CHAPTER: 6. ROTATING BIANCHI TYPE VIII & IX ANALOGUES OF des/tter UNIVERSE

Transcription

1 CHAPTER: 6 ROTATING BIANCHI TYPE VIII & IX ANALOGUES OF des/tter UNIVERSE

2 CHAPTER 6 ROT A TING BIANCHI TYPE VIII & IX ANALOGUES OF oesitter UNIVERSE 6.1 INTRODUCTION Modern astronomical observational data do not exclude the possibility of a rotating universe. The concept of a rotating universe with vanishing expansion and shear appeared with the advent of Godel (1949) universe. But it is known that the universe is in expanding phase. Therefore it seems reasonable to construct and investigate non-static rotating cosmological models. The universe will in general be filled with a cosmic fluid. It is well known that the cosmological constant A is equivalent to i. a vacuum field with p = -p. Let v be four velocity field of the cosmic fluid. Then its acceleration vector f., l the scalar of expansion e, the shear tensor d1.k and the rotation tensor wi.k defined by are k f. = v v 1. i.;k, e* = v 1. ;i 1 di.k = 2 [vi.;k + vk;i.] 1 1 i [f v + 2 f v] - v [gik - vv] l k k I. 3 ;i. l k 1 wi.k - 2 [vi.;k - vk;i.] - [fi.vk - fkvi.]... ( )

3 CHAPTER:6 PAGE: 81 where semicolon indicates covariant derivative. The rotation 0 and the shear d of the flow vector v. can be obtained from ~ ik = wik w ' - d d~k ik... (6.1.2) In the large scale behaviour of the universe, astronomical observations have tended to conform a degree of spatial anistropy. Therefore many investigators have studied spatially anistropic homogeneous world models which exhibit universal rotation. Bradley and Sviestins (1984) have discussed Bianchi type VIII rotating cosmological models filled with perfect fluid which is not thermalised. In the recent years, the investigation of anistropic cosmological models with a non-zero cosmological constant has became important due to the discovery of inflationary universe scenario Guth (1981). The behaviour of Kantowski-Sachs models with ~0 have been discussed by Weber (1984,1985), Gron(1986), Gron and Eriksen (1987) and by Sahani and Kafman (1986). Vaidya (1985) has investigated a general line element which contain two well-known Bianchi type IX metrics as particular cases, viz the desitter metric and the NUT metric with the cosmological constant. The principal aim of the present chapter is to obtain exact solutions of Einstein equations which will describe Bianchi type VIII and IX rotating analogues of the desitter model. A Bianchi type II vacuum metric will also be discussed.

4 CHAPTER:6 PAOE: : THE METRIC AND THE FIELD EQUATIONS Assad and Damiao Soares (1983) have given the unified treatment of Bianchi type II, VIII and IX space times. Taking (~,8, ) as the local co-ordinates on the homogeneity sections t=constant, they have expressed the metric of the space-time in the form 2 d.z = dt - A 2 [d'l' + 4m 2 d 2 ] - B 2 K 2 [cte 2 + Sin 2 e d 2 ]... (6.2.1) where A and B are functions of time t and m and K are functions of e satisfying the differential equations 4m Cosece m' = nk 2... (6.2.2) and K" - ~ K ' 2 + Cote K. - K = rk 3... (6.2.3) where an overhead dash denotes the derivative with respect to e. Here n and r are constants, r being proportional to the curvature of the two dimentional surface with the metric d~ 2 = K 2 [de 2 + Sin 2 e d 2 ] They have shown that the metric (6.2.1) with n ~ 0 (i) represents a Bianchi type II space-time if r=o and K=Cosece. (ii) represents a Bianchi type VIII space-time if r=l and K=tane (iii) represents a Bianchi type IX space-time if r=-1 and K=l. We wish to introduce rotation in space-time described by the metric (6.2.1). In general case, finding exact solutions of

5 CHAPTER:6 PAOE: 83 Einstein field equations describing the rotating universe is connected with many mathematical hurdles. Therefore one must make certain simplifying assumptions. to get such solutions. Consequently we take a rotating generalisation of the metric (6.2.1) in the form... (6.2.4) where B is a function of t, a is a real constant and the function m and K satisfy the equation (6.2.2) and (6.2.3) respectively. The mathematical work needed in deriving our solutions, will be considerably simplified by using differential forms. Introducing the basic one-forms e 3 = BKSine d, 1 - dt + ae... (6.2.5) the metric (6.2.4) takes the simple form... (6.2.6) Here and in what follows the bracketed indices denote tetrad components with respect to the tetrad (6.2.5). Using this tetrad and the Cartan s equations of structure, the tetrad components R b of the Ricci tensor for the metric (6.2.4) can be <a > computed. The surviving R<ab> are listed below easily

6 CHAPTER:6 PAOE: 84 2a [ 2.2 <14> B2 R = - n - B B + B ] 3a 2 B 1 R {11> = B B2 [B.2 B n 2 ] R [a 2.2 3B a = B- B + 2B - {44) B2 2n 2 ] [s8 + a 2.2 2n [.. R R<sa> ] 2n 2 r {22> = = 2B BB + 2B - ] (6.2.7) Here and in what follows an overhead dot indicates the derivative w.r.t t. We shall use the field equation Ri..k = Agi..k. Their tetrad form is R<a.b> = Ag<a.b>" Here A is the cosmological constant. This field equations and the result (6.2.7) imply the following relations: R = 0 <i.4)... (6.2.8) -R - - A (44)... (6.2.9) where R <a. b are given by (6.2.7). >. In the next section we shall discuss solutions of the equations (6.2.8) and (6.2.9).

7 CHAPTE:R:CS PAOE: : SOLUTIONS OF THE FIELD EQUATIONS The differential equation (6.2.8) can be easily integrated for the metric potential B. The solution can be expressed as B = B Cosh(nt/B ) (6.3.1) where B 0 is a constant of integration. Using the value of B given by (6.3.1) in the relation (6.2.9), it is easy to see that, A - 3r r Ct = 1 + 4n 2... (6.3.2) The tetrad components of the four velocity field of the cosmic fluid can be taken as v(a) = (0,0,0,1)... (6.3.3) The components v. of the flow vector can be obtained with the of the relations I. aid <a> <a> a v = e. and e. dx = e. i. I. I. We have computed f.,e*,c? and 0 2 for v given by (6.3.3). They are I. i. given by 2 f. = ( ab, 0, 4m ab, 0) I. c? = 01 n B... (6.3.4)

8 CHAPTER:6 PAOE: 9, by ( ). Here it should be noted that r=o, we get A=O. In this case we obtain an empty space-time with Bianchi II symmetry. The geometry of this rotating Bianchi II space-time is described by the line element... ( ). We have verified that the curvature tensor for the metric (6.3.5) is non-zero. Hence the empty space-time described by (6.3.5) is non-flat. It admits a time like congruence vi. which is rotating, expanding and geodetic. 2 If r=l, a becomes strictly positive. In this case K=tane and the equation (6.2.2) admits the solution m 2 = ~ [ Case + Sece ]... (6.3.6) To have a familiar form of the metric we apply the co-ordinates transformation e = -logcose, = 2... (6.3.7) Using this the rotating Bianchi type VIII desitter like metric can be expressed as

9 CHAPTER:6 ds 2 = [dt + a.b 0 Cosh(nt/B 0 ) [ dl.f + 2nCosh8 d ] ]z PAOE: Bi - B:Cosh 2 (nt/b 0 ) [ [ dl.f+2ncosh8 d d8 2 + Sinh 2 8 d 2 ] ( ) 2 1 Note that here a. = Therefore non-rotating counterpart of 4n 2 the solution (6.3.8) does not exist. If r = -1 and K = 1 the solution (6.2.2) admits the solution z n m = - 2 Case... ( ) Therefore the rotating Bianchi type IX desitter-like metric can be expressed in the final form as 2 ds 2 = [ dt + a.b 0 Cosh(nt/B 0 ) [ dl.f-2ncose d ] ] - B~Cosh 2 (nt/b 0 ) [ [dl.f-2ncose d ] 2 +d8 2 +Sin 2 8 d 2 J... (6.3.10) It should be noted that for the metric (6.3.8)~ A = -~ B 2 and for 4 0 the metric (6.3.10) A Thus the cosmological constant is negative for Bianchi type VIII model and it is positive for Bianchi type IX model. For the metric (6.3.10), z a. = 1 2 Th 2 > n. us n - 4' a. vanishes when n = ± ~. In this case the rotation disappears and the metric (6.3.10) reduces to

10 CHAPTER:6 PAOE: (6.3.11) We wish to cast the metric (6.3.11) in a more familiar form by the introduction of the Cartesian co-ordinates x, y, z defined by Vaidya (1985) x = R 0 Sin(B/2) Cos[( - ~)/2] y = R 0 Sin(6/2) Sin[( - ~)/2] z = R 0 Cos(8/2) Sin[( - ~)/2]... (6.3.12) The line element (6.3.11) then takes the form ds z = d t z -Cosh z ( t/r [ dxz+d z+d z+ [ xdx+ydy+zdz] 2 ) Y ] 0 z R -[ x +y +z 1 The metric (6.3.13) is the well-known desitter metric (6.3.13) Therefore the metric (6.3.10) represents the rotating desitter universe. In all the above mentioned solutions the fluid velocity field vi is expanding, rotating, non-shearing and non-geodetic. The explict expressions for 8 and 0 are e* = ~ tanh(nt/8 0 ) 0 0 = an z Sech (nt/8 0 ) Bz 0... (6.3.14) At the grand unified theory (GUT) time t j_ =1.0x10 there is an effective cosmological constant so that 8 0 = tj_. -35 sec.

11 CHAPTER:6 PAOE: BS:: From the result (6.3.14) it is clear that rotating Bianchi type VIII and IX universes discussed above, shows a transition to exponential expansion at this time, independently"of the magnitude of ~which characterises the rotation. This is a characteristic of inflationary era. The result (6.3.14) also shows that the rotation exhibits a transition to exponential decay at this time. This is an interesting feature of our solutions. 4: CONCLUIDING REMARKS In this chapter, a general metric is considered which can describe Bianchi type II, VIII and IX space-times under certain conditions. Exact soluitons of the field equations Rik= Agik are obtained where A is a cosmological constant. The Bianchi soluiton represents.a homogeneous rotating generalisation type of IX the well-known desitter universe. A non-flat rotating Bianchi type II empty space-time is also discussed. The solutions discussed here are all homogeneous. In the next chapter, we shall investigate some rotating inhomogeneous universes with matter and a scalar field.

12 CHAPTER:6 PAOE :90 REFERNCES: 1. Assad M J D and Damiao Soares I, Phys. Rev., 028, 1858 (1983). 2. Bradley Hand Sviesteins E, Gen.Rel.Grav., 16, 1119 (1984). 3. Godel K, Rev. Hod. Phys., 21, 447 (1949). 4. Gron 0, J.Hath.Phys., 27, 1490 (1986). 5. Gron 0 and Eriksen E, Phys. Lett., A121, 217 (1987). 6. Guth A H, Phys.Rev., D23, 347 (1981). 7. Sahni V and Kofman LA, Phys. Lett., A117, 275 (1986). 8. Sviestins E, Gen.Rel.Grav., II 521 (1985). 9. Vaidya PC, Pramana (J.Phys.), 25, 513 (1985). 10. Weber E, J.Math.Phys., 25, 3279 (1984). 11. Weber E, J.Math.Phys., 26, 1308 (1985).