DIFFERENTIAL GEOMETRY AND MODERN COSMOLOGY WITH FRACTIONALY DIFFERENTIATED LAGRANGIAN FUNCTION AND FRACTIONAL DECAYING FORCE TERM

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1 DIFFERENTIAL GEOMETRY AND MODERN COSMOLOGY WITH FRACTIONALY DIFFERENTIATED LAGRANGIAN FUNCTION AND FRACTIONAL DECAYING FORCE TERM EL-NABULSI AHMAD RAMI Plasma Applcaon Laboraory, Deparmen of Nuclear and Energy Engneerng and Faculy of Mechancal, Energy and Producon Engneerng, Cheju Naonal Unversy, Ara-dong, Jeju , Souh Korea; Emal: Receved Ocober 9, 6 Some neresng aspecs and feaures of geodesc equaon wh fraconal decayng force erm n Remannan dfferenal geomery and modern cosmology were dscussed whn he framewor of fraconally dfferenaed Lagrangan funcon formulaed recenly by he auhor. Key words: fraconal acon-le varaonal approach, Remann geomery on manfolds, 4-dmensonal Lorenzan manfolds, cosmology.. INTRODUCTION I s well beleved oday ha fraconal calculus s a que rreplaceable means for descrpon and nvesgaon of classcal and quanum complex dynamcal sysem wh holonomc as well as wh nonholonomc consrans []. The fraconal dervaves and negrals descrbe more accuraely he complex physcal sysems and a he same me, nvesgae more abou smple dynamcal sysems. Fraconal dervaves and negrals have recenly been appled o many problems n physcs, fnance and hydrology, polymer physcs, bophyscs and hermodynamcs, chaoc dynamcs, chaoc advecon, random Brownan wals, modelng dsperson and urbulence, vscoelascally damped srucures, conrol heory, ransfer equaon n a medum wh fracal geomery, sochasng modelng for ulraslow dffuson, nec heores, sascal mechancs, dynamcs n complex meda, wave propagaon n complex and fracal meda, asrophyscs, cosmology, ec. Dealng wh fraconal dervaves s no more complex han wh usual dfferenal operaors. Today here exs many dfferen forms of fraconal negral operaors, rangng from dvded-dfference ypes o nfne-sum ypes, ncludng Grunwald-Lenov fraconal dervave, Capuo fraconal dervave, ec. bu he Remann-Louvlle Operaor s sll he mos See [] and references heren. Rom. Journ. Phys., Vol. 5, Nos. 3 4, P , Buchares, 7

2 468 El-Nabuls Ahmad Ram frequenly used when fraconal negraon s performed. The sudy of fraconal calculus opened new branches of hough and flls n he gaps of radonal sandard calculus n ways ha as of ye, no one compleely assmlaes or undersands. Alhough varous felds of applcaon of fraconal dervaves and negrals are already well done, some ohers have jus sared n parcular he sudy of fraconal problems of he Calculus of Varaons (COV) and respecve Euler-Lagrange ype equaons s a subjec of curren srong research and nvesgaons. In , F. Rewe used he COV wh fraconal dervaves and consequenly obaned a verson of he Euler-Lagrange equaons (ELE) wh fraconal dervaves ha combnes he conservave and non-conservave cases []. In, anoher approach was developed by M. Klme by consderng fraconal problems of he COV bu wh symmerc fraconal dervaves and corresponden ELE's were obaned, usng boh Lagrangan and Hamlonan formalsms [3]. In, O. Agrawal exended Klme problem and proved a formulaon for varaonal problems wh rgh and lef fraconal dervaves n he Remann-Louvlle sense [4]. In 4 he ELE's of Agrawal were used by D. Balng and T. Avar o nvesgae problems wh Lagrangans whch are lnear on he veloces [5]. In all he above menoned sudes, ELE's depend on lef and rgh fraconal dervaves, even when he problem depend only on one ype of hem. In 5, M. Klme suded problems dependng on symmerc dervaves for whch ELE's nclude only he dervaves ha appear n he formulaon of he problem [6]. The major problem wh all hese approaches s he presence of non-local fraconal dfferenal operaors and he adjon of a fraconal dfferenal operaor used o descrbe he dynamcs s no he negave of self. Oher complcaed problems arse durng he mahemacal manpulaons as he appearance of a very complcaed Lebnz rule (he dervave of produc of funcons) and he non-presence of any fraconal analogue of he chan rule. In general, he physcal reasons for he appearance of fraconal equaons are long-range dsspaon and nonconservaon. Recenly, we proposed a novel approach nown as fraconal acon-le varaonal approach (FALVA) o model nonconservave dynamcal sysems where fraconal me negral nroduces only one parameer α whle n oher models an arbrary number of fraconal parameers (orders of dervaves) appear [7 ]. The derved Euler-Lagrange equaons are smlar o he sandard one bu wh he presence of fraconal generalzed exernal force acng on he sysem. No fraconal dervaves appear n he derved equaons. The conjugae momenum, he Hamlonan and he Hamlon s equaons are shown o depend on he fraconal order of negraon α and vary as nverse of me. In he presen wor, we wll dscuss whn he same framewor, some neresng fraconal feaures of geomery on Remannan and 4-dmensonal Lorenzan manfolds and show he mporance of he fraconal varaonal approach o explan some neresng cosmologcal aspecs

3 3 Dfferenal geomery and modern cosmology 469 n agreemen wh asrophyscal observaons. The paper s organzed as follows: n secon II, we revew he basc posulaes of fraconal funconal acon negral; n secon III, we dscuss a smple applcaon: he problem of arc lengh,.e. he shores lne beween wo pons on a Remann manfold wh a posve defne merc whch enables us o nroduce he fraconal geodesc equaon. In secon IV, we rea he fraconal geodescs n Eucldean space and dscuss some of s mporan feaures. In secon V, we show he mporance of fraconal geodesc equaons n Remann geomery and n parcular, n modern cosmology. Three epochs are dscussed: he maer, he radaon and he nflaon epochs. Fnally, conclusons are gven n secon VI.. THE FRACTIONALLY DIFFERENTIATED LAGRANGIAN FUNCTION APPROACH In hs secon, we revew brefly wo basc pons of he fraconal acon-le varaonal classcal approach n classcal and feld heory: he resuled Euler-Lagrange equaons and he volaon of Noeher's conservaon heorems. The Remann-Louvlle fraconal negral s defned by: α ( = ) τ α I f τ = f ()( τ), Γα < α <. () = Consder a smooh Remann manfold M and le L be an admssble smooh Lagrangan funcon L: R TM R. Gven a Remann-Louvlle fraconally dfferenaed Lagrangan funcon S <α< on he se of pahs q( τ ), τ beween wo gven pons A= q ( ) and B= q(),.e. a funcon on he angen bundle TM. For any pecewse smooh dfferenable pah q: [, ] M, he fraconally dfferenaed Lagrangan funcon S o L s defned by [, ] <α< [ ] α S<α< q = L( q( τ), q( τ), τ)( τ) = L( q, q, τ) dg ( τ), Γα R = = α where L( q, q, τ) s he Lagrangan weghed wh ( τ) Γ( α) () and g α ( ) α wh he scalng properes g α g, Γ α τ = τ μ μτ = μ τ μ >. In realy, we consdered a smooh acon negral (a me smeared The reader s referred o [7 ] and references heren for more deals.

4 47 El-Nabuls Ahmad Ram 4, R ) whch can be rewren as he srcly sngular Remann-Louvlle ype fraconal dervave Lagrangan measure dg ( τ) on he me nerval [ ] β (,)[ ] ( (), (), ) S q= D Lq q = β β β τ ( () () ) ( () () ) = L q, q, L q, q,, and hereby rereved he sandard acon negral or funconal negral. In hs wor, we have β = α, α (, ). Such ype of funconals s nown n mahemacal economy, descrbng, for nsance, a so called dscounng economcal dynamcs. The rue fraconal dervaves are also ofen, nowadays, used for descrbng so called dsspave srucures appearng n nonlnear dynamcal sysems and ec. The problem now s o fnd he pahs q ( τ) for whch mae S <α< saonary. For hs, we consder a one-parameer famly of pahs q= q τ, ξ,,, A= q, ξ o τ β ξ β of class C n ( τ ξ ) from B = q, ( ξ ) n agreemen wh a gven pah = ( τ) q q for ξ =. Consequenly, he fraconal acon a he gven pah q( ξ, τ ) becomes a fraconal funcon of ξ f q q(, ) S an exremum, ha s ( ds d ξ ) = for all τ = τ maes <α< <α< ξ= such one-parameer varaons q( ξ, τ ) of q( τ ). Le x x(, ) pon of q( τξ, ). Se L L( x, ) coordnae sysem on M and ζ = = τξ be he coordnae = ζ evaluaed a ζ=x where = x x s a dx are funcons on angen vecors. Then he nal curves x = x ( τ) sasfes he modfed Euler-Lagrange equaons (MEL) L d L = α L F x ζ τ ζ ζ, α, (3) Proof. We evaluae ds<α< d ζ and we change he order of dfferenaon wh respec o τ and ξ and we negrae by par o fnd easly: [ ] d S<α< L x dζ = τ= = Γα ζ ξ τ= α d L ( ) L α L α τ α τ ( τ) x Γα x ζ ζ ξ The frs negral s zero because of he boundary condons x( ξ,) = x( A) and x( ξ, ) = x( B). In fac d[ S ] d ζ mus vansh a ξ = for all x = x(, ξ) <α<

5 5 Dfferenal geomery and modern cosmology 47 sasfyng he boundary condons. By assumng x ( τξ ) = x ( τ ) ξy ( τ) hen x ξ a ξ = can be any C funcon y mes. Thus y ( τ),, τ vanshng a he boundary α d L ( ) L α τ α ( τ) Γα d = τ ζ ζ L α, τ y τ d τ= x from whch we deduce equaons (3). The new erm F' on he RHS of equaons (3) can be nerpreed as he fraconal frconal force whch are a common ype of non conservave force. Tha s by reang he acon as a fraconal negral, a lnear me-decreasng dsspave force erm appears. When α =, we fall no he sandard acon and he laer fals o descrbe dsspave sysems. The mporan benef of he fraconal acon s ha maes naurally he appearance of he me-dependen dsspave erm n he dynamcal equaons whou nroducng any auxlary coordnae n he Lagrangan or usng Raylegh dsspaon funcon and especally whou allowng he appearance he fraconal dervaves n he Lagrangan and he Hamlonan. If ( x, x ) are holonomc local coordnaes on TM such ha ητ= ( x ( τ) ) and ητ = ( x ( τ) ), hen η s a soluon of he fraconal sysems of nonlnear ordnary dfferenal equaons n one of he followng forms: L j (, j, d L ζ x τ j ) ( ζ, xj, τ L j ) = α ( ζ, xj, τ), x ζ τ ζ (, x, ) (, x, ) (, x, ) L l j j L l j j L j j l l ζ ζ τ ζ ζ τ ζ ζ ζ x x ζ τ α L j ( ζ, x j, τ ) =. τ ζ (4) (5) =,, n = dmm. Do denoes me dervave wh respec o τ. We dscuss now he energy problem whn he framewor of fraconal acon prncple. E E p,v L = L p,v and defned n a The energy = assocaed o coordnae sysem ( x, ζ ) by:

6 47 El-Nabuls Ahmad Ram 6 E = L ζ L. (6) ζ s no a consan of moon. p and v are respecvely he momenum and he velocy. Proof. If x = x ( τ) sasfes he fraconal Euler-Lagrange equaons ζ= hen E = E( x, ζ) : TM R sasfes: represened by (3), and we ae x, de d L d L = x L x L x L x L x d d = d x τ τ ζ τ ζ ζ ζ d L L = x α L x =. x ζ τ ζ The Noeher's symmery heorems are volaed. Conservaon occurs when α = or when τ. In oher words, les assume ha L x =. Ths effecvely means ha he Lagrangan has ranslaon symmery n he (generalzed) coordnae x or n oher words L( x, x, x x ) L ( x, x, x ). Δ = = By fraconal Euler-Lagrange equaons (3), we fnd α p τ where p =L x. Hence fraconal Euler-Lagrange equaons n classcal mechancs mples ha ranslaonal nvarance of a generalzed coordnae mples ha he generalzed momenum p s a no a consan of moon, or a conserved quany. The fraconally dfferenaed Lagrangan funcon () s quas-nvaran under he nfnesmal ε -parameer ransformaons up o a gauge erm Λ unless (,, ) ( q) O τ=τεκ τ, ε, (7) ( τ ) = ( τ ) εω ( τ ) ( ε ) q q, q O, (8) α L τ q τ q τ τ d τ = (9) (,, α α = L τ q τ q ( τ) )( τ ) ε( τ) dλ ( τ, q( τ), q( τ )) O( ε ). If he fraconally dfferenaed Lagrangan funcon () s nvaran up o a gauge erm Λ wh he fac ha ω ( q κ), hen (,, ) ( (,, ) ) L τ q τ q τ = L τ q τ q τ q,

7 7 Dfferenal geomery and modern cosmology 473 (,, ) L τ q τ q τ q ( q q ) (, q ) ω τ (,, ) L τ q τ q τ L( τ, q( τ), q ( τ) ) q κ( τ, q ) q Λ τ, τ, τ, () s a consan of moon. When he Lagrangan s no a funcon of q, he consan of moon oo he form: ( ) ( ) L τ, q τ, q τ L τ, q τ, q τ ( α). q q τ L τ q τ q τ q s consan and L s conserved. For α =,,, () 3. THE PROBLEM OF ARC LENGTH AND THE MODIFIED GEODESIC EQUATION As a smple applcaon and of parcular neres smulaneously s he problem of arc lengh,.e. he shores lne beween wo pons on a Remann manfold wh a posve defne merc. In hs case, we need o ae L = P whch s a funcon on angen vecors evaluaed a he velocy vecor q ( τ) of q( τ ). The correspondng quadrac form P s ds = j gjdx dx where g = g x, xj []. In hs way, he fraconal acon negral j [ ] α S<α< q = P( τ), Γα () s ndependen of he parameerzaon of q( τ ). Consequenly, [ ] d S dζ = = = <α< P α = P ( τ) Γα ξ ξ= ξ= P α ( τ), Γα ξ (3) assumng P for ξ =. Thus, we can replace L = P by L = P = g ζ ζ and he speed P accordng o heorem s no consan. Le M be a smooh j j

8 474 El-Nabuls Ahmad Ram 8 Remann manfold. A pah q= q ( τ) maes he fraconal acon negral () saonary f and only f s paramerc equaon x = x ( τ) n any coordnae sysem ( x ) sasfes he equaon: gj dx dx j d g dx dx = α g. x τ Consequenly, le U be an open subse of R n. Gven a pah γ: [, ] (4) U where we may defne he fraconal lengh of γ as (). Then γ sasfes he dfferenal equaon where γ j α γ γ γ j τ j, d d d d Γ =, (5) Γ r j = gjr jgr r gj g, (6) r s he Chrsoffel symbol. The one for geodesc moon s j d x α dx Γ dx dx. j = τ Wren as a sysem of frs order equaons, he negral curves are: dx = v, dv α v Γ v v j j =. τ Equaon (9) can be wren n fac as: (7) (8) (9) dv = Γ vv j v vv j j α Γ j F, () τ where F α v, () τ s he fraconal decayng forcng erm. From a conrol heory pon of vew, F s he correspondng npu wea decayng vecor feld [3]. In fac, by defnng x σ dxσ dt = yσ, equaon (5) s dencal o a Langevn equaon wh a me-dependen frcon erm n case a random source characerzng he properes of medum where moon occurs s appled (for example a random or sochasc elecromagnec feld).

9 9 Dfferenal geomery and modern cosmology THE MODIFIED GEODESICS IN EUCLIDEAN SPACE As an example, consder he geodescs n Eucldean space wh Eucldean merc In hs way, equaons (4) read ds = dx... dx n. () d x α yeldng x c x, = αdx, τ d τ (3) τ = τ < α <. So he geodescs are no sragh lnes as n he sandard case. I s clear ha hey raverse a an accelerang velocy for < α <. Anoher neresng example concerns a sphere of un radus n R 3 wh merc gven smply by ds = dθ sn θdϕ. (4) The geodescs equaons are he exremum of he fraconal le-energy: [ ] = α E Λ L θ, ϕ, θ, ϕ τ = Γα = α = θ cos θϕ τ. Γα ( )( ) = (5) The correspondng exremum are defned by he modfed Euler-Lagrange equaons (3) and we oban: α T T d dt ( T α ) θ θ θ = cosθsn θϕ, (6) α T α d dt ( T α ) Equaon (7) yelds θϕ = ( TT) cos θϕ =. (7) α cos, T s a consan of negraon consequenly, he dfferenal equaon () became: ( α) T θ α θ anθ ( an θ ) =. T T (8) Noe ha for θ ( T ) =, ϕ= α TT or ϕ. If < α <, han ϕ ncreases slowly wh me. In realy, he modfed equaons of moon of free T α

10 476 El-Nabuls Ahmad Ram es parcles or equvalenly, he geodesc equaons can be easly deermned whn he conex of ordnary vecor analyss [4]. In fac, we choose recangular Gallean coordnaes ( x, τ ), =,, 3 wh he orhonormal spaal e ˆ. The modfed Euler-Lagrange equaons are: bass ( ) L d L = α L, (9) r v τ v where L L( r,v, ) c ( c = τ = u u) s he Lagrangan, u v w s he absolue velocy of he cloc relave o physcal space, c s he speed of lgh, v=dr d τ s he coordnae velocy of he cloc and w=w ( r, τ) s he flow velocy. I s easy o verfy ha = L r L c ( w) u and L v L c = u. Consequenly, equaons (9) gve he modfed geodesc equaon n Gallean coordnaes as s expeced from he prevous argumens: du α u I uu 3 w u =, τ c u< c, (3) where I 3 s he deny ensor n 3-space and I 3 ( uu c ) s he symmerc ensor of moon. The non-relavsc approxmaon of equaon (5) s: du α w d v d u u w α ( v w) w ( v w) =. (3) τ τ Smple mahemacal manpulaons lead o: dv α v w α w w = ( w) v. (3) τ τ τ Ths s o say ha he Newonan force acng on a free es body or a saelle of mass m s: F v saelle = masaelle m d = d τ (33) m w α m ( w v) = m( w ) m( ( w) v ). τ τ The frs erm on he RHS of equaon (33) represens he accelerang force arsng from an explc dependence of he flow on me; he second ern represens he fraconal wea decayng frcon force; he hrd erm represens he gravaonal and generalzed cenrfugal accelerang force and he las erm he generalzed Corols accelerang force. Ths equaon s mporan when we deal wh he problem of wea dsspave moon of es parcles or bodes n roang and acceleraed frames of reference n Newonan mechancs.

11 Dfferenal geomery and modern cosmology THE ROLE OF THE MODIFIED GEODESIC EQUATIONS IN MODERN MATHEMATICAL COSMOLOGY In reurn o dfferenal geomery on manfolds, we would le o revew, clarfy and crcally analyze abou he role of he modfed geodesc equaons n modern mahemacal cosmology, n parcular he Fredmann-Roberson-Waler (FRW) model of general relavsc cosmology. Geomercally, a FRW soropc and homogeneous Remannan manfold model s a 4-dmensonal Lorenzan manfold M whch can be expressed as a warped produc I RΣ, where I s an open nerval of he pseudo-eucldean manfold R, Σ s a complee and conneced 3-dmensonal Remannan manfold and he warpng funcon R s a smooh, real-valued and non-negave funcon upon I [5]. The Lorenzan merc on M s wren as: g ds ds= a ( τ) Ω, (34) where a ( τ) s he scale facor and whch deermnes he me evoluon of he spaal geomery of he unverse and Ω s he merc ensor on Σ whch s consdered as globally soropc. In parcular, every homogeneous Remannan manfold (Σ, Ω) s dffeomorphc o some 3-dmensonal Le group. I s easy o prove ha he modfed geodesc equaons reduce o [8]: dvγ α v c R =, (35) τ c beng he celery of lgh (n naural un), v = dx d τ he parcle velocy and dvγ=γ = 4πG ρ beng he Posson equaon, ρ he mass densy n gravaonal neracon, G s Newon gravaonal consan and γ he acceleraon. I follows ha R = πρ 4 Geff c where G v eff = G α GΔG, 4πρGT (36) s he effecve Newon gravaonal consan. 3 In order now o ae no consderaon he spaceme expanson, we wll mae use of he Raychaudhur expanson scalar facor θ defned by [6]: R ab 3 In fac, he me-componen () of he Rcc scalar curvaure s R Γ [5]. Here = be he Rcc ensor where j g Rajb (,,,, ) j j R = g g g g Γ Γ Γ Γ abcd bc ad bd ac ad bc ac bd ad bjc ac bjd p j pj s he covaran curvaure ensor; Γ = g Γ.,

12 478 El-Nabuls Ahmad Ram The perurbed gravy reduces o v 3θ= 3 aa. (37) ( α) 3 Δ G = 4πρT The cosmologcal models are usually descrbed by he Ensen feld equaons (c = ): R μν gμνr= 8 π GTμν, (39) where R= g ab R ab s he Rcc scalar curvaure and Tμν = ( pρ) uμ uν pgμν s he sress-energy ensor of a perfec flud wh pressure p and densy ρ whch are boh scalar felds on M and are consans on each hypersurface Στ Σ =τ Σ bu whch exhb me-dependence. In wha follows, we wll assume ha he consan seconal curvaure of he 3-dmensonal Remannan space from (Σ, Ω) s zero. Tha s he manfold s oo fla. We wll dscuss n wha follows hree possbles,.e. hree epochs: a. a (38). The Maer Epoch. In he absence of he cosmologcal consan, he so-called modfed Fredmann equaons for dus (p = ) are: a a ( α ) a 8πGρ =, T a 3 (4) a a 4πGρ α =, (4) a T a 3 wh T = τ s a new me change varable. Noce ha equaon (4) can be wren as a 8πGρ Λdecayng =. (4) a 3 3 Λdecayng 6 α aat can be vewed as a decayng cosmologcal consan wh cosmc me. Ths dsspave erm can be nroduced n he same way as n decayng wo-flud hydrodynamcs and obeys he conservave law: Tmaer Tcurvaure T Λ decayng μν = where Tcurvaure μν = ( 8 π G) Gμν, Gμν = Rμν gμνr s he Ensen curvaure ensor. The new decayng erm does no nfluence he maer-conservaon law T ( maer) μν ν ; =. One of he movaons for nroducng a decayng lambda erm s o reconcle he age parameer and he densy parameer μν μν

13 3 Dfferenal geomery and modern cosmology 479 of he unverse wh curren observaonal daa. Ths s o say ha whn he conex of fraconal acon negral, he vacuum energy n Ensen feld heory s replaced by exernal wea decayng force. For a dus-epoch manfold,.e. p =, he dynamcal equaons whn he framewor of fraconal acon negral (geodesc equaon wh npu wea decayng vecor feld) reduce o: ( α ) a 4 a a =, a a T a and consequenly, he scale facor reduces o ( α) (43) at T and corresponds for α < 3/4 o an accelerang manfold expanson [7, 8]. There wll be no need o nvoe any ypes of exoc maer n order o accelerae he unverse; all wha we need s a wea decayng forcng erm added o he geodesc equaon and o rea he unverse as a smple mechancal conrol sysem. For a dus-manfold, s an easy as o show ha he maeral densy decreases wh me as and s posve for α < 3/4. ρ = ( 3 α)( α) p= πgt, (44). The Radaon Epoch. For a radaon-epoch manfold,.e.p = ρ/3, he dynamcal equaons whn he framewor of fraconal acon negral reduces o: a a ( α ) a 8πGρ =, T a 3 ( α ) a a 8πGρ =, a T a 3 and consequenly, he scale facor reduces o ( α) 5 4 (45) (46) at T and corresponds agan for α < 3/4 o an accelerang manfold expanson. Consequenly, he maeral densy decreases wh me as ( α)( α) ρ p=ρ 3 = 6πGT and s posve for α < 3/4., (47) 3. The Vacuum Case or Inflaon. For a vacuum manfold,.e. p= ρ, he dynamcal equaons reduce o: a a ( α ) a 8πGρ =, T a 3 (48)

14 48 El-Nabuls Ahmad Ram 4 ( α ) a a 8πGρ =, a T a 3 and consequenly, he scale facor reduces o at ( mt) (49) exp, m s a consan parameer and corresponds o an nflaonary manfold for m > and deflaonary manfold for m < [7]. The new hng n hs scenaro s ha s ndependen on wheher he densy s consan or varable. The maeral densy vares wh me as ( α ) 3emT mt ρ= e. 8πG T (5) I ncreases rapdly wh me for m > and decays rapdly for m <. Noce ha for a vacuum manfold, he maeral densy could decreases wh me f he gravaonal consan s assumed o ncrease exponenally wh me as G e mt. The presen acceleraed expanson of he unverse may be arbued o hs ever-growng gravy. In fac, he exponenal growh of he gravaonal consan s one of he characerscs of nflaon heory [7, 8]. I s mporan o noe ha a new ype of vacuum s a wor n our fraconal scenaro. 6. CONCLUSIONS Despe he mporan queson ha arses wha does he frconal force mean, n he vacuum? we showed ha he new decayng lambda erm leads o varous modfcaon o he sandard predcon of general relavy. There s nohng n he curren noon of physcal space ha enals he presence of he decayng frcon erm. So long as he homogeneous space s represened by a dfferenal manfold, and mass-energy s represened by felds on he manfold, wll be possble and even realsc o magne an empy regon of space. I may well be rue ha here s no physcal procedure whch can mae a regon of space oally empy, bu hs does no mean ha s mpossble for space o be empy. If he maeral densy decreases wh me, hen he unverse ends o be an empy manfold and consequenly, from equaons (4) and (4), one can easly prove ha he scale facor wll evolve as at T and hs s que neresng. The empy unverse acceleraed wh me,.e. a decayng maeral unverse creaes an empy accelerang unverse or an empy eernal accelerang unverse s creaed from physcal decayng frcon. In Remann geomery on manfolds and he geomery of he 4-dmensonal FRW Lorenzan manfold, ohers applcaons and fraconal feaures are under progress.

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