In this paper, we have studied the Einstein Field equation in Kaluza-Klein space-time in five dimension with the metric dS^2 = dt^2 − A^2(dx^2 + dy^2 + dz^2)− B^2dψ^2 under the assumption that B = αt, where α = Constant and scale factor satisfying the relation R^4 = A^3B with perfect fluid having energy momentum tensor

Tij = (ρ + p)vivj − pgij  

In this paper, we have assumed that G = 1/t and we have found the value of cosmological constant Λ is a function time t in terms of hyper geometric function

Kalpana Pawar

Satish T. Rathod

淡江理工學刊

Cosmological constant as Hyper Geometric function in fivedimensional space-time

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Journal of Applied Science and Engineering, Vol. 27, No 1, Page 1953-1955 1953Cosmological Constant As Hyper Geometric Function InFivedimensional Space-timeSatish T. Rathod and Kalpana Pawar∗Department of Mathematics, Shri R. R. Lahoti Science College, Morshi-444905, Dist. Amravati, M. S., India∗Corresponding author. E-mail: drkalpanapawar75@gmail.comReceived: Jan. 11, 2023; Accepted: Mar. 01, 2023In this paper, we have studied the Einstein Field equation in Kaluza-Klein space-time in five dimensionwith the metric dS2 = dt2 − A2(dx2 + dy2 + dz2)− B2dψ2 under the assumption that B = αt, where α =Constant and scale factor satisfying the relation R4 = A3B with perfect fluid having energy momentum tensorTij = (ρ + p)vivj − pgijIn this paper, we have assumed that G = 1t and we have found the value of cosmological constant Λ is afunction time t in terms of hyper geometric function.Keywords: Kaluza-Klein space-time in five-dimension, perfect fluid, Hypergeometric Function, variable cosmologicalconstant Λ = Λ(t) and gravitational constant G = G(t)© The Author(’s). This is an open-access article distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author andsource are cited.http://dx.doi.org/10.6180/jase.202401_27(1).00121. IntroductionKaluza and Klein consider extra dimension apart fromspace and time in order to unify the gravitation and elec-tromagnetism [1–3]. Many researchers of theoretical physi-cist and mathematician tried to the unified theory of grav-ity and electromagnetism before Kaluza and Klein [1–8].Alan Chados and steven Detweiler in his paper where hasthe fifth dimension gone? consider the five-dimensionalspace-time model in vacuum [9]. Let t0 is the reference cos-mological time. The Kasner solution in five-dimensionalspace time is given by dS2 = −dt2 +(tt0) [∑3i=1(dxi)2]+(t0t) (dx5)2. If t = t0 then one can observed that the uni-verse is spatially flats and isotropic. If t ≪ t0 then thevalue of(tt0)goes nearer to zero and the value of(t0t)goes aways from zero sufficiently. From this fact, we can ob-serve that dS2 = −dt2 +(t0t) (dx5)2 and as t approachesto infinity then the cosmos has only one dimensions. Ift ≫ t0 then the value of(tt0)goes away from zero and thevalue of(t0t)goes towards zero sufficiently. If we consider0 ≤ xi ≤ L, where L is the length. Then the distance in fifthdimensions shrunk to√(t0t)L and the space dimensionsincrease to√(tt0)L. As the t approaches to infinity i.e. uni-verse is sufficiently old, the fifth dimension not observe [6].J.Demart &J.-L.Hanquin used the results of Fees G7G8 andG11 Lie Algebra to study the homogeneous and anisotropiccosmological models satisfying five dimension Einstein’sfield equations [7]. The above explanation motivates us tostudy the five-dimensional cosmological model.P.A.M Dirac in his letter Cosmological Constant in 1937,study the behavior of cosmological constant. He observedthat the cosmological constant is depend on cosmologicaltime [10].Sanjay oli studied the five-dimensional cosmologicalmodel with variable cosmological constant and Gravita-tional constant [11].In [11], Sanjay oli considered the metric dS2 = −dt2 +X2(dx2 + dy2 + dz2)+ A2dΨ2 with the relation AX3 = U.He found the value of Λ and G for U = c, U = t, U = tn1954 Satish T. Rathodand conclude that the value of Λ and G decreases andincreases with time respectively [11]. In [12], R.K. Tiwariet.al studied the five-dimensional cosmological model withthe metric dS2 = dt2 − A2(dx2 + dy2 + dz2)− B2dΨ2 withrelation R4 = A3B, where R is the scale factor. In theirinvestigation, they observed that G proportional to t−(1−ω),where ω is term in equation of states p = ωρ with 0 ≤ ω ≤1 [12].In this paper, we have studied the Einstein Field equa-tion in Kaluza-Klein space-time in five dimension with themetric dS2 = dt2 − A2(dx2 + dy2 + dz2)− B2dψ2 underthe assumption that B = αt, where α = Constant and scalefactor satisfying the relation R4 = A3B with perfect fluidhaving energy momentum tensor Tij = (ρ + p)vivj − pgijIn this paper, we have assumed that G = 1t and we havefound the value of cosmological constant Λ is a functiontime t in terms of Hyper geometric function.2. Five-dimensional einstein field equationIn Kaluza-Klein five-dimensional cosmological model, con-sider the line elementdS2 = dt2 − A2(dx2 + dy2 + dz2)− B2dψ2 (1)where A, B is the function of cosmological time t and thespatial average scale factor R(t) satisfies the relation R4 =A3BSuppose universe contain perfect fluid with energy mo-mentum tensorTij = (ρ + p)vivj − pgij (2)Let assume that the matter satisfies the equation of statesp = ωρ, 0 ≤ ω ≤ 1 (3)where p is the pressures and ρ is the energy density ofcosmic matters.The five-dimensional Einstein Field equation which con-tain Λ and G as a function of time is given as follows [12]:Rij − 12Rgij = −8πGTij − Λgij (4)By using line element (1), one can express five-dimensional Einstein Field equation (4) as3Ȧ2 AB + 3ȦḂA2R4= 8πGρ + Λ (5)2ÄA+BB+2ȦḂAB+Ȧ2A2= −8πGp + Λ (6)3ÄA + 3Ȧ2A2= −8πGp + Λ (7)By assuming the energy conservation in general rela-tivity, the covariant derivative in Einstein Field equationgivesρ̇ + (ρ + p)(3ȦB + ḂAAB)+ĠGρ = − Λ8πG(8)Equation (8) gives [13]ρ̇ + (ρ + p)(3ȦB + ḂAAB)= 0 (9)andΛ̇ = −8πρĠ (10)3. General solution of five-dimensional einsteinfield equationEquation (3), (5)-(7) and (9) contain five independent equa-tions with six unknowns , B, p, ρ, G, Λ. To find the solution,we have assumed thatB = αt (11)where α is the constant. From equation (6) and (7), we getȦA− BB=k1A3B(12)where k1 is the constant of integration.A =(−k1 + e3k1ct3) 13α13(13)From equation (11) and (12), we get following line ele-mentds2 = dt2 −(−k1 + e3k1ct3) 13α13 (dx2 + dy2 + dz2)− (αt)2dψ2(14)where c is the constant.For the model (14), the spatial volume V, matter den-sity ρ, pressure p, gravitational parameter G, cosmologicalparameter Λ are given byV =(−k1 + e3k1ct3) 13α133(αt) (15)ρ = e(1+ω)(log(t)− 13 α log(−k1+e3ck1 t3 ))α c1 (16)p = ωe(1+ω)(log(t)− 13 α log(−k1+e3ck1 t3))α c1 (17)Journal of Applied Science and Engineering, Vol. 27, No 1, Page 1953-1955 1955P.A.M Dirac say that G is proportional to t−1. AssumethatG =1t(18)SR = Hypergeometric2 F1[1 + ω3,−1 + ω + α3α,−1 + ω − 2α3α,e3ck1t3k1] (19)Λ =c2 −8πt−1+ω+αα(−k1 + e3ck1t3) 13 (−1−w)(1 − e3ck1t3k1) 1+ω3αc1SR1 + w(20)where c2, k1, c1 are constants.4. Conclusion and discussionP.A.M Dirac say that G is proportional to t−1. In this paper,we can observe that when we consider five dimensionalKaluza-Klein cosmological models, then we found that Λis not constant and it involve Hypergeometric function.References[1] T. Kaluza, (1921) “Sitzungsber" Abhandlungender Königlich Preussischen Akademie der Wis-senschaften 996: 1921.[2] O. Klein. “QUANTUM THEORY AND FIVE-DIMENSIONAL RELATIVITY THEORY”. In: TheOskar Klein Memorial Lectures, 67–80. DOI: 10 .1142 / 9789814368728 _ 0006. eprint: https : / /www. worldscientific . com / doi / pdf / 10 . 1142 /9789814368728_0006.[3] F. R. MILLER and N. B. LAUGHTON, (1926) “TheRôle of the Cerebellum in the Co-ordination of AnimalMovement" Nature 118(2971): 516–517. DOI: 10.1038/118516b0.[4] P. G. Freund, (1982) “Kaluza-Klein cosmologies" Nu-clear Physics B 209(1): 146–156. DOI: https://doi.org/10.1016/0550-3213(82)90106-7.[5] T. Dereli and R. Tucker, (1983) “Dynamical reductionof internal dimensions in the early universe" PhysicsLetters B 125(2): 133–135. DOI: https://doi.org/10.1016/0370-2693(83)91252-2.[6] E. Alvarez and M. B. Gavela, (1983) “Entropy fromExtra Dimensions" Phys. Rev. Lett. 51: 931–934. DOI:10.1103/PhysRevLett.51.931.[7] R. Bergamini and C. A. Orzalesi, (1984) “Towards acosmology for multidimensional unified theories" PhysicsLetters B 135(1): 38–42. DOI: https://doi.org/10.1016/0370-2693(84)90449-0.[8] S. Randjbar-Daemi, A. Salam, and J. Strathdee, (1984)“On Kaluza-Klein cosmology" Physics Letters B 135(5):388–392. DOI: https : / / doi . org / 10 . 1016 / 0370 -2693(84)90300-9.[9] A. Chodos and S. Detweiler, (1980) “Where has thefifth dimension gone?" Phys. Rev. D 21: 2167–2170.DOI: 10.1103/PhysRevD.21.2167.[10] P. A. M. DIRAC, (1937) “The Cosmological Constants"Nature 139(3512): 323. DOI: 10.1038/139323a0.[11] S. Oli, (2014) “Five-Dimensional Space-Times with aVariable Gravitational and Cosmological Constant" Jour-nal of Gravity 2014: 874739. DOI: 10 . 1155 / 2014 /874739.[12] R. K. Tiwari, F. Rahaman, and S. Ray, (2010) “FiveDimensional Cosmological Models in General Relativity"International Journal of Theoretical Physics 49(10):2348–2357. DOI: 10.1007/s10773-010-0421-3.[13] S. T. Rathod and K. Pawar, (2007) “Five-DimensionalCosmological Model in presence of Perfect Fluid with G isproportional to t−1" Romanian Journal of Physics 52:445.

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Cosmological constant as Hyper Geometric function in fivedimensional space-time

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