Einstein s Field Equations in Cosmology Using Harrison s Formula

Transcription

1 Ein s ild Equaions in Cosmoloy U Haison s omula Ioannis Ialis Haanas, Dpamn of Physis and sonomy Yo Univsiy, Tno, Onaio, Canada ioannis@you.a bsa Th mos impoan l fo h sudy of h aviaional fild in Ein s hoy of aviy is his fild quaions. In his sho pap, w dmonsa h divaion of Ein fild quaions fo h dman osmoloial modl u h obson- Wal mi, and fuhmo Haison s fomula fo h ii nso. Th diffn is ha Haison s fomula is an aually sho way of obainin h fild quaions. Th advana is ha h Cisoffl symbols do no hav o b dily alulad on by on. This an aually b a vy usful dmonsaion fo sombody who would li o undsand a slihly diffn bu fas way of divin h fild quaions, somhin ha is aually aly sn in many of undadua and vn adua xbs. Inoduion In 95 Ein pu h finishin ouhs o h nal Thoy of laiviy. Th famous Shwazshild soluion was h fis physially sinifian soluion of h ild Equaions of nal laiviy. I had showd how spa-im is uvd aound a sphially symmi disibuion of ma. This poblm was solvd by Shwazshild and is aually a loal poblm, in h sns ha h disoions of spa-im omy fom h Minowsi omy of Spial laiviy adually diminish o zo as w mov fuh and fuh away fom h aviain sph. In nal, any spa-im omy nad by suh a loal disibuion of ma is xpd o hav h sam popy. I is also nown ha Nwonian aviy podus an analus sul. Th obson-wal mi and h Ein quaions Th obson-wal mi o lin lmn is fundamnal in h sandad modls of osmoloy. Th mahmaial famwo in whih h obson-wal mi ous is ha of nal laiviy and as h fom: ν ( x dx dx ds ( ν ν whih an b fuh win as:[] d ds d ( ( d dφ ( wh: x, x, x, x φ, and ( is h sal fao of h univs ofn alld h xpansion fao and has h dimnsions of h, an hav h valus of, -, ospondin o h h diffn ind of mis. On of h mos impoan quaniis ha hy hav o b alulad in nal laiviy is h ii nso whih is dfind blow:

2 λ λ ν λ λ σ σ λ ν ν λσ λ νσ ( λ ν x x wh: s a h so alld Cisoffl symbols of h sond ind, whih a dfind as: σ σν σλ νλ νλ σ (4 λ ν x x x If w dno h dminan of ν as a maix hn h ii nso an b win as follows: λ [ ] [ ] λ σ ν ν λ νσ (5,λ,ν Bu ou mi ν onains only diaonal lmns ( ν (5 i an b fuh win as: λ [ ] [ ] λ σ λ,λ, σ (6 s a final sp w wi h omponns in ms of h appopia s and h dminan wih ou indis unnin h valus,,,. So w hav: [] [] o o o ( ( ( ( ( (6a f obainin h omponns ν h sala uvau invaian an also b dmind: ν ν (7 inally h Ein nso an b fomd:[4],[5] 8π δ T,,,, 4 (8 wh: δ is h Con dla, T T, T, T, T a h diaonal lmns of h ny momnum nso, and will b dfind la. In h as of vauum T ν. o Haison s omula In Landau and Lifshiz i is ivn ha fo a mi in whih ν fo ν whih w an psn h lmns as follows: and, and o l l,,. (9 Haison s fomula now ivs h omponns of h ii nso in h followin laion: [6]

3 ( [ (] l,, l, l l l,, l ( l and ( l is ivn by: [6] ( l, l ( l, l, l, l, l, l m, l m, l subsips pdd by a omma a dnoin odinay diffniaion wih sp o h ospondin dina. 4. pplyin Haisson s omula L us fis dfin h omponns of h mi nso appain in (: (, ( (,, ( Thn fom (9 w also hav ha: ( (,, Th omponn of h ii nso an b win as: ( ( (, ( ( ( ( ( ( Th ( l offiins boms: ( ( ( ( ( Tain h appopia divaivs, as symbolizd, w an wi: (4 (5 Symbolizin h divaivs w obain: (6

4 4 [ ] [ ] ( ( ( ( ( ( [ ] s ( ( ( ( ( ( (7 and (7, boms: (8 and simplifyin w obain: (9 I should b nod ha sands fo ( vywh in ou alulaions. Similaly h omponn of h ii nso boms: (. ( Similaly as bfo h l offiins bom:

5 5 ( ( uhmo: ( ( ( Tain h divaivs as indiad and muliplyin by h ( found abov w obain: ( whih as h fom of: (4 In h sam way w an wi down h wo mainin omponns of h ii nso: ( ( ( (5 and

6 6. ( ( ( (6 inally h omponn boms: os os ( (7 Nx h omponn will b: ( ( ( (8 Similaly h offiins fo his omponn a ivn by: ( os os ( ( (9 Thfo: (

7 7 5 Calulaion of h sala Now h sala quaniy an b alulad: o o ( and so w obain: ( 6 Calulaion of h Ein Tnso Componns Th mixd omponns of h Ein nso a ivn by: 4 T 8π δ ( Thfo w obain hm fo h fou diffn omponns: [7] T and [7] T, 8 [7], p T T ρ ε πt πt (4 7 Conlusions By main us of Haison s fomula h ovaian omponns of h ii nso w divd. On hi xpssions w obaind h mixd omponns of h Ein nso w also alulad. Th mi usd was h sandad obson-wal mi of h dman osmoloial modl. In my opinion his mhod is fas and has h advana of allowin fo h alulaion of h omponns of h ii nso in suh a way ha alulaion of vy individual Cisoffl symbol is dily avoidd. This mhod lavs lss m fo ompuaional o in an ohwis mo hy

8 8 alulaion. inally i an b said ha h quaions divd a wih hos of sandad osmoloy. fns [] D Sabbaa V., aspini M., Inoduion o aviaion, Wod Sinifi, 985, pa 7-7. [] Islam J., N., n Inoduion o Mahmaial Cosmoloy, Cambid Univsiy Pss, 99, pa 4. [] Islam J., N., n Inoduion o Mahmaial Cosmoloy, Cambid Univsiy Pss, 99, pa 5. [4] Nalia J., V., Inoduion o Cosmoloy, Cambid Univsiy Pss, 99, pa 6-7. [5] Sioff I., S., Tnso nalysis, Thoy and ppliaions, John Wily and Sons, In., 95, pa: 75. [6] L. D. Landau and E. M. Lifshiz, Th Classial Thoy of ilds, Pamon Pss, 98, pa 67. [7] Nalia J., V., Inoduion o Cosmoloy, Cambid Univsiy Pss, 99, pa 7.