ETERNAL

AND PULSATING

EINSTEIN-CARTAN

M. J. D. A S S A D

UNIVERSES

IN

SPACE-TIME

and C. R O M E R O

Departamento de Fisica, CCEN, Universidade Federal da Paralba, Jo~o Pessoa, PB, Brazil

(Received 18 July, 1991) Abstract. We exhibit a class of closed (Bianchi type IX), homogeneous and isotropic universes generated

by an unpolarized spinning perfect fluid on the basis of Einstein-Cartan theory. The models are non-singular and pulsate around a static configuration in an endless number of cycles. All of them undergo periodic phases of accelerated (inflationary) expansion and may be reduced to the dynamics of an anharmonic oscillator.

1. Introduction

Although the Friedmann-Robertson-Walker (FRW) models are the most studied universes of the General Relativity (GR) theory, its evolutionary features are still scarcely known in the Einstein-Cartan (EC) theory (cf. Hehl et al., 1976). In spite of the fact that the flat F R W - E C models were already extensively discussed (cf. Trautman, ~.973; Kopczynski, 1973; Kuchowicz, 1976; Gasperini, 1986), the evolution of the corresponding closed and open universes still deserves a deeper analysis. It was Kerlick (1975) who proved that one cannot incorporate a polarized perfect fluid (aligned distribution of spins) as source of a closed (Bianchi type IX) cosmology. Such a kind of model can only be generated by a perfect fluid in which the spins of its particles are randomly oriented. In this way, Kuchowicz (1978) exhibited F R W - E C exact solutions, among them a closed model which is non-singular and oscillates between a maximum and a minimum radius. However, he obtained this solution at the expense of getting rid of the spin conservation law in EC theory. The late assumption makes Kuchowicz's particular model inconsistent with the field equations, as it will be seen. On the other hand, Nurgaliev and Ponomariev (1983) (NP) obtained a class of closed F R W - E C solutions, which are static and stable under homogeneous and isotropic infinitesimal perturbations. Like the papers by Kuchowicz (1978) or Nurgaliev and Ponomariev (1983), we discuss the class of closed F R W - E C universes by assuming that the source of the gravitational field is an unpolarized spinning perfect fluid. By applying the methods of dynamical systems theory we will show that: (a) There exists a class of closed F R W - E C solutions which exhibit the essential features of Kuchowicz's one, without the need of assuming the violation of the spin conservation law. (b) The class of static NP models is a special solution pictured in the phase space as an equilibrium point (a center). (c) The infinitesimally perturbed NP solutions appear in the phase portrait as elliptical orbits in the neighbourhood of the equilibrium point. (d) For finite perturbations of the static solution, the orbits are still closed, but no longer reproduce the harmonic behavior Astrophysics and Space Science 191: 289-297, 1992. 9 1992 Kluwer Academic Publishers. Printed in Belgium.

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M . J . D . ASSAD AND C. ROMERO

described by NP. (e) In both cases (infinitesimal and finite perturbations), the strong spin-spin repulsive effects overwhelms the gravitational attraction in the stages of huge density of matter, preventing the collapse to a singularity (the Universe reaches its minimum radius) and re-accelerating the Universe backwards to an expansion era. (f) Also, for large values of the expansion factor, the gravitational attraction would lead the Universe to recolapse (after reaching a maximum radius), since expansion weakens the spin terms. (g) The models evolve through cyclic inflationary epochs and one obtains such a scenario by either considering the source terms in the form of the postulated Weyssenhoff semi-classical energy momentum tensor (cf. Weyssenhoff and Raabe, 1947), as particularly done by Kuchowicz (1978) or Nurgaliev and Ponomariev (1983), or the Ray- Smalley improved energy momentum tensor (RS-EMT) &matter with spin (cf. Ray and Smalley, 1982, 1983). (h) By using the conformal form of the FRW metric the EC field equations reduce to the form of that an anharmonic oscillator: a simple model consistent with the phase portrait numerically integrated.

2. The Field Equations We shall investigate the cosmological solutions of the EC field equations (units c = 8rcG = 1) ....... ical G a B = R a B - 51g A u R = T --AB J a B c -- TAB c +

26c~,~Tmo ~ = SAc c ,

(la) (lb)

where GAB is the Einstein tensor of the Riemann-Cartan space-time (U4) and Y]~ ~176 is the canonical energy momentum tensor for matter with spin. TAB c is the torsion, JAB c is the modified torsion, and S A ~ c is the spin density tensor. Capital latin indices refer to a holonomic basis. When one adopts the Weyssenhoff-Raabe (1947) semi-classical description for a perfect fluid with spin, as done in Kuchowicz (1978) or Nurgaliev and Ponomariev (1983), the spin density tensor is factorized in the form SAB c = SABU c ,

(2a)

S A B "~ = S~Bid ~ = O ,

(2b)

g A B = '~

(2c)

Idc S D '

where S A 8 = - S B A , ldA is the normalized four-velocity of the fluid, S ~ is the pseudovector of spin density and eo~23 = 1. The phenomenologically assumed canonical energy-momentum tensor is given by --ABT . . . . . . . ical =

TABF 4;- 2 ( T c S A D C ) I d D b I B ,

(2d)

.g

where 7 c - 7 c + 2 T c o D, and TAFB is the usual energy momentum tensor for a perfect fluid with energy density p and pressure P T AgB = (P + P) uA uB -- P g A B "

(2e)

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If we use the U4 geometrical identity G[ABI = 7 c.~ABc , we get the equation of motion for spin with the help of (1) and (2d) #

*

gr

VCXAB c =

-

(%XB,,C)u~

(2f)

The earlier studies ofEC cosmologies (Trautman, 1973 ; Kopczynski, 1973; Kuchowicz, 1976) used the spin conservation law VcSABC= 0 instead of (2f). Of course, all solutions of this law solve (2f), but the converse is not generally true. A criterion to decide whether the above equation of motion must be used, or the conservation law is a sufficient one, is given by Obhukov and Korotky (1987). They proved that, if the # conditions (2a, b) are satisfied, the conservation of spin occurs, i.e., 7CSAe c = 0, if and only if the elements of the fluid move everywhere without acceleration in U4, i.e.,

0 c= u~

c= O.

Indeed, this will be the case considered. We assume the space-time metric in the FRW form ds 2 = + dt 2 _ R2(t) [dz 2 + o'2(Z) (d02 + sin20d02)] ,

(3a)

cr(z ) _ sinw/e ,,~ Z ,

(3b)

where

with e - 0, _+ 1, standing for the curvature parameter. Equations (2b) and (tb) shows ,g

that the torsion tensor is trace free. Hence, .~B c = TABc =SAe c, 7 c = 7 c and the U4 connection satisfies FooC= O. In consequence, the evolution of the pseudo-vector of spin density is given by s{,

S'(t)=R3,

(i= 1 , 2 , 3 ) ,

(4)

where S~ are integration constants. The squared spin terms scale: with R - 6 , as we exactly determined from the field equations, i.e., from the spin conservation law. Hence, to abandon (4), as Kuchowicz (1978) did by assuming $2oc R-2a, b = 2, leads to an inconsistency both with the field equations and the FRW geometry generated by a Weyssenhoff fluid. To obtain solutions in the case e = + 1, we must assume that the source of the gravitational field is an unpolarized perfect fluid. Even though the spins are randomly oriented, the average of the spin-squared terms does not vanishes in general. Then we have (SAB)

= 0, (5)

1

S ~ -- ~(SA~S AS) r O. The field equations (1) furnish

3

P = )s

+

+

So

(6a)

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M. J. D. A S S A D A N D C. R O M E R O

P=

-

R

~,R- )

- R~ + R ~

(6b)

"

Before discussing the solutions of (6), let us consider the same problem on the basis of the self-consistent Einstein-Cartan theory, which makes use of the R S - E M T . According to this point of view, the spin density is prescribed by SA C

=

S~BU c

= ~1R S A B U

(7a)

C,

where SAB = ]c(EIAEaB

-

EIBE2A)

(7b)

,

is the specific spin tensor density and k is a function. A set of tetrads E " A is chosen by identifying one of them with the matter four-velocity (Greek indices referring to a anholonomic basis)

(%)

u A = Eo A .

Hence, it follows that the fluid obeys the condition (2b). The self-consistent form of the EC field equations with the R S - E M T , for the trace free torsion case, is GAB - V c ( T A B c -- TACB + TCAB) = T(ASB s) ,

(Sa)

which, due to Equation (1), can be written in the form GA ,

....... ioal = = V --A B

T(A~ s) +

2VcSC(AB)

+ VcSA, c

9

(8b)

The R S - E M T consists of the perfect fluid part (2a) and a intrinsic spin part T AB

=

pu(AsB)Cs

+ Vc[pU(BS A)c] + pU(AsB)CWcD uD -- p W C ( ~ s B ) c , (9)

The spin angular velocity is W4B = E"aE~B, where an overdot represents the proper time covariant derivative E " A = u B 7 8 E " A. Then we have R

w~B = w~B - s ~ ,

(10)

R

with WAR standing for that part of the spin-angular velocity depending only on the Christoffel symbols. Moreover, conservation of spin ensures that the canonical energy-momentum tensor in (8b) is actually a symmetrical one. Introducing (9) in (8b), we note that the terms containing derivatives cancel due to (7a). Hence, only the two last terms in (9) survive besides the perfect fluid part in the R S - E M T , since/~c = 0. A closer look into these R

remaining terms reveals that only WcD contributes to the third term of (9), due to (2b). Taking into account a random distribution of spins, the EC field equations with the R S - E M T in the case of spin conservation are GAB = TFB -- 2 ( S A C S B c )

.

(11)

ETERNAL AND PULSATING UNIVERSES IN EINSTEIN-CARTAN SPACE-TIME

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We restrict our considerations to the case wherein an equation of state of the form p = (2 - 1)p, 1 _< y _< 2, holds. Then, the EC field equations fttrnish RR" + c~k 2 + f i r m + F = 0,

(12a)

where ~Z=(~),

F = ( 3 7 2 2-) e,

and

fi=-(~Y)So

2

if assuming (2d). Otherwise,

in the case of (8a). In any case fi ~< 0, for 1 _< 7 < 2. If m r - 4, we will get rid of the spin conservation equation in the sense alluded above. A first integral for (12a) is R2=p2 R

2~

2fi R, ~ _ _ , F 2~+m c~

(12b)

where p2 is a positive integration constant. Substituting (12b) in (12a) one gets fi" R

~'P~ + f l ( 2 a z l ~ R m R 2~ \2a + m/

"

(12c)

Note that this equation does not explicitly depend on the curvature parameter e. As a matter of fact, the above equation was discussed by Gasperini (1986), who demonstrated the existence of consecutive phases of superinflation, exponential and power-law inflation in flat F R W - E C models generated by a perfect fluid with a random distribution of spins. The same analysis is applied here. 3. The Phase Portrait

A general exact solution for (12b) is not yet found. However, defining R = x, and R = y, one obtains from (12a) the plane autonomous system • = y = F(x,y),

(13a)

9 = - 1 (c~y2 + fix m + F) = H ( x , y ) .

(13b)

X

The qualitative behaviour of the solutions of (13) is better investigated if one constructs the phase diagrams of the system. The integral curves appearing in these diagrams represent cosmological models evolving in time. However, special solutions corresponding to the equilibrium points of the dynamical system (points of the plane where the right-hand side of Equation (13) vanishes simultaneously) describe static

294

M, J, D. A S S A D A N D C. R O M E R O

models. A brief look into these equations reveals that these static solutions lie on the x-axis (y = 0) and correspond to the roots of the equation (14)

x m = - r/ft.

Once the equilibrium points are found, then their topological nature is determined by the standard methods of the dynamical system theory (Andronov e t al., 1973). Note that the system (13) is not defined for x = 0. Also, since x < 0 has no physical meaning, we restrict ourselves to the analysis of the region x > 0. Now, for e = - 1, it turns out that the system has no equilibrium points, static solutions with open FRW metric are not permitted in Einstein-Cartan theory. Let us consider ~ = 1. As we mention earlier, the spin conservation condition implies that the only value to be assigned to the parameter m is - 4. In this case, (14) has only one real and positive root, namely, x o = ( - f l / F ) 1/4. The exact location of this point on the x-axis depends clearly on the values chosen for So2, as well as on the equation of state of the fluid. It is worthwhile mentioning that this solution is formally identical to Einstein's static solution in G R theory, although we are in presence of matter with spin. To proceed the determination of the topological nature of the equilibrium point x o we must evaluate the trace I and the determinant A of the matrix M

M =

(3-X1C~c~ 8ySF(~=~o,

cy

(15) y=o)

evaluated at x = x o and y = O. Straightforward calculation gives us (for arbitrary m) I = 0 and A = - ( m F / x ~ ) . For m < 0 and ~ = 1, we get A > O, characterizing a configuration of closed curves around x o (a c e n t e r ) . Figure 1 depicts the corresponding

Y

Fig. 1. Closed curves around the center x = xo.

ETERNAL AND PULSATING

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phase portrait, numerically integrated with the aid of P H A S E R (cf. Kogak, 1986) by assuming Xo = 1 (point P in the figure). Looking into this diagram we are led to the conclusion that the closed curves represent a class of cyclic and eternal universes whose radii oscillates between a minimal (Rmin) and a maximal value (Rmax) passing through the intermediary value Ro = Xo (which corresponds to the radius of the static universe). The explicit calculation of Rmin and Rma~can be done by the use of(12b). A typical solution of this class alternates expansion and contraction eras indefinitely. Thus, we have a universe which bounces forwards and backwards during an endless number of cycles. At this point, it should be noted that this bouncing effect is due only to the presence of spin. Moreover, if we disregard spin, then we are left with Friedmann closed model (e = 1) - a tmiverse which undertakes no more than just one cycle from the initial Big Bang to the final collapse, as shown in Figure 2.

\ /

x

Fig. 2. Friedmann's one cycle model. The existence of accelerated (or inflationary) phases in each of the curves in Figure 1 may be seen just by considering the region of the phase plane defined by - ~ < y < and x " < - F/ft. If ~ is chosen sufficiently small then R" - y > 0, and inflation occurs immediately after the Universe has attained its minimum value Rmln. At this stage, the spin density is maximal and a repulsive spin-spin interaction arises, which is responsible for the accelerated expansion. This effect becomes more significant as we go away from the inner to the more external curves, since in this process Rmln gets smaller. In this way, we see that both phenomena (the avoidance of gravitational collapse and the existence of an inflationary phase) are a consequence of the repulsive spin-spin force acting as negative pressure at the epochs the Universe experiences its minimum size. 4. Final Remarks

Let us examine the stability of the static solution under both finite and infinitesimal metric perturbations. Here, by finite perturbations of the static Einstein-type solution

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M. J. D. A S S A D A N D C. R O M E R O

(represented in the phase diagram as the equilibrium point xo) we mean just going from x 0 to one of the closed curves around x o. Since the behaviour of these curves is periodic and oscillatory, we conclude that the static solution is stable under finite perturbations. To see what happens if we perform infinitesimal perturbations in the static solution (R --, R + 5R), we expand the right-hand side of Equation (13) around x o keeping only the linear terms. Then, we have (~x) + o92 5x = O,

(16)

where 6x = (x - Xo) and w2 = fimx[~~ - 2 > 0. We conclude that the perturbed universe oscillates harmonically around the static configuration, as obtained by NP. Using the conformal form of the F R W metric one gets a simple model consistent with the phase portrait shown in Figure 1. After introducing the conformal time coordinate dt = R dr/, we make the scale transformation (Kamke, 1977) R ~: '

(~#o),

z = lnR,

(~ = 0),

Z=

(17)

to get from (12a) Z"

+ c~FZ :

- ~I3Z ~. . . . )/~,

Z" + F = -fie "'z ,

(c~ # 0 ) ,

(~

=

(18)

0),

where Z ' = d Z / d t 1. The first corresponds to the classical motion of a particle under the asymmetric potential valley

v ( z ) : re% z2 + __~'2B zc2~+~/~ 2

2c~+m

which has a minimum at Z "'/~ = - F / ~ . Though this is a slight generalization of a well-known behaviour of the FRW universes in G R theory (cf. Assad and de Lima, 1988), it constitutes a simple example that quantum effects, even if classically considered, may drastically change our picture of the evolution of the Universe.

References Andronov, A. A., Lentovich, E. A., Gordon, I. I., and Maier, A. G,: 1973, Qualitative Theory of Second-Order Dynamical Systems, John Wiley and Sons, New York. Assad, M. J. D. and Sales de Lima, J. A.: 1980, Gen. Rel, Gray. 20, 535. Gasperini, M.: 1986, Phys. Rev. Letters 56, 2873. Hehl, F. W., vonder Heyde, P., Kerlick, G. D., and Nester, J. M.: 1976, Rev. Mod. Phys. 48, 393. Kamke, E.: 1977, Differentialgleiehungen: Lgsungs-Methoden und L6sungen, B. G. Teubner, Stuttgart. Kerlick, G. D.: 1975, Doctoral Thesis, Princeton University. Kodak, H.: 1986, Differential and Difference Equations Through Computer Experiments, With Diskettes ContainO~gPhaser : an Animator/Simulatorfor Dynamical Systems for the IBM Personal Computers, Springer-Verlag, New York. Kopczynski, W.: 1973, Phys. Letters 43A, 63.

ETERNAL AND PULSATING UNIVERSES IN EINSTEIN-CARTAN SPACE-TIME Kuchowicz, B.: 1976, Astrophys. Space Sci. 39, 157, Kuchowicz, B.: 1978, Gen. Rel. Gray. 9, 51I. Nurgaliev, and Ponomariev: 1983, Phys. Letters 130B, 378. Obukhov, R, N. and Korotky, V. A.: 1987, Class. Quantum Grav. 4, t633. Ray, R. and Smalley, L. L.: 1982a, Phys. Rev. Letters 49, 1059. Ray, R. and Smalley, L. L.: 1982b, Phys. Rev. D26, 2615. Ray, R. and Smalley, L. L.: 1982c, Phys. Rev. 26, 2619. Ray, R. and Smalley, L. L.: 1983, Phys. Rev. Letters 50, 623(E). Trautman, A.: 1973, Nature Phys. Sci. 242, 7. Weyssenhoff, J. and Raabe, A.: 1947, Acta Phys. Pol. 9, 7.

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