Volume 149B, number 1,2,3 PHYSICS LETTERS 13 December 1984

KALUZA-KLEIN-BIANCHI-KANTOWSKI-SACHS COSMOLOGIES

D. LORENZ.PETZOLD Fakultitt fur Physik, Universitiit Konstanz, D-7750 Constance, Fed. Rep. Germany

Received 31 July 1984 Revised manuscript received 4 September 1984

Einstein's field equations axe solved for the class of higher-dimensional Bianchi-Kantowskl-Sachs space-times of the Kaluza-Klein type. The solutions given axe cosmological models filled with pressure-free matter, in general with a non- vanishing cosmological constant.

1. Introduction. Recently several solutions of higher-dimensional gravity have been studied. Chodos and Detweiler [1 ] discussed a generalized Bianchi type I vacuum solution of the Kasner-type [2] in (4 + D) dimensions. It has been shown that the "extra" dimensions in such generalized Kaluza-Klein-type theories may contract to an unobservable scale [3,4]. The Chodos-Detweiler solution has been extended by Freund [5 ] to the higher-dimensional version of the scalar-tensor theory of gravitation due to Brans and Dicke [6] (see also ref. [7]). Such solutions are close- ly related to exact solutions for N = 1 supergravity in ten and eleven dimensions and are therefore of much interest [2,8,9].

However, all of these models are free of matter and therefore cannot describe the behaviour of matter in the universe. A first step into the direction of con- struction of some more realistic models has been recently made by Bergamini and Orzalesi [10] (see also ref. [ 11 ]). The resulting model is a Bianchi type I dust solution in (4 + D) dimensions. Another anise- tropic solution was given by Sahdev [12] for the radia- tion-dominated Kantowski-Sachs universe in (1 + d + D) dimensions, which is closely related to the Bianchi type III space-time [13]. The same space-time in (2 + D) dimensions has been recently discussed also by Ishihara et al. [14], including a non-vanishing cosmo- logical constant A in the vacuum case. Such an effec- tive cosmological term has been considered by many authors in relation to the phase transition in an early stage of the universe [15].

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Tins work is a continuation of our recent study [16] of higher-dimensional vacuum Bianchi cosmolo- gies with (and without) a cosmological constant. We consider the Bianchi-Kantowski-Sachs models filled with dust in (1 + 1 + D) dimensions. Our models are characterized by two time-dependent cosmic scale functions with different behaviour on M 1 and M D.

2. Field equations and solutions. The field equa- tions to be considered are

Ruu = [ 2 / ( N - 2)1Aguv + [ 1 / ( N - 2)](e - P ) g u u

+ (e + p)uuuu, ( la)

where Ruv denotes the Ricci tensor, guu the metric tensor, A the N-dimensional cosmological constant, u u the velocity four-vector and e and p are, respective- ly, the energy density and pressure of perfect fluid matter. The N-dimensional gravitational constant G N = GO[2d+D-3 has been absorbed in the definition of e [12,17]. We consider only non-tilted models, i.e. u u = 80. In the following we are interested in perfect fluid solutions obeying the equation of state p = 0 (dust). In general, we assume a metric of the form

- 1

g,v = r2gmn , ( lb)

R 2

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Volume 149B, number 1,2,3 PHYSICS LETFERS 13 December 1984

where r = r d = rd(t), R = R D = R D ( t ) are the cosmic scale funct ions , gmn, gMN the B i a n c h i - K a n t o w s k i metr ics on M a × M D wi th curvature constants k = kd ,

K = K D (wi th values (0, > 0 , < 0 ) ) and #, v = 0, 1 . . . . , d + D; m , n = 1, . . . , d ; M , N = d + 1 . . . . . d + D. The corresponding field equat ions to be solved are

+ h(dh + D H ) + 2 k / r 2 = ( 2 A + e - p ) / ( N - 2) , (2a)

f I + H ( d h + D H ) + 2 K / R 2 = (2A + e - p ) / ( N - 2) , (2b)

(dh + D H ) 2 - dh 2 - D H 2 + 2kd / r 2 + 2 K D / R 2

= 2(A + e ) , (2c)

~- + (e + p ) (dh + D H ) = 0 , (2d)

where h = (In r) ' , H = (In R)" are the Hubble parame- ters, e and p are the dens i ty and the pressure o f perfect fired ma t t e r and ( )" = d/dt. In order to decouple the field equat ions (2) we assume d = 1, D = n (i.e. N = 2 + n) and k = 0.

The field equat ions can be decoup led to give

2 n f I + n(n + 1)H 2 + 2 n K / R 2 - 2A = 0 , (3a)

h H = f l + H 2 + m/ rR n, e = n m / r R n , (3b)

where rn = const . Af te r solving eq. (3a) for R = R ( t ) the solut ions are comple t ed b y solving eq. (3b) for r = r(t) . The first integral o f (3a) is given b y

n 2 - cR -(n+l) - 2A/n(n + 1) + [2K/(n - 1)]R - 2 = 0,

where c = const . (4) Fo r c = 0 we can ob ta in general solut ions vafid for

all d imensions n / > 2. I f A 4= 0 we obta in : Solu t ion (i). K = - 1 , A < 0:

R = ( l / a ) sin at , a 2 = - 2 A / n ( n + 1), K = 2 K / ( n - 1 ) ,

r = C cos at + ma n - 2 [(sin at) - n - (cos a t ) 2 F n ] ,

F n = (cosec at) n

l -1 + ~ ( 2 l - 1) ... ( 2 t - 2 k + 1) ( coseca t )2 t_2k

k = l 2 k ( l - 1 ) . . . ( l - k )

+ n ( 2 l - - 1)!! (In tan ~at) (cos a t ) - l , n = 2 l , 2 l l !

F n = (cosec at) n

l -1 2 k (l - 1) ... (l - k ) (cosec at) 2 t - 2 k - 1 ,

+ k = l ~ (2 [ - -3 )~ . . - (2 - l - - - ~ 1)

n = 2 l - I . (5)

Solu t ion (ii). K = - 1 , A > 0:

R = ( l / a ) s i n h a t , a 2 = 2A/n(n + 1),

r = C cosh at + ma n - 2 [ - ( s i n h at) - n + (cosh at) 2 G n ],

G n = - ( c o s e c h at) n

I -1 + ~ ( _ l ) k _ 1 (2 l -- 1) ... ( 2 l - 2 k + 1)

k = l 2k(l - 1) ... ( l - k)

X (cosech at) 21-2k

t _ l ~ l ( 2 / - - 1)!! (In t anh ½at)(cosh a t ) - l , + n , . , (2/)! !

n = 2 l ,

G n = - ( c o s e c h at) n

1-1 2 k (l - 1) ... ( l - k )

+ ~ ( - - 1 ) / - 1 ( 2 / - - 3 ) . . . ( 2 / 2 k 1) k = l -- --

X (cosech at) 2 1 - 2 k - 1 , n = 2l - 1 . (6)

So lu t ion (iii). K = 1, A > 0:

R = ( l / a ) c o s h a t , a 2 = 2A/n(n + 1) ,

r = C sinh at + ma n - 2 [ - ( c o s h at) - n + (sinh at) 2 H n ],

H n --- (sech at) n

1-1 + ~ ( 2 l - 1) ... ( 2 l - 2 k + 1) (sech at) 2 l -2k

k = 1 2 k (l -- 1) . . . ( I - k)

( 2 l - - 1 ) Tw 1 + n ( 2 O H ' " (a rc tanh(s inh a t ) ) ( s inh a t ) - ,

n = 2 I ,

H n = (sech at) n

l -1 + ~ 2 k ( l - 1) ... ( l - k) (sech at~ 21-2k-1

k = 1 ( 2 1 _ 3) . . . ( 2 1 _ 2 k _ l ) , , ,

n = 2l - 1. (7)

So lu t ion (iv). ~" = 0, A > 0:

R = a e x p ( b t ) , b E = 2A/n(n + 1 ) ,

r = B exp(b t ) - [m/anbE(n + 1)] e x p [ - b ( n + 1)] . (8)

I f A = 0 we have:

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Volume 149B, number 1,2,3 PHYSICS LETTERS 13 December 1984

Solution (v). K --- - 1 , R = t ,

r = b +m(ln t ) , n = 2 ,

r = b + (m/2 - n) t 2-n , n 4: 2 . (9)

We now turn to the general case c 4: 0. I f n = 2 we obtain the four-dimensional LRS (locally rotationally symmetric) Bianchi type I, type III and Kantowski- Sachs dust solutions discussed by us previously [ 1 8 - 20]. I f n = 3 we obtain the following new five-dimen- sional solutions:

Solution (vi). ~" = - 1, A < 0:

R 2 = (2/a2)(1 + b sin a t ) ,

r2 = (a2/2) [C + (m/b) tan at] 2 (cos at) 2 1 + b sin at

a 2 = - ] A , c = ( b 2 - 1 ) / a 2 . (10)

Solution (vii). K = 1, A > 0:

(a) R 2 = (2/a2)(1 + b sinh a t ) ,

r2 = (a2/2) [C + (m/b) tanhat ] 2 (cosh at) 2 1 + b sinh at

a 2 = ] A , c = ( b 2+1) / a 2 . (11)

(b) R 2 = (2/a2)(1 +b cosha t ) ,

r2 = (a2/2) [C - (m/b) coth at] 2 (sinh at) 2 1 + b cosh at

a 2 = ] A , c = (1 - b2)/a 2 . (12)

(c) R z = (2/a2)[1 + b exp(at)],

r2 = (a2/2) [C exp(at) - ( m / Z b ) exp(-a t ) ] Z 1 + b exp(at)

a 2 = ] A , c = l / a 2 . (13)

Solution (viii). K = 0, A > 0:

R 2 = 2a sinh b t ,

r 2 = [C + (re~b) tanh bt] (cosh bt) 2 sinh bt '

b 2 = ] A , a2b 2 = c . (14)

Solution (ix). ~-2 = 1, A = O:

R 2 = a - ~'t 2, r 2 = (Ct + mK)2/(a - K t 2 ) ,

c = a ~ ' . (15)

In addition to the time-dependent solutions r = r(t) and R =R(t) we obtain also the following vacuum solu- tion with one internal radius R = const,:

Solution (x).

R 2 = n ( n - 1 ) ~ ' / 2 A , m = 0 .

r = u exp(bt) + o e x p ( - b t ) ,

b 2 = 2 A / n . (16)

In eqs. (5) - (16) we have B, C,a, b, c, u, o = const.

3. Conclusions. In conclusion we have given a com- plete discussion of the higher-dimensional Bianchi- Kantowski-Sachs space-times of type I and type III both in the dust as well as in the vacuum (m = 0) case. Our solutions (5 ) - (16) exhibit the interesting feature that the evolution o f the "radius" R is independent of perfect fluid matter p = 0. If in addition m = 0 (vacuum) we have r ~ /~ , i.e. the size o f ordinary space (n = 3) is changing very slowly (or rapidly), the size of the extra space M d = M 1 is small (or large). The detailed behav- iour of our new solutions will be discussed in a forth- coming paper [21 ].

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(World Scientific, Singapore, 1984). [4] W. Mecklenburg, Fortsehr. Phys. 32 (1984) 207. [5] P.G.O. Freund, Nucl. Phys. B209 (1982) 146. [6] C. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 925. [7] M. Nishioka, Nuovo Cimento 74B (1983) 27. [8] B. Dolan, Phys. Lett. 140B (1984) 304. [9] D. Lorenz-Petzold, J. Math. Phys. (1984), to be published.

[10] R. Bergamini and C.A. Orzalesi, Phys. Lett. 135 (1984) 38. [11] E. Alvarez and M.B Gavela, Phys. Rev. Lett. 51 (1983) 931. [12] D. Sahdev, Phys. Lett. 137B (1984) 155. [13] M.P. Ryan and L.C. Shepley, Homogeneous relativistic

cosmologies (Princeton U.P., Princeton, NJ, 1975) [14] H. Ishihara, K. Tomita and H. Nariai, Prog. Theor. Phys.

71 (1984) 859. [15] G.W. Gibbons, S.W. Hawking and S.T.C. Siklos, The very

early universe (Cambridge U.P., Cambridge, 1983). [16] D. Lorenz-Petzold, Phys. Lett. 148B (1984) 43. [17] P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [18] D. Lorenz, Phys. Lett. 92A (1982) 118. [19] D. Lorenz, J. Phys. A15 (1982) 2997. [20] D. Lorenz, J. Phys. A16 (1983) 575. [21] D. Lorenz-Petzold, Universitiit Konstanz (1984), in

preparation.

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