Finsler Geometry & Cosmological constants

Transcription

1 Avaabe oe at Peaa esearch Lbrary Advaces Apped Scece esearch, 0, (6):44-48 Fser Geometry & Cosmooca costats. K. Mshra ad Aruesh Padey ISSN: CODEN (USA): AASFC Departmet of Mathematcs, SLIE Deemed Uversty, Loowa, Puab, Ida ABSAC It has bee otced that Fser Geometry fas to sove the probem of eometrzato of the Maxwe Eectrodyamcs 4- dmeso despte the fact that the Fsera metrc tesor s a fucto of the oca state of the physca system. he terest exampe s obvousy ophysca Fsera cocuso about the Laraa of the mass pot, t may be + deree of homoeety fucto of the mass pot veocty. Here the preset commucatos we are aree wth the resuts pubshed by Sye, V.waer, L.Berwad, E.carta, H. Bssema, ad H.ud whch dcate the deep coecto of emaa Geometry wth Fser Geometry athouh ther approaches are dfferet but wth the hep of Fsera metrc tesor we may suest the sht modfcato cosmooca term as dscussed by us ao wth other peer researchers. Key Words : Cosmooy, Cosmooca costat, Fser Geometry Cassfcato:083,085. INODUCION As we ow that Fser Geometry s a theory of spaces where eths are measured sma steps ad the scae of measuremet depeds o the pot of the space ad the choce of drecto at the pot. Whe that the ftesma dstace ds betwee the adacet pots, whose coordates ay system are x ad x +dx (,,3 ) s coected by the reato ds dx dx (,,,3,...). Where the coeffcets are fucto of the coordates x. he quadratc dffereta form the secod member of above reato s caed as emaa metrc ad eometry based upo a emaa metrc s caed emaa eometry. o vestate the coecto of eatvty & Fser eometry frst of a we have to address the probem of eatvty of the Fser Geometry as suested by Sye []. It has bee otced that Fser Geometry fas to sove the probem of eometrzato of the Maxwe Eectrodyamcs 4- dmeso despte the fact that the Fsera metrc tesor s a fucto of the oca state of the physca system. he terest exampe s obvousy o- physca Fsera cocuso about Peaa esearch Lbrary 44

2 . K. Mshra et a Adv. App. Sc. es., 0, (6):44-48 the Laraa of the mass pot, t may be + deree of homoeety fucto of the mass pot veocty. Here we are aree wth the defto that Fser Geometry s a eometry of metrc spaces possess a trsc oca space whose matrces do ot reduce to a quartc form the dfferetas of the coordates. But after certa vestato we ca say that the resuts pubshed by Sye, V.waer, L.Berwad, E.carta, H. Bssema, ad H.ud dcate the deep coecto of emaa Geometry wth Fser Geometry athouh ther approaches are dfferet but wth the hep of Fsera metrc tesor we may suest the sht modfcato the Cosmooca term. hs modfed cosmooca term may hep to dscuss the umerous cosmooca cosequeces wth varabe cosmooca ad ravtatoa costats as pubshed by ahma, Berma, Padey, Chadra & Mshra, Johar & Chadra [-0] Este s Fed equatos wth Laraa he fudameta metrc tesor whch we used the Este s fed equato π (,,,3,4 ) () 8 G c 4.. G May be expressed as... 4 G Ad the C- tesor C off 4 May be wrtte as () (3) ( ) b ( ) bb... (4) GC ( )( )[ b bb b b bb + bb b ] (5) Here the metrc tesor G s caed reuar of the basc tesor has the o- vash determat. & Here the coservato aw s same as Geera eatvty, we have ; 0 (,,, 3, 4 ) (6) o avod the creato or vash of the eery the uverse. he equato (6) whe apped to the fed equato w ve 8πG c ; 4 + Λ ; 0 (,,, 3, 4 ) (7) Whch overs the varato of G & Λ. he fed equatos (6) & (7) are ot derved from the Laraa formuato. If oe attempts to derve the fed equatos va Laraa formuato, oe has to add the term cossts of the frst order parta dervatve of G ad Λ the Laraa desty ad the resut fed equato cotas udetermed costats. Peaa esearch Lbrary 45

3 . K. Mshra et a Adv. App. Sc. es., 0, (6):44-48 N-DIMENSIONAL EINSEIN S EQUAIONS he metrc for a N-dmesoa space tme wth fat N dmesoa foato s: ds dx dx (8) where [ α ( y) β, ± ] (9) β s the -dmesoa metrc for fat Eucda/Mows space & y s a space e/tme e Gaussa orma coordate for upper/ower ss so that ds < 0 s the tme e case. he Chrstoffe symbos are ve by a ac b ( cb, + c, b b, c) (0) So that us metrc equato we have.. r c,, 0, ± () r c r We have 0, 0, 0 α β, 0 Ad α ±α αβ, [ ± α ± α m () α ] β (3) α α α α Where dash represets dervatve w. r. t. Y β (4), Here ema curvature tesor s defed by: σ b σ b, + ba b a (5) σ σ σ a ν ν a a, ν So that (6) N[ ± α α ± α ] β ± Nα β ± α β N α α α α β N β N β α α α α ab he cc tesor s the cotracto of the curvature tesor o the frst ad thrd dces, derved as a (8) a (7) Peaa esearch Lbrary 46

4 . K. Mshra et a Adv. App. Sc. es., 0, (6):44-48 Us these reatos we have m m α α m α β m ± ± N δ α α (9) α ± N α (0) he cc scaar s defed as: () Us above reato we have α α ± N ± N ( N + ) () α α Sce the Este tesor s defed as: G (3) So, G (4) Due to symmetry, G he eery mometum tesor for a scaar fed φ ad poteta V (φ ) s ve by G G (5) p φ φ [ φ pφ + V ] δ (6) So that for φ φ( y), [ ± φ V ] δ + (7) ESULS AND DISCUSSION (a)he hyper surfaces wth costat cosmc tme are maxmay symmetrc subspaces of the whoe space-tme. (b)-not oy the metrc but a the cosmc tesors such as are form varat wth respect to sometrcs of the subspaces. Wth the hep of the resut dscussed above. Peaa esearch Lbrary 47

5 . K. Mshra et a Adv. App. Sc. es., 0, (6):44-48 we may aso cocude that the coordate trasformato must eave the metrc ds dx dx (,,,3,...) form varat. herefore the coordate trasformatos are purey the spata coordate s trasformato as: t t, x x ( x ) (,, 3) I fact the spata coordate s trasformato we may tae the or x 0 to aother pot of the space. c) A mportat cocuso of the above equato (7) s that the cosmooca costat Λ depeds upo G as we as eery mometum cotaed the uverse. Cotract the equato (7) we w obta 8π Λ G, (,,,,3, 4 ) Putt the vaue of ; c 4 from (6), we et Uder dfferet assumptos as pubshed the papers [4-9] the authors have aready costructed sutabe cosmooca modes wth dfferet assumptos of varabe cosmooca costats. Accord to the assumptos fat mode of the uverse s possbe for dfferet postve vaues of ad ope mode & cosed modes are possbe oy for. we have aso cacuated the umerca vaues of costats costats from the observato ad estmated data. EFEENCES [] Sye J.L. Amsterdam, North-Hoad pubsh compay, 958 [] Berma, M. S. Geera. eatvty & Gravtato. 99, 3, pp [3] ahma A., Geera. eatvty & Gravtato., (990),, 655. [4] Pade, H. D., Chadra,. ad Mshra, av Kat, J. Nat. Acad. Math., 997, Vo., pp. 8-. [5] Pade, H. D., Chadra,. ad Mshra, av Kat, Ida J. pure app. Math., 000, 3(), pp. 6. [6] Mshra,. K. & Sh, Amrtbr, Varahmhr Joura of Mathematca Sceces, 008, vo. 8,, pp [7] Mshra.K, Sh A Europea Astroomca Socety pubcatos Seres, (EDP Sceces) 009, Vo. 36, pp. 0-04, [8] Mshra.K, Sh A, Advaces Apped Scece esearch vo., (6), December 0.(o appear) [9] Mshra, av Kat, Sh, Amrtbr & Padey, A. K., Goba Joura of Pure ad Apped Mathematcs, 009, Vo. 5,, pp [0] Johr, V. B. ad Chadra,., J. Math. Phys. Sc. Ida, 983, 7,73. Peaa esearch Lbrary 48