Psychoanalysis of Centrally Symmetric Gravitational Field Using a Centrally Symmetric Metric

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1 Psyhoanalysis of Cntrally Symmtri Gravitational Fild Using a Cntrally Symmtri Mtri Muhammad Sajjad Hossain, M. Sahabuddin and M. A. Alim Abstrat Th urrnt study is ondutd to solv som quantitis,, and whih ar funtions of th radial oordinat, and tim oordinat, mploying a ntrally symmtri mtri. In this study, a numbr of nw osmologial solutions hav also bn alulatd. Th physial signifian of th analysis is dissusd in dtaild. Th rsults of this invstigation illustrat that th ntrally symmtri gravitational fild with onstant dnsity automatially tas th shap of th stady stat li univrs. Indx Trms ii tnsor, gnral rlativity, Einstin fild quations, osmology. I. INTODUCTION A ntrally symmtri gravitational fild an b produd by any ntrally symmtri distribution of mattr[]. Th ntrally symmtry of th fild mans th spa tim mtri, that is, th xprssion for th intrval ds, must b th sam for all points loatd at th sam distan from th ntr. In Eulidan spa this distan is qual to th radius vtor. But in a non- Eulidan spa, in th prsn of a gravitational fild, thr is no quantity whih has all th proprtis of th Eulidan radius vtor. Thrfor th hoi of a radius vtor is now arbitrary. Fidlr and Shimming [] prformd th singularity-fr stati ntrally symmtri solutions of som fourth ordr gravitational fild quations. Th fourth ordr fild quations proposd by Trdr with a linar ombination of Bah's tnsor and Einstin's tnsor on th lft-hand sid admit stati ntrally symmtri solutions whih ar analytial and non-flat in som nighborhood of th ntr of symmtry. Vyblyi [3] onsidrd th gnral ntrally symmtri xtrior solutions of th fild quations of th rlativisti thory of gravitation for th stati Shwarzshild and issnr Nordstrom stati filds. Partiular intrior solutions ar found and thy ar fittd to th xtrior solutions. Gn [] invstigatd th strong ntrally symmtri vauum fild in th rlativisti thory of gravitation. In this invstigation, an xat stati solution of th quations of th rlativisti thory of gravitation in vauum for th as of sphrial symmtry that gnralizs Fo s harmoni intrval and also radial motion of tst partils in th gnralizd harmoni mtri ar onsidrd. Th ondition that th fild b physial is usd to find th rgion of appliability of that solution. Th authors Manusript rivd Dmbr 0,0, rvisd Marh 30, 0. M. S. Hossain and M. Sahabuddin ar with th Dpartmnt of Arts and Sins, Ahsanullah Univrsity of Sin and Thnology, Dhaa-08, Bangladsh (-mail: msh80_du@yahoo.om) M. A. Alim is with th Dpartmnt of Mathmatis, Bangladsh Univrsity of Enginring, Dhaa-000, Bangladsh (-mail: a0alim@gmail.om) prsntd that th pitur of th motion of th partils for a distant obsrvr diffrs fundamntally from th on usually adoptd (in Fo s mtri) and dos not orrspond to th pitur of asymptoti slowing down of th ollapsing body. Th nrgy dnsity of th gravitational fild in this as was alulatd. Vry rntly, Samsonov and Ptrov [5] mad an invstigation on th physial intrprtation of th ntral symmtri gravitational-fild singularitis. Th authors showd that th xistn of singularity in th ntrally symmtri gravitational fild, whih is intrprtd as a surfa with unusual physial proprtis, follows from quations for th ation and th nrgy of a tst partil not using Einstin quations and thir solutions. In addition, a bla hol is tratd as a physial modl of th singularity in qustion. To th bst nowldg of th authors, no attntion has bn paid to solv th quantitis and using a ntrally symmtri mtri. Th ntrally symmtri gravitational fild with onstant dnsity automatially tas th shap of th stady stat li Univrs. This ntrally symmtri gravitational fild also satisfis th rigorous thorm nown as th Birhoff s thorm [6], whih stat that any sphrially symmtri vauum solution of Einstin quation is nssarily th Shwarzshild solution that is, stati. This thorm implis that if a sphrially symmtri sour li a star undrgos pulsations or hangs its shap, whil maintaining th sphrially symmtry, it annot radiat any disturbans in th xtrior, namly, Shwarzshild xtrior [7] solution an b usd to dsrib th outsid mtri for svral situations as sphrially symmtri star is ithr stati or it undrgos radial sphrially symmtri gravitational ollaps [8]. II. MATHEMATICAL ANALYSIS Th sphrial spa oordinats r, θ, ϕ ar onsidrd for this prsnt invstigation. Most of th gnral ntrally symmtri xprssion for ds [9] is as follows: (, ) (, ) ds = h r t dr + r t ( sin θdϕ + dθ ) (, ) (, ) + l r t dt + a r t drdt whr a, h,, l ar rtain funtions of th radius vtor r and th tim t. But baus of th arbitrarinss in th hoi of a rfrn systm in th gnral thory of rlativity, w an still hang th oordinats to any transformation whih dos not dstroy th ntral symmtry of ds,this implis, w an transform th oordinats r and t aording to th formula r = f ( r, t ) and t = f ( r, t ), whr f, f ar funtions of th () 98

2 nw oordinats r and t.w ma us of th two possibl transformation of th oordinats r, t in th lmnt of intrval. Firstly, ma th offiint a(r,t) of drdt vanish, and sondly ma th radial vloity of th mattr vanish at ah point ( baus of th ntral symmtry th othr omponnts ar not prsnt). Aftr that, r and t an still b subjtd to an arbitrary transformation of th form r = r (r ) and t = (t ). W now assum th radial oordinat and tim oordinat by and τ and th offiints h,, l by λ,, ν, rsptivly [whr λ,,and ν ar funtions of and τ ]. Thn th quation-() givs ( sin ) ν λ ds = dτ d dθ + θdϕ () Th nrgy momntum tnsor for prft fluid is as follows: ( ) T = p+ uu pg ij i j ij whr p is th prssur and is th nrgy dnsity. In th ommoving rfrn systm th omponnts of th nrgy- momntum tnsor ar: ε, 3 T = T = T = T = p Th non-zro mtri tnsor an b found from quation () as follows ν 00 g00 = g = λ λ g = g = g = g = 33 g33 = sin θ g = sin θ Now w ar intrstd to find out th non-zro valus of Γ as follows, i il Γ = g ( gil, + g j l, j g j, l ) () Substituting i = j = = 0 in quation (), on may gt 0 ν Γ 00 = & [Using quation (3)] Using quations (3) - (), th following rsults an asily b found 0 0 Γ =Γ = ν, Γ =Γ = λ & λ 0 & λ 0 & Γ =, Γ =, 0 & Γ 33 = sin θ, λ λ λ, 33 Γ = Γ = Γ = & Γ 0 =Γ 0 =, Γ =Γ = (3) (5) 3 3 & 3 3 Γ 03 =Γ 30 =, Γ 3 =Γ 3 = (6) 3 3 ν λ ν Γ 3 =Γ 3 = ot θ, Γ 33 =sinθos θ, Γ 00 = Th ii Tnsor is dfind as i i m i m i j j, i j, i j mi ji m =Γ +Γ Γ Γ +Γ Γ With th aid of quations (5), (6) and (7), th ii Tnsor tas th form as blow: νλ & ν & λ & ν = + + νλ νλ νλ ν + νλ ν && λ & ν + & νλ& && ν + && λ && λ λ ν λ & λ & λ& = λ & λν& λ ν ν + + λ ν λ λ && & = + λ & ν& & λ& λ + + λ ν λ && sin θ sin θ = + & ν& sin θ & λ& sin θ λ & sin θ ν sin θ + λ λ λ sin θ sin θ + sin θ (7) (8) (9) (0) () 99

3 0 & ν & λ & = + On th othr hand th ii salar is alulatd as follows: λ λ ν ν λν & λν = ν + && & λ+ && + λ ν λ ν & λ 3 & λ λ & ν& + & λ& + λ λ ν + Th Einstin fild quation is dfind as g = T ij ij ij Th following rsults an b found taing suitabl valus of i and j in quation (3) as: 00 g00 = T 00 g = T g = T 33 g33 = T 33 0 g0 = T 0 () (3) () Thrfor th fild quations ar obtaind from quations (8)-() and () as follows: λ & + & λ& + λ 3 ε + λ = ν ν λ ν λν && & & & (5) && λ & λ & λ & ν& λν& (6) = p && & & ν& & λ& + + λ λ ν + + = p (7) λ + ν 3& && + & ν& (8) = p & & ν & λ + & = 0 as λ 0 (9) Th symbol prim dnots diffrntiation with rspt to whil dot rprsnts diffrntiation with rspt to τ. Th Einstin s fild quation for th mtri g ij in th prsn of mattr an b dfind as folows: ij gij = T ij (0) Equation (0) givs, i i Gj = T j () Diffrntiat quation () with rspt to i, w gt [5] i 0 = Tj, i For onsrvation th rlation is as follows: i T = ji, 0 i i.. T, = 0 Givs, ( gt i ) gl l T = 0 i g x x () Thrfor for = 0 th quation () givs th following rsult: ( & λ+ & + & ν) ε + & ε ( & νε & λp & p) = 0 & & ε i.. λ+ & = (3) p + ε If w onsidr th dnsity is onstant vrywhr, i.. = onstant so that = & 0, th quation (3) boms, & ξ + & = 0 i.. ξ f () t + = () But in th bginning of th univrs i.. at tim t = 0, th quation () boms, ξ + = 0 (5) W nd on mor quation whih is providd by th quation of stat P = P(), in whih th prssur is givn as a funtion of th mass-nrgy dnsity. This quation of stat is as follows: 00

4 ( ξ ) P=, whr ξ (6) Now for = th first part of quation (3) is givn by, ( λ ν ) + + p + p And th nd part of th quation (3) an b xprssd as, νελp p. Hn for = th rsult of quation () an b writtn as: ( λ ν ) + + p+ p νελp p = 0 So, w gt, p ν = p + ε (7) If p is nown as a funtion of thn quation (3) an b intgratd in th form dε λ+ = + f ( ) (8) p + ε whr th funtions f ( ) an b hosn arbitrary in viw of th possibility mntiond abov of maing arbitrary transformations of th form = ( ). Now, at th initial momnt t = 0 th quation (8) givs f ( ) = 0 III. FINDINGS Cas-I: Whn ξ =, p= 0 i.. for dust li sphr or th prssur lss mattr, th quation (8) boms, λ+ = d i.. = ξ+, whn ξ + = 0. Thrfor, = onstant (9) Cas-II: Whn ξ = 3, P = ε i.. for radiation dominant univrs, 3 th quation (8) boms, d λ+ =3 =, whn ξ + = 0. 3 i.. ξ+ Thrfor, = onstant (30) Cas-III: Whn ξ =, P = i.. for stiff fluid, th quation (8) boms, d λ+ = ( ) i.. = ξ+, whnξ + = 0. Thrfor, = onstant (3) Again, for th quation (7), w gt th following ass: Cas-I: Whn ξ =, p= 0 i..for dust li sphr or th prssur lss mattr, th quation (7) boms, ξ = Constant (3) Cas-II: Whn ξ = 3, P = ε i.. for radiation dominant univrs, 3 th quation (7) boms, α. ξ This implis, = onstant (33) Cas-III: Whn ξ =, P= i.. for stiff fluid, th quation (7) boms, α ξ This implis, = onstant (3) IV. CONCLUSION W hoos th intrval ds in th form of quation (), thr still rmaind th possibility of an arbitrary transformation of th tim of th form t = f(t) li (). Suh a transformation is quivalnt to adding to ν an arbitrary funtion of th tim, and with its aid w an always ma f(t) in quation () and vanish if w onsidr t = 0. And so without any loss in gnrality, w an gt ξ +µ=0. Not that th ntrally symmtri gravitational fild with onstant dnsity in ah solution from quations (9) to (3) automatially tas th shap of th stady stat Univrs. Sin th Univrs is xpanding, th prinipl dmands that nw mattr must b ratd to maintain a onstant dnsity of th Univrs [0]. Th most rmarabl fatur of th thory is that th nw mattr (blivd to b hydrogn atom) is supposd to b ratd out of nothing in a ration fild alld th C fild, that is, Mattr rquirs to b ontinuously ratd in th Univrs aording to this thory []. EFEENCES [] L. D. Landau, E. M Liftshitz., Th lassial thory of Filds, Pragamon prss, 975, unsolvd problm pp

5 [] B. Fidlr,. Shimming, Singularity-fr stati ntrally symmtri solutions of som fourth ordr gravitational fild quations, Astronomish Nahrihtn, vol. 30, No.5, pp. 9, 983. [3] Yu. P. Vyblyi, Cntrally symmtri solutions of th quations of th gravitational fild in th rlativisti thory of gravitation, Tort. Mat. Fiz., vol. 88, No., pp. 35 0, 99. [] A. V. Gn, Strong ntrally symmtri vauum fild in th rlativisti thory of gravitation, Tort. Mat. Fiz., vol.87, No., pp. 30 0, 99. [5] V. M. Samsonov, E. K. Ptrov. On th physial intrprtation of th ntral symmtri gravitational-fild singularitis, Physis of Partils and Nuli Lttrs, vol. 8, No., pp. 8-, 0. [6] G. D Birhoff, lativity and Modrn Physis, Harvard Univrsity Prss, Cambridg M.A., 93. [7] S. Winbrg,, J Wily, Sons, Gravitation and Cosmology; Prinipls and appliations of th gnral thory of lativity, 9, pp-333. [8] P. S. Joshi.. Global Aspts in Gravitation and Cosmology, Oxford Univrsity Prss ln, Nw Yor, 993. [9] M. Ali, Studis on Global Aspts in Gravitation and Gravitational Wavs. M. S. Thsis Dpartmnt of Mathmatis, Univrsity of Chittagong, Chittagong, Bangladsh, 00. [0] B. lyr ; N. Muunda, and C.V. Vishvrshwasa, Gravitation, Gaug thoris and th Eashy Univrs, Kluwr Aadmi Publishrs, th Nthrlands, 988, pp. 3-. [] B. Basu. An introdution to Astrophysis. Prnti Hall of India Privat limitd Nw Dlhi-000, 006. Muhammad Sajjad Hossain has bn srving as a Lturr in Mathmatis with th Dpartmnt of Arts and Sins, Ahsanullah Univrsity of Sin and Thnology (AUST), Dhaa, Bangladsh. H joind at AUST in Sptmbr 00. Bfor joining AUST h was a tahr of Univrsity of Asia Paifi, Dhaa, Bangladsh for mor than 0 (two) yars. Part-tim faulty of Prmir Univrsity, Chittagong, Bangladsh, AM of Azim Group, Chittagong and also S.O. of Agrani Ban, Artilari Branh, Bangladsh. H also arnd thr Gold Mdals and UGC mrit sholarship. H has alrady ompltd his MBA from UAP. Now h is ontinuing his M. Phil in BUET. M.Sahabuddin has bn srving as a Profssor of Mathmatis with th Dpartmnt of Mathmatis Ahsanullah Univrsity of Sin and Thnology (AUST), Dhaa, Bangladsh. M. A. Alim has bn srving as a Assoiat Profssor of Mathmatis with th Dpartmnt of Mathmatis, Bangladsh Univrsity of Enginring and Thnology (BUET), Dhaa, Bangladsh. 0

Lecture 14 (Oct. 30, 2017)

Lecture 14 (Oct. 30, 2017) Ltur 14 8.31 Quantum Thory I, Fall 017 69 Ltur 14 (Ot. 30, 017) 14.1 Magnti Monopols Last tim, w onsidrd a magnti fild with a magnti monopol onfiguration, and bgan to approah dsribing th quantum mhanis