Shear viscosity in Friedmann Robertson Walker Universes

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1 Physica A 286 (2000) Shear viscsity in Friedmann Rbertsn Walker Universes Alfred Sandval-Villalbaz a;, L.S. Garca-Cln b; 1 a Departament de Ciencias, Universidad Iberamericana, Prlngacin Pase de la Refrma 880, Cl. Lmas de Santa Fe , Mexic DF, Mexic b Departament de Fsica, Universidad Autnma Metrplitana-Iztapalapa, Av. Pursima y Michacan S=N, Cl. Vicentina 09340, Mexic DF, Mexic Received 1 June 2000 Abstract An irreducible representatin f entrpy prductin is used t shw that shear viscsity must be taken int accunt in the analysis f dissipative Friedmann Rbertsn Walker Universes. The resulting eld equatins are built in the lcal equilibrium regime and sme f its implicatins are discussed. c 2000 Elsevier Science B.V. All rights reserved. Keywrds: Bulk and shear viscsities; Friedmann Rbertsn Walker Universes; Lcal equilibrium; Curie s principle; Entrpy prductin 1. Intrductin It is an established fact that irreversible thermdynamics must have played a decisive rle in the physics f the early universe. It is knwn that dissipative prcesses in istrpic hmgeneus Universes have restrictins due t the symmetry f the metric tensr, s that the intrductin f such eects must be dne in a rather careful manner. In this cntext, it is ften assumed that bulk viscsity is the nly dissipative mechanism admissible in Friedmann Rbertsn Walker (FRW) Universes [1 4]. The purpse f this brief cmmunicatin is t shw that a precise analysis f the entrpy prductin in a FRW metric, assuming a nn-equilibrium regime, leads t a nn-negligible shear Crrespnding authr. Tel./fax: addresses: alfred.sandval@uia.mx (A. Sandval-Villalbaz), lgcs@xanum.uam.mx (L.S. Garca-Cln). 1 Als at El Clegi Nacinal, Luis Gnzalez Obregn 23, Centr Histric 06020, Mexic DF, Mexic /00/$ - see frnt matter c 2000 Elsevier Science B.V. All rights reserved. PII: S (00)00337-X

2 308 A. Sandval-Villalbaz, L.S. Garca-Cln / Physica A 286 (2000) viscsity cecient whse csmlgical implicatins may be relevant at certain stages f the evlutin f the Universe. 2. Bulk and shear viscsity in FRW metrics The starting pint f the analysis is the cnserved stress-energy tensr T = u u p + p c 2 u u + ; (1) which is assumed, as usual, t be prprtinal t the Einstein tensr G, s that G = T : Here, is the uid density, p represents the pressure, u is the cvariant velcity fur-vectr, g is the metric tensr, and stands fr the viscsity tensr, which is suppsed t be symmetric. The tensr (1) is slightly mre general than the ne used by ther authrs [2] wh assume that has a special frm, namely, = ( + (1=c 2 )u u ), where is a scalar bulk viscsity: In the fllwing frmalism, such assumptin is dismissed. The FRW slutin fr the system (2) is based n the prpsal f a hmgenus istrpic metric, which fr a clsed universe reads ) 1 ds 2 = (1 r2 dr 2 + r 2 d 2 (c dt) 2 : (3) a 2 Here, r is the radial crdinate, is the slid angle, and a = a(ct) represents the scale factr, which can be identied with the radius f the universe. The furth crdinate crrespnds t ct. It is well knwn that the fur divergence f the Einstein tensr in Eq. (2) vanishes, s that T ; =0: Eq. (4) cntains the balances fr linear mmentum and mechanical energy, and the prjectin t the velcity fur vectr u T ; = 0 (5) leads, by means f Eq. (1) t the cntinuity equatin [5,6] D () D + ()v; = p () c 2 g v ; 1 c 2 v ; : (6) This last equatin is well-knwn by csmlgists, and leads, in the absence f viscsity and pressure, t the [a(ct)] 3 dependence f density in the dust case. As we have been insisting in the past, the lcal equilibrium assumptin has been arbitrarily rejected by many practitiners f relativistic irreversible thermdynamics, mainly by arguing that csmlgical prcesses take place far frm equilibrium [2 4]. We disagree with such a statement. Mrever, it shuld be remembered that the (2) (4)

3 A. Sandval-Villalbaz, L.S. Garca-Cln / Physica A 286 (2000) cncept f thermal equilibrium, which in the absence f chemical reactins is synnymus f lcal equilibrium has been shwn t be a relativistic invariant [7]. As we have clearly shwn in the previus wrk, nthing mre is needed in additin t the cnservatin equatins, plus linear cnstitutive equatins t derive relativistic transprt equatins cnsistent with causality. We take the same psitin here. Therefre, the entrpy prductin may be written as [5] = J [Q] ; 2 u ; ; (7) where and represent entrpy prductin and abslute temperature, respectively. It is ften taken fr granted that heat must be neglected in the FRW metric, since nn-diagnal elements in the stress energy tensr (Eq. (1)) seem t arise in the presence f this quantity. Althugh it has been shwn in an earlier paper that heat may enter the transprt equatins in an indirect frm, withut disturbing the symmetry f the FRW space time [7], we will nt g deeper in that discussin and nly cncentrate in the secnd term at the right-hand side f Eq. (7). This term invlves the cvariant derivative f the fur velcity, u;, in a FRW metric. In general, this bject is irreducibly represented as the sum f fur terms [8], but in an istrpic and hmgenus Universe, ne simply gets ȧ=a u; 0 ȧ=a 0 0 = 0 0 ȧ=a 0 = 3 h ; (8) where ȧ=da=st, =u ; =3ȧ=a and h = +(1=c 2 )u u. This bject is a purely spatial prjectin, i.e., u u; =0. Nw, Curie s principle, the thermdynamical argument in which this cmmunicatin is based, implies that, in rder t build suitable relatins between the thermdynamical frces and uxes invlved in the entrpy prductin (Eq. (7)), expressin (8) shuld be decmpsed as [12]: ȧ=4a ȧ=4a u; 0 ȧ=4a 0 0 = 0 0 ȧ=4a ȧ=4a ȧ=4a ȧ=4a ȧ=4a =[u;] s ; s that the entrpy prductin becmes (9) = J [Q] ; 2 [u ;] s ; (10)

4 310 A. Sandval-Villalbaz, L.S. Garca-Cln / Physica A 286 (2000) where = 1 4 is the trace f the viscsity tensr, = is its the traceless part and [u;] s is the traceless symmetric part f the fur-gradient f the velcity, explicitly displayed in Eq. (9). We are nw able t prpse cnstitutive relatins fr thermdynamical uxes and frces using Curie s principle [9]. This principle states that, given an expressin such as (10) fr the entrpy prductin in an istrpic system, nly tensrs f equal rank may be cupled. Accrding t this idea, the mst simple admissible cnstitutive relatins that preserve the entrpy prductin psitive-semidenite are J [Q] = ; ; (11) = u ; (12) and = [u ; ] s : The transprt cecients intrduced here are, the heat cnductivity, ; the bulk viscsity and, the shear viscsity, as understd in nn-equilibrium thermdynamics. It is imprtant t stress ut that the cecient is nt related t the shear cntributin that appears in the standard decmpsitin f u; mentined earlier [8], such a relatin wuld imply directin-dependent velcity elds, an inexistent feature in FRW Universes. Rather, is micrscpically related t peculiar velcity averages [10], and it des nt aect the physical features f this type f metric. Cnstitutive equatins (12), (13) can be expressed in terms f the scale factr by and = 3ȧ a (13) (14) ȧ=4a ȧ=4a 0 0 = 0 0 ȧ=4a 0 : (15) ȧ=4a Therefre, if Eqs. (14), (15) are intrduced int the stress energy tensr f the eld equatins (2), the standard methds f general relativity fr calculating the Einstein tensr fr a given metric [8] lead t the independent equatins 1+ȧ +2a a = p(ct) ȧ ( ) a 4 +3 ; (16) 3(1+da(ct)=d(ct)) 2 [a(ct)] 2 = c 2 (ct) ȧ ( ) 3 a 4 +3 ; (17) clearly exhibiting that shear viscsity has nt aected any kind f symmetry in the FRW metric, since Eq. (15) invlves a diagnal tensr. Thus, signicant cntributins

5 A. Sandval-Villalbaz, L.S. Garca-Cln / Physica A 286 (2000) frm this variable may arise in the study f the large-scale structure f the Universe. Dissipative eects need nt be assciated with a bulk viscsity nly. Shear viscsity has been cnsidered earlier in the study f astrphysical uids by Maartens and thers [3]. Hwever, thse studies have nt been extended t the cntext f general relativity and csmlgy. T the knwledge f the authrs, Eqs. (16), (17) are the rst expressins established in a purely phenmenlgical manner that shw that shear viscsity des have an imprtant rle in the study f FRW Universes. Als Eq. (6), which is frequently used t identify the relatin between the scale factr and the thermdynamical variables is clearly aected by the presence f bth viscsity cecients. It is imprtant t emphasize that viscus eects in the cntext shwn here are strictly related t the dynamical structure f the scale factr a. In nn-relativistic physics, Eqs. (12), (13) vanish in the cmving frame, since in that case, the hydrdynamic three-velcity is exactly zer and n eects f the cvariant derivatives exist. On the ther hand, bth shear and bulk viscsity must be calculated by a suitable kinetic thery. Such a thery must als relate the curvature eects invlved in a in terms f a micrscpical mdel. This is a task that has nly been partially accmplished at the present time [11], and is part f future wrk. References [1] W. Zimdahl, Phys. Rev. D 53 (1996) [2] Ju. J. Casas-Vazquez, G. Lebn, Extended Irreversible Thermdynamics, 2nd Editin, Springer, Berlin, 1996 (Chapter 12). [3] R. Maartens, R. Class, Quantum Grav. 12 (1995) [4] D. Pavn, J. Bafalvy, D. Ju, Class. Quantum Grav. 8 (1991) 347. [5] A. Sandval-Villalbaz, L.S. Garca-Cln, Physica A 240 (1997) 480. [6] A. Sandval-Villalbaz, L.S. Garca-Cln, Csmlgical implicatins f irreversible thermdynamics, J. Gen. Relat. Grav. 31 (1999) 781. [7] E. Piña, R. Balescu, Acta Phys. Austriaca 28 (1968) 309. [8] H. Stephani, General Relativity 2nd Editin, Cambridge University Press, New Yrk, [9] S.R. de Grt, P. Mazur, Nn-Equilibrium Thermdynamics, Dver Publicatins, New Yrk, [10] E.W. Klb, M.S. Turner, The Early Universe, Addisn-Wesley Publishing Cmpany, USA, [11] J.L. Andersn, Relativity, Plenum Press, New Yrk, 1970, pp [12] G.B. Arfken, H.J. Weber, Mathematical Methds fr Physicists, 4th Editin, Academic Press, Lndn, 1995, p. 143.