Scale. Abstract. 1. Introduction. ction. equations for studied. Rao et al. [16,17] [2], Wagoner. [1], Nordvedt. scale invariant theory.

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1 J. Mod. Phys..,,, do:.436/jmp..37 Publshed Onlne Augus (hp:// Sale Inaran Theory of raaon n Ensen-Rosen Spae-Tme Budua Mshra, Pradyumn Kumar Sahoo, Addepall Ramu Mahemas roup, Brla Insue of Tehnology and Sene, Hyderabad Campus, Andhra Pradesh, Plan, Inda E-mal: {budua, sahoomaku@redffmal.om Reeed June 9, ; resed July 9, ; aeped July 6, Absra In hs paper, we hae suded he perfe flud dsrbuon n he sale naran heory of graaon, when he spae-me desrbed by Ensen-Rosen mer wh a me dependen gauge funon. The osmologal equaons for hs spae-me wh gauge funon are soled and some physal properes of he model are suded. Keywords: Sale Inaran, Spae-Tme, Perfe Flud. Inroduon Seeral new heores of graaon hae been formulaed whh are onsdered o be alernaee o Ensen s heory of graaon. In alernae heores of graa- on, salar ensor heores proposed by Brans and Dke [], Norded [], Wagoner [3], Ross [4], Dunn [5] and Saez and Balleser [6] are mos mporan among hem. In he heory proposed by Brans and Dke [] here ex- ss a arable graaonal parameer. Anoher heory, whh adms a arable, s he sale oaran heory of Canuo e al. [7]. Dra [8,9] rebul he Weyl s un- an addonal gauge funonn. A sale naran ara- on prnple was proposed from whh graaonal and eleromagne feld equaons an be dered. I s on- luded ha an arbrary gauge funon s neessary n all sale-naran heores. fed heory by nrodung he noon of wo mers and I s found ha he sale naran heory of graaon agrees wh general relay up o he auray of ob- seraons made up now. Dra [8,9], Hoyle and Narlkar [] and Canuo e al. [7] hae suded seeral aspes of he sale naran heory of graaon. Bu Wesson s [,] formulaon s so far he bes one o desrbe all he neraons beween maer and graaon n sale free manner. Mohany and Mshra [3] hae sudedd he feasbly of Banh ype VIII and IX spae-mes wh a me de- penden gauge funon and a maer feld n he form of perfe flud. In ha paper, hey hae onsrued a ra- dang model of he unerse for he feasble Banh ype VIII spae-me. Mshra [4] has onsrued he non- sa plane symmer Zeldoh flud model n hs heory wh a me dependen gauge funon. Reenly, Mshraa [5] has onsrued sa plane symmer Zel- sym- doh flud model n sale naran heory. Rao e al. [6,7] hae dsussed ylndrally mer salar meson felds and Brans-Dke salar felds. I s found from he leraure ha he sale naran heory of graaon has no been suded so far n Enswe hae en-rosen spae-me. Hene, nn hs paper, aken an aemp o sudy he ylndrally symmer spae-me n he sale naran heory of graaon. A osmologal model has been presened.. Feld Equaons Wessonn [,] formulaed a sale naran heory of graaon usng a gauge fun x, where, x,,,3,4 are oordnaes n he four-dmensonal spae-me and he ensor feld s denfed wh he meand sale r ensor g. Ths heory s bohh oordnae naran n naure. The feld equaons formulaed by Wesson [,] for he ombned salar and ensor felds are: ;,, j ab, a, b ab ; ab 4 g g g T () wh R Rg () Here s he onenonal Ensen ensor nolng e, on Copyrgh SRes.

2 86 B. MISHRA ET AL. g. Semolon and omma respeely denoe oaran dfferenaon wh respe o g and paral dfferenaon wh respe o oordnaes. The osmologal erm g of Ensen heory s ransformed o Λ Λ g n sale naran heory wh a dmensonless onsan Λ. T s he energy momenum ensor of he 8 maer feld and. 4 The lne elemen for Ensen- Rosen mer wh a s. gauge funon dsw dse (3) wh A B ds e d dr B B r e d e dz (4) E where A and B are funons of only, and s he eloy of lgh. Here we nend o buld osmologal models n hs spae-me wh a perfe flud hang he energy momenum ensor of he form m m m j m T p UU p g (5) j ogeher wh guu where U s he four-eloy eor of he flud; p m and p m are he proper sorop pressure and energy densy of he maer respeely. The non anshng omponens of onenonal Ensen s ensor () for he mer (4) an be obaned as B 4 (6) 4 A4 r (7) A 44 (8) 33 A 44 4 (9) B () 44 4 Here aferwards he suffx 4 afer a feld arable denoes exa dfferenaon wh respe o me only. Usng he omong oordnae frame where U 4, he non-anshng omponens of he feld Equaon () for he mer (3) an be wren n he followng expl form: AB A B pe m A4 e () () 4.e. A k, where k s an negrang onsan. pe m B e 33 (3) AB AB 4 pme B e (4) AB AB AB 4 4 AB me 3 e (5) Equaon () redues he aboe se of Equaons ()-(5) as k B k B pe m e (6) k B 33 pme k B (7) e 4 k B 44 m e 4 4 k B (8) 3 e Now, Equaon () and Equaons (6)-(8) (Wesson []) sugges he defnons of quanes p (auum pressure) and p (auum densy) ha noles neher he Ensen ensor of onenonal heory nor he properes of onenonal maer. These wo quanes an be obaned as: k B e p (9) k B e p () 4 4 k B 4 3 e () I s eden from Equaons (9) and () ha B k sne 4 () where k s an negrang onsan. Usng Equaon () n Equaons (9)-(), he pressure and energy densy for auum ase an be obaned as Copyrgh SRes.

3 B. MISHRA ET AL k k p k e k (3) e 4 k k 3 4 k e k (4) e Here p and p relae o he properes of auum only n onenonal physs. The defnon of aboe quanes s naural as regards o he sale naran properes of he auum. The oal pressure and energy densy an be defned as p pm p (5) (6) m Usng he aforesad defnons of p and p, he feld equaons n sale naran heory.e. (6)-(8) an now be wren by usng he omponens of Ensen ensor (6)-() and he resuls obaned n Equaons ()- (4) as: 3. Soluon k B p e (7) k B k B e (9) B B p e (8) From Equaons (7) and (9), we obaned he equaon of sae p (3) Usng Equaon (7) n Equaon (8), we obaned B d d (3) where d and d are negrang onsans. Subsung Equaon (3) n Equaon (7) and Equaon (9) respeely, he oal pressure p and energy densy an be obaned as: d p d (3) qe where k k q e s a onsan. The realy ondon demands ha d. Usng Equaon (3) n Equaons (3) and (4) respeely and akng, he pressure and energy den- sy orrespondng o auum ase an be alulaed as: p q q (33) q 3 4 q (34) In hs ase, when here s no maer and he gauge funon s a onsan, one reoers he relaon p 8.e. p, whh s he equaon of sae for auum. Here R = onsan, s he osmologal onsan n general relay. Also p beng dependen on he onsans, and, 4 ΛR s unform n all dreons and hene sorop n naure. The osmologal model wh hs equaon of sae s rare n leraure and s known as auum or false auum or degenerae auum model [8-], he orrespondng model n he sa ase s a well known de-ser model. Now he maer pressure and densy an be obaned as: q d pm p p kdd (35) q e 3 q d m 4 kdd (36) q e Now, we hae m as and m as. Also when, m onsan. I s neresng o noe ha he model free from sngulary. So, he Ensen-Rosen ylndrally symmer model n sale naran heory of graaon s gen by he Equaons (), (3) and (3) and he mer n hs ase s dsw k d d d d d d e d dr r e d e dz (37) 4. Some Physal Properes of he Model QT The salar expanson, U; 3 for he model Q gen by Equaon (37) akes he form d dk de (38) Thus, we fnd ( dk de ) as and as. If, d and d k he model represens expandng one for k d ( ). d I s also obsered ha as m onsan as and m. Thus he unerse onfrms he as homogeney naure of he spae-me. R Copyrgh SRes.

4 88 B. MISHRA ET AL.. Followng Rayhaudhur [], he ansoropy an be defned as g,4 g,4 g,4 g33,4 g33,4 g,4 g g g g33 g33 g (39) Consequenly for he model (37), 8 3 dd. So he shear salar remans onsan for and beomes ndefnely large for. The rao of ansoropy o expanson 8 e k d for. Thus here s a sngulary of 3 for k d s no ery large. Moreoer, he model s soropy for fne and does no approah soropy for large alue of. I s obsered ha he ory w anshes whh ndaes ha u s hypersurfae orhogonal. As he aeleraon u found o be zero, he maer parle follows geodes pah n hs heory. 5. Conlusons Eery physal heory arres s own mahemaal sruure and he aldy of he heory s usually suded hrough he exa soluon of he mahemaal sruure. In hs heory blak holes do no appear o exs. If he exsene of blak holes n naure s onfrmed, wll represen a grea suess of general heory of relay. Sne here s no onree edene a presen for he exsene of blak holes, one an ake a sand pon ha blak holes represens a famlar onep of spae me. Therefore he sale naran heory noles gauge heores as relaes o graaonal heores wh an added salar feld. The sgnfane of he presen work deals wh he modfaon of graaonal and geomeral aspes of Ensen s equaons. These are ) sale naran heory of graaon whh desrbes he neraon beween maer and graaon n sale free manner; and ) he gauge ransformaon, whh represens a hange of uns of measuremens and hene ges a general salng of physal sysem. The naure of he osmologal model wh modfed gray ha would reprodue he knemaal hsory and eoluon of perurbaon of he unerse s nesgaed. Here, ylndrally symmer sa zeldoh flud model s obaned n he presene of perfe flud dsrbuon n sale naran heory of graaon. As far as maer s onerned he model does no adm eher bg bang or bg runh durng eoluon ll nfne fuure. The model appears o be a seady sae. 6. Aknowledgemens The auhors are ery muh graeful o he referee for hs aluable suggesons for he mproemen of he paper. 7. Referenes [] C. H. Brans and R. H. Dke, Mah s Prnple and a Relas Theory of raaon, Physal Reew A, Vol. 4, No. 3, 96, pp [] K. Norderd, Jr., Pos Newonan Mer for a eneral Class of Salar Tensor raaonal Theores and Obseraonal Consequenes, The Asrophysal Journal, Vol. 6, 97, pp [3] R. V. Wagoner, Salar Tensor Theory and raaonal Waes, Physal Reew D, Vol., No., 97, pp [4] D. K. Ross, Salar Tensor Theory of raaon, Physal Reew D, Vol. 5, No., 97, pp [5] K. A. Dunn, A Salar Tensor Theory of raaon, Journal of Mahemaal Physs, Vol. 5, No., 974, pp [6] D. Saez and V. J. Balleser, A Smple Couplng wh Cosmologal Implaons, Physs Leers A, Vol. A3, No. 9, 985, pp [7] V. Canuo, S. H. Hseh and P. J. Adams, Sale Coaran Theory of raaon and Asrophysal Applaons, Physs Reew Leers, Vol. 39, No. 88, 977, pp [8] P. A. M. Dra, Long Range Fores and Broken Symmeres, Proeedngs of he Royal Soey of London, Vol. A333, 973, pp [9] P. A. M. Dra, Cosmologal Models and he Large Number Hypohess, Proeedngs of he Royal Soey of London, Vol. A338, No. 65, 974, pp [] F. Hoyle and J. V. Narlkar, Aon a a Dsane n Physs and Cosmology, W. H. Freeman, San Franso, 974. [] P. S. Wesson, ray, Parle and Asrophyss, D. Redel, Dordreh, 98. [] P. S. Wesson, Sale Inaran ray A Reformulaon and an Asrophysal Tes, Monhly Noes of he Royal Asronomal Soey, Vol. 97, 98, pp [3]. Mohany and B. Mshra, Sale Inaran Theory for Banh Type VIII and IX Spae-mes wh Perfe Flud, Asrophyss and Spae Sene, Vol. 83, No., 3, pp [4] B. Mshra, Non-Sa Plane Symmer Zeldoh Flud Model n Sale Inaran Theory, Chnese Physs Leers, Vol., No., 4, pp [5] B. Mshra, Sa Plane Symmer Zeldoh Flud Model n Sale Inaran Theory, Turksh Journal of Physs, Vol. 3, No. 6, 8, pp Copyrgh SRes.

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