We have obtained cosmological solutions in five-dimensional space-time-mass theory of gravitation by assuming components of energy momentum tensor, pressure p=0 and the role of p4 as a cosmological constant. The behaviour of the solution is discussed for the cases in which k=-1,0,+1.Izveli smo kozmološka rješenja u petdimenzijskoj prostor-vrijeme-masa teoriji gravitacije pretpostavljajući komponente tenzora energije-impulsa, tlak p = 0, te uzevši p4 u ulozi kozmološke konstante. Raspravljamo značajke rješenja za k = +1, 0, −1

Khadekar, G. S.

Patki, V.

Radha, R.

English

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ISSN1330–0016CODEN FIZBE7COSMOLOGICAL SOLUTIONS IN VARIABLE REST MASS THEORY OFGRAVITATIONG. S. KHADEKARa, VRISHALI PATKIa and R. RADHAbaDepartment of Mathematics, Nagpur University, Nagpur, IndiabDepartment of Mathematics and Statistics, Hyderabad University, Hyderabad, India.Received 30 July 2004 Accepted 20 December 2004Online 7 February 2005We have obtained cosmological solutions in five-dimensional space-time-mass the-ory of gravitation by assuming components of energy momentum tensor, pressurep = 0 and the role of p4 as a cosmological constant. The behaviour of the solutionis discussed for the cases in which k = −1, 0,+1.PACS numbers: 04.20.Ex, 04.40.-b UDC 530.12Keywords: cosmology, five dimensional space, Wesson theory, Kaluza-Klein theory1. IntroductionMany different theories of gravity alternative to Einstein’s general general the-ory of relativity have been proposed in which either the gravitational constant Gand the rest masses of the object vary with time. Wesson [1,2] discussed the dif-ficulties encountered by these different approaches and proposed a variable-masstheory of gravity where the mass is regarded as a geometrical coordinate in a contin-uum 5-dimensional (5D) space-time-mass (STM). In some sense, the 4 dimensionalEinstein’s theory would be embedded in it. In this Kaluza-Klein type theory, thefifth coordinate is closely related to the mass m through x4 = Gm/c2, where thegravitational constant G and the velocity of light c are true constants.Although the addition of a fifth dimension to the usual four dimensions doesnot alter the numerical size of the line element for local problems, it might havenoticeable consequences for cosmological problems because the x4 coordinate growslarger relative to the space coordinates. Such a possibility leads some authors tostudy cosmological solutions in vacuum. Wesson [3] found a vacuum solution witha vanishing cosmological constant. Chatterjee [4] and Fukui [5] obtained solutionsFIZIKA B 13 (2004) 4, 711–718 711khadekar et al.: cosmological solutions in variable rest mass . . .in which the space-time properties depend both on time and rest mass. In thesesolutions, the fifth coordinate has been introduced as a time-like coordinate. Asthis would allow the existence of closed time-like orbits in the time-mass plane,Gron [6] considered a Bianchi type-I form of the metric with a space like fifthcoordinate to study the inflationary cosmology. Ma [7] interpreted the rest massas the length of the fifth-dimension subspace. Berman and Som [8] studied thecosmological consequences of a perfect fluid and the role of the fifth componentconsidered as a cosmological constant and obtained an infrastationary model.In the present paper, we have generalized the work of Ma [7] and obtained thecosmological solution in a five dimensional STM theory of gravitation by assumingpressure p = 0 and p4 as a cosmological constant.2. Field equations and solutionsWe assume that the cosmological principle could be extended to the 5D space-time-mass and choose co-moving coordinates with u0 = 1 and uµ = 0 (µ = 1, 2, 3, 4)forui =dxidτ.We consider the line elementds2 = dt2 − a2(t)(dr21 − kr2+ r2dθ2 + r2 sin θdφ2)+ µ2(t)dψ2, (1)where a(t) is a spatial scale factor, µ(t) is the mass scale factor, k = −1, 0,+1 andunits are chosen such that c = 1. The energy-momentum tensor for a perfect fluidis taken in the form suggested by Gron [6]T ij = diag(ρ,−p,−p,−p,−p4). (2)We restrict ourselves to the case p = 0. The 5D gravitational field equationsGij = −8πGTijcan be written as3(ȧ2a2+ka2+ȧµ̇aµ)= 8πGρ, (3)2äa+µ̈µ+ȧ2a2+ka2+ 2ȧµ̇aµ= 0, (4)äa+ȧ2a2+ka2= −p43, (5)712 FIZIKA B 13 (2004) 4, 711–718khadekar et al.: cosmological solutions in variable rest mass . . .where overhead dot represents the differentiation with respect to t. Similarly, theexpansion factor θ and the scalar shear σ are given byθ = 3ȧa+µ̇µ, (6)σ2 =38(ȧa−µ̇µ)2. (7)The covariant energy conservation law T ij;j = 0 gives the equationρ̇+ ρ(3ȧa+µ̇µ)=3Λ8πGµ̇µ, (8)where Λ = −p4/3, which can also be derived from Eqs. (3)–(5). The solution ofEqs. (4) and (5) are given bya2(t) =c1e√2Λ t + c2e−√2Λ t +kΛ, Λ > 0 ,c1 cos(√−2Λ t)+ c2 sin(√−2Λ t)+kΛ, Λ < 0 ,(9)µ(t) =(c3 cos(√Λt)+ c4 sin(√Λt))a−1(t), Λ > 0 ,(c3e√−Λt + c4e−√−Λt)a−1(t), Λ < 0 ,(10)where c1, c2, c3 and c4 are arbitrary constants of integration.Using Eqs. (9) and (10) in Eqs. (3) and (8), we getρ =ka2(11)+Λ√2µa3(c1e√2Λ t−c2e−√2Λ t)(c4 cos(√Λ t)−c3 sin(√Λ t)), Λ > 0 ,(c3e√−Λ t−c4e−√−Λ t)(c2 cos(√−2Λ t)−c1 sin(√−2Λ t)), Λ < 0 ,FIZIKA B 13 (2004) 4, 711–718 713khadekar et al.: cosmological solutions in variable rest mass . . .where ρ = 8πGρ/3 andσ2 = (12)3Λ8[√2a2(c1e√2Λt − c2e−√2Λt)−c4 cos(√Λt)− c3 sin(√Λt)c4 sin(√Λt)+ c3 cos(√Λt)]2, Λ > 0 ,−3Λ8[√2a2(c2 cos(√−2Λt)− c1 sin(√−2Λt))−c3e√−Λt − c4e−√−Λtc3e√−Λt + c4e−√−Λt]2, Λ < 0 ,θ=√2Λa2(c1e√2Λt − c2e−√2Λt)+√Λaµ(c4 cos(√Λt)− c3 sin(√Λt)), Λ > 0 ,√−2Λa2(c2 cos(√−2Λt)−c1 sin(√−2Λt)+√−Λaµ(c3e√−Λt−c4e−√−Λt),Λ < 0 ,(13)The constants c1, c2, c3, c4 can be determined using the initial conditions at t = t0such that a(t0) = α, ȧ(t0) = β, µ(t0) = γ and µ̇(t0) = δ. Without loss of generality,we can choose t0 = 0, i.e, a(0) = α, ȧ(0) = β, µ(0) = γ and µ̇(0) = δ. Hence, inview of the dimensionless variables asR(x) = a(t)/α, A(x) = µ(t)/γ, x = βt/α, (14)we get ȧ(t) = βR′(x), ä(t) = β2R′′(x)/α, µ̇(t) = (βγ/α)A′(x), µ̈(t) =β2γA′′(x)/α2, where prime represents the differentiation with respect to the vari-able x. Thus we have the solution in the form of dimensionless variables asR2(x) =c1e√2λ x + c2e−√2λ x +kλβ2λ > 0 ,−kβ2x2 + c1x+ c2 λ = 0 ,c1 cos(√−2λx)+ c2 sin(√−2λx)+kλβ2λ < 0 ,(15)A(x) =(c3 cos(√λx)+ c4 sin(√λx))R−1(x) λ > 0 ,(c3x+ c4)R−1(x) λ = 0 ,(c3e√−λ x + c4e−√−λ x)R−1(x) λ < 0 ,(16)where λ =−p4α23β2=Λα2β2.714 FIZIKA B 13 (2004) 4, 711–718khadekar et al.: cosmological solutions in variable rest mass . . .The constants c1, c2, c3, c4 in Eqs. (15) and (16) are to be determined usingthe initial conditions asR(0) = 1, R′(0) = 1, A(0) = 1, A′(0) =αδβγ, (17)when λ > 0 :c1 =12+k2λβ2+1√2λ, c2 =12+k2λβ2−1√2λ, c3 = 1, c4 =1√λ(1 +αδβγ),when λ = 0 :c1 = 2, c2 = 1, c3 = 1 +αδβγ, c4 = 1,when λ < 0 :c1 = 1+kλβ2, c2 =√−2λ, c3 =12+12√−λ(1+αδβγ), c4 =12−12√−λ(1+αδβγ).Now, R′ = 0 leads to the maximum of R, say Rm at x = xm, wherexm =12√2λlogc2c1,R2m = 2√c1c2 +kλβ2, (18)for λ > 0, andxm =1√−2λtan−1c2c1R2m = c1 cos(tan−1c2c1)+ c1 sin(tan−1c2c1)+kλβ2, (19)for λ < 0.3. ConclusionIn this section, we have given some physical properties of the model. Whenthe universe is spatially closed, i.e. k = 1, A(x) contracts and R(x) expands andvice-a-versa. R(x) has maximum value Rm where R2m = 2√c1c2 + k/Λβ2. Whenc2 c4 < 0 and c3c1 < 0, then the energy condition ρ̄(t) must be positive.The ratio σ/θ tends to a finite limit as t → ∞. Therefore the model is highlyanisotropic for large t.FIZIKA B 13 (2004) 4, 711–718 715khadekar et al.: cosmological solutions in variable rest mass . . .Equations (4)–(6) are integrated in view of Eqs. (14) and (17) for dimensionlessvariables R, A, ρ as given in Eq. (14). The behaviour of the variables R, A, ρ versusx are depicted in Figs. 1–8. Figs. 1–3 and Figs. 4–6 exhibit the behaviour of R(x)and A(x) respectively for different values of λ and k, where as Figs. 7 and 8 showthe nature of density for k = 0 and k = +1.Fig. 1 (left). Behaviour of R(x) for k = −1 and different values of λ.Fig. 2. Behaviour of R(x) for k = 0 and different values of λ.Fig. 3 (left). Behaviour of R(x) for k = 1 and different values of λ.Fig. 4. Behaviour of A(x) for k = −1 and different values of λ.716 FIZIKA B 13 (2004) 4, 711–718khadekar et al.: cosmological solutions in variable rest mass . . .Fig. 5 (left). Behaviour of A(x) for k = 0 and different values of λ.Fig. 6. Behaviour of A(x) for k = 1 and different values of λ.Fig. 7 (left). Behaviour of ρ̄(x) for k = 0 and different values of λ.Fig. 8. Behaviour of ρ̄(x) for k = 1 and different values of λ.It can be observed from the figures that as the cosmological constant λ, i.e. −p4,increases, the region of existence of R(x), A(x) and ρ decreases for k = −1, 0,+1.For the same value of λ and k, one can notice that if R(x) increases then A(x)decreases as x increases and vice-a-versa. Figures 7 and 8 exhibit a rapid decreaseof density as λ increases.FIZIKA B 13 (2004) 4, 711–718 717khadekar et al.: cosmological solutions in variable rest mass . . .AcknowledgementsThe author wishes to thanks University Grant Commission (UGC) New Delhi,India for the financial support under the minor research project No. Dev/RAD/270.References[1] P. S. Wesson, Astron. Astrophy. 119 (1983) 145.[2] P. S. Wesson, Gen. Rel. Grav. 16 (1984) 196.[3] P. S. Wesson, Astron. Astrophy. 165 (1986) 1.[4] S. Chaterjee, Astron. Astrophy. 179 (1987) 122.[5] T. Fukui, Gen. Rel. Grav. 19 (1987) 43.[6] O. Gron, Astron. Astrophy. 193 (1988) 1.[7] Guang-wen Ma, Physics Letters A 143 (1990) 4.[8] M. S. Berman and M. M. Som, Astrophysics and Space Science 207 (1993) 105.KOZMOLOŠKA RJEŠENJA TEORIJE GRAVITACIJE S PROMJENLJIVOMMASOM MIROVANJAIzveli smo kozmološka rješenja u petdimenzijskoj prostor-vrijeme-masa teoriji gravi-tacije pretpostavljajući komponente tenzora energije-impulsa, tlak p = 0, te uzevšip4 u ulozi kozmološke konstante. Raspravljamo značajke rješenja za k = +1, 0,−1.718 FIZIKA B 13 (2004) 4, 711–718

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