th omputr mthods [7]. 3.. Th alulatons ordr. Is onsdrd th moton o N j dsrt tst partls layrs rom th xtrnal sura to th shll. Th moton s dsrt; th numbr o stps s N q. For vry layr poston th alulatons mad or N N n ponts. Th vry pont poston dtrmnd n oordnat systm x. For vry pont P(n th volum lmnt s bult at th vtors r r 1 3 d = d( dx d x d x ) =13. Lt ths vtors rat th oordnat systm d ( n j) dx = = B (17) On sd (d d 3 ) o ths lmnt s dsposd at th layr poston q and th oppost sd at th layr poston q+1. Th vtor dl r dsrb th all o partl rom pont P(n up to pont F('n'q+1) at th layr poston q+1. For vry pont P(n and vry layr j ar alulatd rom (15) th ld nry drntal dδe and mass dnsty σ('n'q+1j) or pont P('n'q+1). Atrwards th layr j arrv th poston q = N q mtr omponnts alulatd or all ponts P(n. Th mthod o suh alulaton no onsdrd baus that dos not mattr or th purpos o ths papr Th ravtatonal ld ntral nry and nry dnsty nvaran. Lt th volum lmnt s bult at th vtors d. Th mass o th partls roup passd throuh ths lmnt s qual ( n j) = δσ ( n j) ds( n q j) δ m (18) whr δσ(nj) s th mattr dnsty n th partls layr j and ds(nj) th ara o th lmnt (d d 3 ). Th mass δm o ths lmnt whh s onsdrd as n th on pont onntratd all rom pont P(n n pont F(nq+1) wth oordnats + dl. Componnts dl an b alulatd n oordnats d rom (1) (17). Componnt dl 1 = d 1 and

4 dl p 1 p = dl =3 (19) 1q q By mans ntrpolaton an b alulat th mass δm(nq+1j) and mass dnsty δσ(nq+1j) or ponts P(nq+1). Consdr th sussv pass o th layrs throuh th lmnt o ara ( d d 3 ) wth pont P(n. From (15)(17) atr stp j = N j th ld nry han n volum lmnt s qual N j ds δσ dl s = s j n j ( n whr [] dpnd on (nj). Th quantts undr th symbol Σ ar th nvarants thror (n s nvarant. Th sum o nry n all ponts o th ld s also nvarant. Th nry dnsty n pont P(n s vn by ( n w ( n = dv ( n dv () 1 3 = я d d d (1) whr dv(n s th volum o th volum lmnt bult at th vtors r r r d d d or stp j=n j ; s dtrmnant o th mtr omponnts. Th quantty ( 1 ) 3 dv s salar thror w(n s nvarant. Thus th approah basd on th onsdraton o th sothrmal omprsson o th layr onsstd o th nohrnt mattr an b usd or asymmtral systms. 4. Th transormaton o th ormulas or ld nry dnsty o th symmtral systms. Th ormulas or ths quantts n th papr [1] mantan bsds th mtr tnsor omponnts th ld sour mass M and th dstan to symmtry ntr R. As th mtr tnsor omponnts ar th untons o M and R t s possbl to xpt M and R rom ths ormulas Th ylndral symmtry. In [1] thr ar th ormulas a R 4GM = R a = () whr R radus R radus o th ld sour M th lnar mass dnsty. From (3) ollow GM = (3) R 1 4 and nry dnsty GM w = πr 4 = 3 G π 4.. Th sphral symmtry. From [1] n ths as 1 GM = 1 ; dx 1 =dr; dx =Rdθ; dx 3 =RSnθdφ (5) R and nry dnsty (4) с = GM GM ln πGR R R w (6)

5 or rom (5) 4 4 с 1 1 w = ln + 1 (7) 16πGR 3π 4.Rrns ( ) [ ] 1 G 1.R.Sosnovsy.r-q 5716.K.S.Vrbhadra.A ommnt on th nry-momntum psudotnsor o Landau and Lsht. Phys. Ltt.A 157(1991)195 3.J.Kat. r-q N.V.Mtvtsh. Physal lds n nral rlatvty. Naua Mosow J.L.Martn. Gnral Rlatvty. N.Y A.Lounov. Lturs n rlatvty and ravtaton. A modrn Loo. Naua Mosow L.Lhnr. r-q 167