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Non-chaotic Kaluza-Klein cosmology
Volume 192, number 3,4
PHYSICS LETTERS B
2 July 1987
NON-CHAOTIC KALUZA-KLEIN COSMOLOGY Lars Gerhard J E N S E N CERN, CH-1211 Geneva 23, Switzerland
Received 20 March 1987
We investigate the evolution of higher-dimensional, homogeneous, anisotropic cosmologies. The natural extension to higher dimensions of the three-dimensional mixmaster cosmologies are analysed, and it is found - contrary to the three-dimensional case - that the higher-dimensional cosmologiesexhibit no chaotic behaviour.
1. Introduction. In the past few years higherdimensional cosmologies have received a great deal o f attention [ 1 ]. This is due to the theoretical possibility of a unification of the fundamental forces in a higher-dimensional spacetime. In these theories the extra dimensions are usually assumed to compactify at some energy scale of the order of the Planck mass, while the remaining dimensions expand into our observable universe. In the type o f model commonly considered, the extra dimensions are curled up in a maximally symmetric n-sphere. This rather arbitrary assumption is usually made because o f the simple geometry of maximally symmetric spaces. It is well known from 3 + 1 dimensions that even if one considers a less symmetric geometry, there are interesting models, such as the mixmaster cosmologies [2], which can evolve into our observable universe but which, close to the initial singularity, enter an infinite succession of Kasner epochs during which the three scale factors oscillate with arbitrary large amplitude and frequency. It is also well known that this type of chaotic behaviour is exhibited in the most general (inhomogeneous) cosmologies close to the initial singularity [3]. Recently it has been shown that the same result is true for spatial dimensions between four and nine, and that the chaotic behaviour disappears for dimension ten and above [4]. For (n + 1 )-dimensional cosmology with n/> 10, the general approach to the singularity is the generalized Kasner solution. In this case the universe enters a finite succession of Kasner epochs, of which the last
one is stable and determines the approach to the singularity. For homogeneous cosmologies the problem of the approach to the singularity is still unsolved for 4 ~
2. H o m o g e n e o u s cosmology. The extension to n + 1 dimensions of the usual (3 + 1 )-dimensional Bianchi models consists o f the cosmologies with a oneparameter family of n-dimensional spatial hypersurfaces with a Lie group acting simply transitively, i.e., the group motion may carry any point into any other point. The Lie group G has n generators Xa, a = 1..... n and corresponding invariant forms e a - e~a dx, = 1, ..., n, which satisfy d e " = -~,~bc~Pa ~b ^~c, where Cgc are the structure constants of G. We shall assume that the metric is diagonal in the ea-basis:
ds 2 = d r 2-Ra2(ea) 2 . Then we find the following 0 - 0 and a - a Einstein equations in vacuum:
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
315
Volume 192, number 3,4
~
1(
2~/g
~
PHYSICS LETTERS B
=0,
(la)
-p~
, 1 1" R~ Pg=~(C~,)
(no sum),
(lb)
1 R] (C,a)2 2R2R~
C~d)
+ R 7 2 C'd~Cd,.,+ 2R j 2 C',:d
(no sum over i) .
(2)
Now, if the right-hand side of eq. (lb) can be ignored, then we recover the famous Kasner solution: taking the trace of eq. ( I b) we find g ~ t 2. Using this again in eq. (lb) we find that Ra~t p" with Yp~= 1. From eq. (la) it then follows that Z p ] = 1; this is the Kasner solution. A closer look at eqs. (1) shows that the Kasner solution is a good approximation as long as t2Pg remains small for small times t. We notice that this is always the case for the two last terms in t2Pg since these have the form t 2(1 P'). The first two terms, however, have the form t 2""'' with cGb,,=l+ Pa--P~,--Pc and could possibly be negative. It is therefore interesting to investigate which constraints one must impose to d e m a n d that the aat,,, be positive. If all the aa~,,, remain positive for arbitrary small times, then the approach to the singularity is that o f the Kasner solution, while if negative a~,~ exist in t2p~ the corresponding term grows indefinitely for small t and the universe will " b o u n c e " and enter a new Kasner epoch described by new p~. Now, notice from eq. (2) that each term containing t 2c~"'' is proportional to Cg,, i.e., only terms with Cgc ~ 0 appear in PaJ. We may then ask which groups have structure constants such that for a suitable choice of the p~ all the a,b,, corresponding to non-vanishing Cg~, are positive. We shall show that if one can find a m o n g the generators X~ of G a closed subset of generators a subalgebra - of dimension d with 2 < d~< n - 1, 316
then it is possible to find a region of finite measure in p-space such that all the aabc in t2P} are positive for Pa in this region. This shows that the approach to the initial singularity in this case cannot be chaotic after going through a (perhaps large) number of Kasner epochs the universe will eventually bounce into the region with all the a ~ positive. Once described by a set ofpa from this region the universe evolves for arbitrary small times according to the Kasner solution corresponding to this particular set of p~. -
"~
where g = R ] . . . R ] and Pat, is the n-dimensional Ricci tensor. We find that
-
2 July 1987
3. The approach to the initial singularity. Let us assume it is possible to divide the generators Xa into a set {XA}, A = 1, ..., d and {X~}, a = 1, ..., D, where D - - n - d and where the XA span a d-dimensional subalgebra o f G, i.e., C~8 = 0 for allA, B, a. Now, we shall provide a set ofpa such that all a~bc corresponding to non-vanishing Cf,c are positive: let Pa be the Kasner index corresponding to X,, and define PA =P, A = 1..... d and p~ = q, a = 1, ..., D, where P-
1/dD+ ( 1 / d ) x / 1 / d + 1 / O - 1/dD 1/d+ 1/D '
1 / d D - ( 1 / D ) x / 1 / d + 1 / D - 1/dD q1/d+ 1/D Here p and q are defined as the solution to the equations d p + D q = d p 2 + D q 2 = l (i.e., 5 7 p a = Z p ] = l ) . Let us now check the possible values o f aabc: aABC = 1 - - p ,
aABa = aAoB = 1 -- q ,
aA~,p=l+p--2q,
OLaAp=aapA=l--p,
aap~ = 1 - q . These are all positive since q < p < 1; aoAB does not appear in t 2Pi, since C~8 = 0. Now, since all the aabc are positive for this particular choice of p, they are also positive for a whole surrounding region I of finite measure in p-space. This shows that the region corresponding to all the ceabo being positive has finite measure. Therefore, we can conclude that models which allow for diagonal solutions, and for which the generators can be split as devised above, are nonchaotic and the general approach to the initial singularity is the Kasner solution. All the models already analysed allow such a split of the generators [ 5], so
Volume 192, number 3,4
PHYSICS LETTERS B
it follows from our simple argument that these are non-chaotic, in agreement with the earlier analysis. Any Lie algebra L can be split into two parts L = Q + M, where M and Q are subalgebras with Q semisimple a n d M solvable with [ M , Q ] c M [6]. If neither Q nor M are trivial, then the generators of L m a y be chosen according to the split into Q and M, and it follows from our result, that any existing diagonal solution close to the singularity is the (non-chaotic) Kasner solution. If Q is trivial then L = M is solvable, so L ~ [L, L], i.e., L m a y be split into L' = [L, L] plus r e m a i n i n g generators, a n d our result shows also in this case that the general solution is non-chaotic. Finally if L = Q , then L = L , ~ . . . ~ L k with Li being simple. Obviously if k>_-2 we m a y split the generators and draw the same conclusion. If k = 1, then L is simple; but the only simple, real algebras without subalgebras are S O ( 3 ) and SO (2,1 ), i.e., the Bianchi models IX a n d VIII. These are therefore the only candidates for chaotic b e h a v i o u r (it is well known that these models exhibit the chaotic mixmaster b e h a v i o u r close to the singularity). The natural candidates as higher-dimensional m i x m a s t e r cosmologies, namely the higher-dimensional simple Lie groups, all contain subalgebras, i.e., it is possible to split the generators as divised above. Therefore even for these groups the chaotic b e h a v i o u r - which is characteristic for the three-dimensional simple groups - is absent. In conclusion, we have shown that no obvious generalization to higher d i m e n s i o n s o f the usual threed i m e n s i o n a l diagonal m i x m a s t e r cosmologies exist. We have shown that whenever diagonal solutions
2 July 1987
exist they will eventually evolve into the K a s n e r solution corresponding to a set o f Pa with all the a,,bc positive.
Finally, thanks are extended to the m e m b e r s o f the Theory Division at C E R N for their hospitality and to J a i m e Stein-Schabes.
References [ 1] S. Randjar-Daemi, A. Salam and J. Strathdee, Phys. Lett. B 135 (1984) 388; D. Sahdev, Phys. Lett. B 137 (1984) 155; E.W. Kolb, D. Lindley and D. Seckel,Phys. Rev. D 30 (1984) 1205; E.W. Kolb, Cosmology and extra dimensions, FERMILAB Pub 86-138 (1986). [2] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 19 (1970) 525; C.W. Misner, Phys. Rev. Lett. 22 (1969) 1071; J.D. Barrow and F.J. Tipler, Phys. Rep. 56 (1979) 372. [3] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31 (1982) 631. [4] J. Demaret, M. Henneaux and P. Spindel, Phys. Lett. B 164 (1985) 27; A. Hosoya, L.G. Jensen and J.A. Stein-Schabes, Nucl. Phys. B283 (1987) 657. [5] T. Furusawa and A. Hosoya, Progr. Theor. Phys. 73 (1985) 467; J.D. Barrow and J.A. Stein-Schabes, Phys. Rev. D 32 (1985) 1595; A. Tomiatsu and H. Ishihara, Gen. Rel. Grav. 18 (1986) 161; H. Ishihara, Progr. Theor. Phys. 74 (1985) 490; P. Halpern, Phys. Rev. D 33 (1986) 354. [6] V.S. Varadarajan, Lie groups, Lie algebras and their representations (Prentice-Hall, Englewood Cliffs, N J, 1974).
317
PHYSICS LETTERS B
2 July 1987
NON-CHAOTIC KALUZA-KLEIN COSMOLOGY Lars Gerhard J E N S E N CERN, CH-1211 Geneva 23, Switzerland
Received 20 March 1987
We investigate the evolution of higher-dimensional, homogeneous, anisotropic cosmologies. The natural extension to higher dimensions of the three-dimensional mixmaster cosmologies are analysed, and it is found - contrary to the three-dimensional case - that the higher-dimensional cosmologiesexhibit no chaotic behaviour.
1. Introduction. In the past few years higherdimensional cosmologies have received a great deal o f attention [ 1 ]. This is due to the theoretical possibility of a unification of the fundamental forces in a higher-dimensional spacetime. In these theories the extra dimensions are usually assumed to compactify at some energy scale of the order of the Planck mass, while the remaining dimensions expand into our observable universe. In the type o f model commonly considered, the extra dimensions are curled up in a maximally symmetric n-sphere. This rather arbitrary assumption is usually made because o f the simple geometry of maximally symmetric spaces. It is well known from 3 + 1 dimensions that even if one considers a less symmetric geometry, there are interesting models, such as the mixmaster cosmologies [2], which can evolve into our observable universe but which, close to the initial singularity, enter an infinite succession of Kasner epochs during which the three scale factors oscillate with arbitrary large amplitude and frequency. It is also well known that this type of chaotic behaviour is exhibited in the most general (inhomogeneous) cosmologies close to the initial singularity [3]. Recently it has been shown that the same result is true for spatial dimensions between four and nine, and that the chaotic behaviour disappears for dimension ten and above [4]. For (n + 1 )-dimensional cosmology with n/> 10, the general approach to the singularity is the generalized Kasner solution. In this case the universe enters a finite succession of Kasner epochs, of which the last
one is stable and determines the approach to the singularity. For homogeneous cosmologies the problem of the approach to the singularity is still unsolved for 4 ~
2. H o m o g e n e o u s cosmology. The extension to n + 1 dimensions of the usual (3 + 1 )-dimensional Bianchi models consists o f the cosmologies with a oneparameter family of n-dimensional spatial hypersurfaces with a Lie group acting simply transitively, i.e., the group motion may carry any point into any other point. The Lie group G has n generators Xa, a = 1..... n and corresponding invariant forms e a - e~a dx, = 1, ..., n, which satisfy d e " = -~,~bc~Pa ~b ^~c, where Cgc are the structure constants of G. We shall assume that the metric is diagonal in the ea-basis:
ds 2 = d r 2-Ra2(ea) 2 . Then we find the following 0 - 0 and a - a Einstein equations in vacuum:
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
315
Volume 192, number 3,4
~
1(
2~/g
~
PHYSICS LETTERS B
=0,
(la)
-p~
, 1 1" R~ Pg=~(C~,)
(no sum),
(lb)
1 R] (C,a)2 2R2R~
C~d)
+ R 7 2 C'd~Cd,.,+ 2R j 2 C',:d
(no sum over i) .
(2)
Now, if the right-hand side of eq. (lb) can be ignored, then we recover the famous Kasner solution: taking the trace of eq. ( I b) we find g ~ t 2. Using this again in eq. (lb) we find that Ra~t p" with Yp~= 1. From eq. (la) it then follows that Z p ] = 1; this is the Kasner solution. A closer look at eqs. (1) shows that the Kasner solution is a good approximation as long as t2Pg remains small for small times t. We notice that this is always the case for the two last terms in t2Pg since these have the form t 2(1 P'). The first two terms, however, have the form t 2""'' with cGb,,=l+ Pa--P~,--Pc and could possibly be negative. It is therefore interesting to investigate which constraints one must impose to d e m a n d that the aat,,, be positive. If all the aa~,,, remain positive for arbitrary small times, then the approach to the singularity is that o f the Kasner solution, while if negative a~,~ exist in t2p~ the corresponding term grows indefinitely for small t and the universe will " b o u n c e " and enter a new Kasner epoch described by new p~. Now, notice from eq. (2) that each term containing t 2c~"'' is proportional to Cg,, i.e., only terms with Cgc ~ 0 appear in PaJ. We may then ask which groups have structure constants such that for a suitable choice of the p~ all the a,b,, corresponding to non-vanishing Cg~, are positive. We shall show that if one can find a m o n g the generators X~ of G a closed subset of generators a subalgebra - of dimension d with 2 < d~< n - 1, 316
then it is possible to find a region of finite measure in p-space such that all the aabc in t2P} are positive for Pa in this region. This shows that the approach to the initial singularity in this case cannot be chaotic after going through a (perhaps large) number of Kasner epochs the universe will eventually bounce into the region with all the a ~ positive. Once described by a set ofpa from this region the universe evolves for arbitrary small times according to the Kasner solution corresponding to this particular set of p~. -
"~
where g = R ] . . . R ] and Pat, is the n-dimensional Ricci tensor. We find that
-
2 July 1987
3. The approach to the initial singularity. Let us assume it is possible to divide the generators Xa into a set {XA}, A = 1, ..., d and {X~}, a = 1, ..., D, where D - - n - d and where the XA span a d-dimensional subalgebra o f G, i.e., C~8 = 0 for allA, B, a. Now, we shall provide a set ofpa such that all a~bc corresponding to non-vanishing Cf,c are positive: let Pa be the Kasner index corresponding to X,, and define PA =P, A = 1..... d and p~ = q, a = 1, ..., D, where P-
1/dD+ ( 1 / d ) x / 1 / d + 1 / O - 1/dD 1/d+ 1/D '
1 / d D - ( 1 / D ) x / 1 / d + 1 / D - 1/dD q1/d+ 1/D Here p and q are defined as the solution to the equations d p + D q = d p 2 + D q 2 = l (i.e., 5 7 p a = Z p ] = l ) . Let us now check the possible values o f aabc: aABC = 1 - - p ,
aABa = aAoB = 1 -- q ,
aA~,p=l+p--2q,
OLaAp=aapA=l--p,
aap~ = 1 - q . These are all positive since q < p < 1; aoAB does not appear in t 2Pi, since C~8 = 0. Now, since all the aabc are positive for this particular choice of p, they are also positive for a whole surrounding region I of finite measure in p-space. This shows that the region corresponding to all the ceabo being positive has finite measure. Therefore, we can conclude that models which allow for diagonal solutions, and for which the generators can be split as devised above, are nonchaotic and the general approach to the initial singularity is the Kasner solution. All the models already analysed allow such a split of the generators [ 5], so
Volume 192, number 3,4
PHYSICS LETTERS B
it follows from our simple argument that these are non-chaotic, in agreement with the earlier analysis. Any Lie algebra L can be split into two parts L = Q + M, where M and Q are subalgebras with Q semisimple a n d M solvable with [ M , Q ] c M [6]. If neither Q nor M are trivial, then the generators of L m a y be chosen according to the split into Q and M, and it follows from our result, that any existing diagonal solution close to the singularity is the (non-chaotic) Kasner solution. If Q is trivial then L = M is solvable, so L ~ [L, L], i.e., L m a y be split into L' = [L, L] plus r e m a i n i n g generators, a n d our result shows also in this case that the general solution is non-chaotic. Finally if L = Q , then L = L , ~ . . . ~ L k with Li being simple. Obviously if k>_-2 we m a y split the generators and draw the same conclusion. If k = 1, then L is simple; but the only simple, real algebras without subalgebras are S O ( 3 ) and SO (2,1 ), i.e., the Bianchi models IX a n d VIII. These are therefore the only candidates for chaotic b e h a v i o u r (it is well known that these models exhibit the chaotic mixmaster b e h a v i o u r close to the singularity). The natural candidates as higher-dimensional m i x m a s t e r cosmologies, namely the higher-dimensional simple Lie groups, all contain subalgebras, i.e., it is possible to split the generators as divised above. Therefore even for these groups the chaotic b e h a v i o u r - which is characteristic for the three-dimensional simple groups - is absent. In conclusion, we have shown that no obvious generalization to higher d i m e n s i o n s o f the usual threed i m e n s i o n a l diagonal m i x m a s t e r cosmologies exist. We have shown that whenever diagonal solutions
2 July 1987
exist they will eventually evolve into the K a s n e r solution corresponding to a set o f Pa with all the a,,bc positive.
Finally, thanks are extended to the m e m b e r s o f the Theory Division at C E R N for their hospitality and to J a i m e Stein-Schabes.
References [ 1] S. Randjar-Daemi, A. Salam and J. Strathdee, Phys. Lett. B 135 (1984) 388; D. Sahdev, Phys. Lett. B 137 (1984) 155; E.W. Kolb, D. Lindley and D. Seckel,Phys. Rev. D 30 (1984) 1205; E.W. Kolb, Cosmology and extra dimensions, FERMILAB Pub 86-138 (1986). [2] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 19 (1970) 525; C.W. Misner, Phys. Rev. Lett. 22 (1969) 1071; J.D. Barrow and F.J. Tipler, Phys. Rep. 56 (1979) 372. [3] V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31 (1982) 631. [4] J. Demaret, M. Henneaux and P. Spindel, Phys. Lett. B 164 (1985) 27; A. Hosoya, L.G. Jensen and J.A. Stein-Schabes, Nucl. Phys. B283 (1987) 657. [5] T. Furusawa and A. Hosoya, Progr. Theor. Phys. 73 (1985) 467; J.D. Barrow and J.A. Stein-Schabes, Phys. Rev. D 32 (1985) 1595; A. Tomiatsu and H. Ishihara, Gen. Rel. Grav. 18 (1986) 161; H. Ishihara, Progr. Theor. Phys. 74 (1985) 490; P. Halpern, Phys. Rev. D 33 (1986) 354. [6] V.S. Varadarajan, Lie groups, Lie algebras and their representations (Prentice-Hall, Englewood Cliffs, N J, 1974).
317