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Phenomenology of Late Stages of Kaluza-Klein Cosmologies.

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Annalen dcr Physik. 7. Folge, Band 46, Heft 5, 1989, S. 385-388
VEB J. A. Barth, Leipzig
Phenomenology of Late Stages of Kaluza-Klein Cosmologies
By LJ. RLEYER
Einstein-Laborntorium fur Theoretische Physik der
dkitdemie der Wissenschaften der DDR, Potsdam
and D.-E. LIEBSCHER
Zentralinstitut fur Astrophysik der Akademie der M’issenschnften der DDR, Potsdam
A b s t r a r t . We consider the arguments t o additional space dimensions in
phenomenological rosmology, and shorn, that strong conditions are t o be met.
R
general scheme of
Phanomenologie spatcr Stadien von Kaluza-Klein-Kosmologien
I n h t ~ l t s t i b e r s i c h t Wir
.
betrachten die Argument? zur R a g e zusatzlicher Raumdimensionen
i n elnem allgemeinen Schema der phiinomcnologischen Kosmologie und zeigen, da13 starke einschrankende Bedingnngen niiftreten.
1. Introduction
The names of T. Kaluza and 0. Klein stand for all theories with additional dimensions
of space-time, which are hidden from usual measurement by their presumably microscopic size. These additional diinensions are introduced for different reasons, part,ly for
unification of gravitation theory with the theory of other fields, partly for saving quantization procedures from singularities. Additional dimensions produce some “classical”
quantization, which give them a special appeal.
We want t o show t,hat classical cosmology requires very strong conditions t o be met
h p the matter fields in the limit of the phenomenological matter tensor. Avoiding the
clearly involved discussion about the development in the very early universe, we restrict
ourselves to later times, aft’eriiucleosynthesis say, when we are pretty sure phenomenological matter tensors are given by a mist’ure of some ideally fluid components, which
are adiabatically isolated except for some comparatively rapid quasichemical reactions
changing their relative abundances.
2. The Cosmological Model
We start wit,h t,he assumption that, the additional dimensions form lioniogeneous and
isotropic subspaces j u s t as the usual three space dimensions do. The line element which
we intend to consider shall be given by
a
as2
= at2 -
2 R$(t)ds$,
i=O
di
Ann. Physik Leipzig 46 (1989) 6
386
The Einstein equations for this metric are fount1 t o be
2
( z d i 2 ) -
a
2
di (q
R; - (dj - 1)
/
'
(3)
j=O
Of course, n e have to assume tlie matter tensor to be tliagonalizetl, and the pressures
being equal in any of the different invariant subspaces. The system (4) consists of
ix
1 equations, the first integral being eq. (3), equivalent to the Bianchi identitys
of continuity of the matter teiisor
+
3. Cold Matter
Most of the tinie, the matter components of the universe expand adiabatically, not
reacting effectively to disturb the concentrations. If the component in question is relativistic, its partial pressure is one third of its partial clensity, if it is non-relativistic,
its partial pressure is effectively zero coniparecl rn ith its partial density. The time between
is comparatively short, and for overall discussions it may be replaced by a fast transition
from the relativistic equation of state to the pressure zero.
More unconventional equations of state, ultrastiff niatter ( p = Q), gas of walls
2
2) = - l i e ) , gas of strings ( p = -p/3), vacutiiii energy (2, = -e), are all of the same
(
type :
rl being the dimension of tlie space in question. The notion of temperature plays no role
in the equation of state of these coniponents, so we may call them cold. In a usual Frietlniaiin univei'se, they have an especially simple expansion law: By force of equation (5)
we fiiitl
for a niixture of such coniponeiits. The M7*are constants, ancl the continuity equation
lioltls for every component separately. The Jf, represent the constitution of the mixture,
a w l may change in the short periods of rapid processes. If these processes take place
at a moment of tinie for which R has a given value, these changes of the M , are subject
o f the coiitlition
also tlerivate of equation (5). We have to note, that we may represent the curvature of
space here, 1.e. on the right-haiirl side of the equations of motion of the cosmological
model, just by putting it into tlie component with rn = 2, taking into account, that no
reactions change its contribution. This is all we need for tayloriiig models like Petrosian
[ I I.
We niay generalize this model of a mixture of colt1 matter coniponents to tlie case of
the Kalnza-Klein niotlels of the type given by the metric, eq. ( I ) . We only consider
U. BLEYER
itnd D.-E. LIEBSCHER,
Kaluzn-Klein Cosmologiw
387
equations of state like eq. (6) also for the pressiire in the internal spaces
giving by equation ( 5 ) just
I t is no probleni to consider now mixtures of such components, and to see, that quasiclieniical reactions are noM subject of a condition analogous to eq. (8). Again, the cliffeI ent curvatures are just represented by matter components with every index mi zero
ekcept the one corresponding to the internal space i n cluestion, which is 2. We note,
that different field-theoretical models produce different behaviour of the "vacuum".
F o instance,
~
in a space-time with do = 3, d, = d , iy = 1, Moss [2] fiiitls the vacuum to
be iiitlesetl by (d
4, 0), Salidev [3] by (3 /(d 3)
3, d/(d 3) 4- d ) ) , Taylor [4]
by (0, d
4). Salani [5] finds a two-component vacuum with (0, d
4) and (4, d ) . A
1) behaves like dust in the ordinary space
trace-free component indexed by (3, d
[ (5- 81.
The iliain advantage uf the restriction to cold matter lies in the possibility to reduce
the probleni fornially to that of a zero energy particle in a formal potential in a formal
space of iy
1 dimensions. To this end we only have t o choose ti = In Ri for coordinates,
the quantity
+ +
+
+
+
+
+
+
@ =
esp ( 2
\
2 dlEJ @(to, . .., E,)
j=o
(1 1)
I
as potential, where p is now including all curvature terms, aiitl the time T by
dT
= esp
(-
djti) at.
(12)
i=o
The eqiiations (Y), ant1 (4) n o w read
a
1
..
.
dj5; = 7 m..t.t.
= 2@,
1 1 2 . 1
$
*
( 1 .
80
7 ))l,.. - t - iY
=o
& I d z' i - ati '
'
a
(13)
6.
= - 6,.
1
nT 1'
The forninl inass matrix in these equat'ions is given by
m . . = d.d. - (1.8...
17
I. J
a ti
(14)
Tliese equation allow the general discussion of the behaviour of solut'ions by the analogy
to particle mot'ion. We have t,o observe, that t.he quasi-mass mat.ris rnii usually is indefinite, and that we find the corresponding interpretat'ion.
4. Coiiclusioiis
We may choose as initial contlitions to-day
Ri = 1 j = 0, ..., a ,
R 7. = 0 j = I ,
R,
= 1.
...,&,
Ann. Physik Leipzig 46 (1989) 6
388
Then, of course, all the values uf tlic interiial curvatures Mo,...,o,e,o,,..,o
( 2 not at, t,he
first place) have to be large as presiiniecl. In spit’e of this, we have t o have a value of @
of order unity, and very small values of 8 @ / 8 t i , j = I, ..., LX,in order t o avoid large
changes in Ei in the time froni the priniortlial nucleosynthesis till now, and a small value
of a@/a[, t o get no extraordinarily large value of the tleceleratiun parameter. Hence,
with respect t’othe large curvature C(Jl1l1xJIlt?ntS necessary to make the additional spaces
unobservable, the potential @ has t u have an estmnirim and a zero at the origin in the
t p p a c e . This are
1 conditions to he met. It follows, that at least two different, and
large matter coniponents have t o be present in o u r niisture. That means, one simple
vacuum model nil1 never do. J n order t o iiialte t lie Kaloza-Klein cosniology viable in
the present universe, we have t o ensure
+
1. large internal curvatures
2. a ) at least’ two “ v a c u ~ i n ~coniponents
”
also large to meet tlie required condition OIL
the potential @, or
b ) usual matter coniponents having internal pressures always equal minus the
density (to be not affected by a change in the size of the internal spaces).
In older t o circumvent t’heseconditions one rnay think of higher-order ternis in the curvature [9, 101, or of high shear in tlie ( I W W anisotropic) internal spaces [Ill.
A c k n o w 1e cl g e 111e n t s
We are indebted t o H.-J. Tretler for inviting us t u present these result a t the Conference 011 Physical Cosmology 1988.
References
[l] PETROSIAN,
V.: Kcittire Y9S (1!)82) 805.
[ 2 ] Moss, I. G.: Phys. Lett. 140 B (1984) 29.
[ 3 ] SAHDEV,
D. : Phys. Lett. 137 B (1954) 155.
[4] GLEISER,M.; E ~ J P O O T
S.;, T ~ Y L OJR. ,G.: Higlicr climensioiinl cosmologies, King’s C’ollege
preprint, 1984.
[5] RANDJBAR-DBEMI,
8.; SALAX,A.; STRATHDEE,
J.: Pliys. Lett. 135 (1Y84) 388.
[6] BLEYER,
U.; LIEBSCHER.
D.-E.: GRG 17 (1983) 989.
[7] Car, 9.; Hu, S.;Xu, Y.: Phys. Lett. B 201 (1988) 34.
[8] GUENDELMANN,
E. I.: Phys. Lett. B Y O 1 (1988) 39.
[9] FARINA-BUSTO,
L.: Phys. Rev. D 38 (1958) 1741.
[lo] SHAPI,Q.; WETTERICH,C.: Nncl. Phys. B BS9 (1987) 787.
[I11 TOMIMATSU.
A.; ISHIH4RA, H.: GRG Journal 1s (1986) I(i1.
Bei der Redaktion eingegmgen a m
(i. Uezemher
1!)88.
Ansclrr. d. Verf.: Dr. U. BLXYEB
Einstein-LaborHtorium fur Theoretische I’liysik der
Akademie der Wissenscliaftcn der DDR
Rosa-Luxembnrg-Str. 1 7 a
Potsdam, DDR-1.590
Prof. Dr. D.-E. LIEBSCHER
Zentralinstit t i t fur Astrophysik der Akademie der Wissenschaften der DDR
Rosa-Luxemburg-Str. 17 :i
Potsdam, DDR-1590
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