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M. I. WANAS Department of Astronomy, Faculty of Science, Cairo University, Cairo, Egypt
(Received 26 October, 1988) Abstract. The field equations of the generalized field theory constructed by Mikhail and Wanas have been applied to a well-established geometrical structure given earlier by H. P. Robertson in connection with the cosmological problem. A unique solution, representing a specified expanding Universe (with qo = 0, f~o = 0.75, k = - 1) has been obtained. The model obtained has been compared with cosmological observations and with FRW-models of relativistic cosmology. It has been shown that the suggested model is free of particle horizons. The existence of singularities has been discussed. The two cases, when the associated Riemannian-space has a definite or indefinite metric are considered. The case of indefinite metric with signature ( + - - - ) is found to be characterized by k = - 1, while the case of + ve definite metric is characterized by k = + 1. Apart from that difference, the two cases give rise to the same cosmologicalparameters. It has been shown that energy conditions are satisfied by the material contents in both cases. 1. Introduction
One of the main aims of cosmology is to construct a mathematical model capable of describing the general structure of the Universe as is observed today; and which can tell us something about the history of the Universe or what its future will be. This problem has been tackled by different authors since the formulation o f the theory of general relativity in 1916. Different field theories have been used as basis for the construction of such a model. The most famous models were those depending on the general theory of relativity. After the discovery of the cosmic microwave background radiation in 1965 by Penzias and Wilson, it was generally believed by cosmologists that the Universe must have gone through a hot phase. This restricts, to some extent, our choice o f the cosmological model. One of the successful models in this respect, is that k n o w n as 'The H o t Big-Bang Model' or 'The Standard Model'. This model is supported by the agreement between the predictions of light elements synthesis 3He, 4He, D, 7Li, and the observed abundances (cf. Longair, 1982), since the abundance of light elements can be interpreted cosmologically rather than astrophysically. In spite o f the success of the standard model, some questions remained still without an answer. For instance: (a) Is the Universe open or closed? (b) What can be said about the big-bang singularity? The theory o f general relativity gives no definite answer, while relevant observations are affected by uncertainties. Some authors believe that the standard model cannot represent the history of the Universe for all times, because o f the existence, in such a model, o f a global singularity (cf. Novello and Salim, 1983). Others pointed out that the prediction o f space-time singularities indicate that the classical theory will break-down Astrophysics and Space Science 154: 165-177, 1989. 9 1989 Kluwer Academic Publishers. Printed in Belgium.
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and that quantum gravitational effects will dominate. Furthermore, they have suggested that quantum effects would eliminate all the singularities, and modification is certainly needed near the big-bang (cf. Hawking, 1982; McCrea, 1985). Modifications may be carried out by following a purely geometrical scheme, or by using a quantum approach, or by following a somewhat intermediate way mixing quantum and geometry. We believe that geometrical trials have not been exhausted. The generalized field theory constructed by Mikhail and Wanas (1977) may form one of the possible ways to solve the problem geometrically. The results obtained so far, using this theory, are promising (cf. Mikhail and Wanas, 1981; Wanas, 1981, 1985). It is the aim of the present work to apply that theory to the cosmological problem)We hope that this will throw some light on the questions raised in connection with the cosmological problem. In Section 2 the absolute parallelims (AP) space for cosmological applications is given. Section 3 comprises the field equations and their solution giving rise to a new workable model for the Universe. In Section 4, a brief review of the cosmological quantities relevant to the present work, is given. Section 5 represents a summary of cosmic observations, which will be used in discussing the model. A comparison with the scheme of relativistic cosmology is given in Section 6. The work is discussed in Section 7 and concluded in Section 8.
2. AP-Space for Cosmological Applications The generalized field theory, which is applied in the present work, is based on a 4-dimensional AP-space. The structure of such AP-space is defined completely by a tetrad vector field. So, instead of the ten field variables of general relativity (manifested in the metric tensor g,v), we have sixteen field variables (the contravariant components of the tetrad field 2") for the generalized field theory. We believe that the results of any geometrical field theory depend not only on the structure of the theory itself, but also on the choice of the particular geometrical model used for its applications. Hence, it becomes necessary to find out some general rules to select a certain model suitable for a certain type of application. In fact, the author in an earlier paper (Wanas, 1986) succeeded to discover a set of conditions to be satisfied by any geometrical model (AP-space) which will be suitable for cosmological applications. The geometrical model satisfying these conditions will be capable of representing a homogeneous, isotropic, electrically neutral, and non-empty universe. Furthermore, it is also adequate to describe a strong gravitational field. It has been found (Wanas, 1986) that one of the AP-spaces constructed earlier by Robertson (1932), satisfies these specified conditions. The structure of this space is given by the following tetrad which is written in Cartesian-like coordinates (x ~ - t, x 1 = x, x 2 ----y, x 3 --- z) in the form 2 ~ = 1,
20 = 2 ~ = O,
0
a
0
A(t)2 ~ = b~(1 - 88 2) + 89
(2.1) a + kl/2e~,axa ;
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where e, a = 1, 2, 3; k is the curvature constant (-- + 1, 0, - 1); e ~ , skew-symmetric with respect to all its indices (e123 = + 1); and A(t), an unknown function of (t). In the spherical polar coordinates (t, r, 0, q~), the contravariant components of the tetrad (2.1) can be written in the form 2" = {1, O, O, 0}"
(2.2a)
0
2" = {0, L + sin 0 cos q~/4A, (L - cos 0 cos ~p - 4kl/2r sin O)/4Ar, 1
- (L - sin ~b + 4kl/2r cos 0 cos c~)/4Ar sin 0}, 2" = {0, L + sinOsin(~/4A, (L 2
(2.2b)
cos0sinq~ + 4kl/2rcos(o)/4Ar,
(L - cos cp - 4kl/2r cos 0 sin ~))/4Ar sin 0}, 2" = {0, L + cos O/4A, - L - sin O/4Ar, kl/2/A} 9 3
(2.2c) (2.2d)
where/~ ( = 0, 1, 2, 3) represents the coordinate components, and
L • -- 4 + kr 2 .
(2.3)
For every AP-space, there is always an associated Riemannian space whose metric tensor, in terms of the tetrad vectors, is defined by gUv~ f ) f ,~v i i'
(2.4)
def
g.v--
the 2,'s being the covariant members of the tetrad. The metric tensor corresponding to the tetrad (2.2) is given by gOO = 1,
gal = (L+/4A)2,
g22 = g l l / r 2 '
goo= 1,
gll = ( 4 A / L + ) 2,
g22=g11 r2,
g33 = glair2 sin20,
(2.5)
g33=gla rzsin20,
which is clear to form a + ve definite metric. If the metric is required to be indefinite with the signature ( + - - - ) , we change 2" given by (2.2) to 2" such that (cf. i McCrea and Mikhail, 1956) i 2~ 0
0
~
2 ~= 2 ~ 0
a
2 ~ = i 2 ~', a
a
(2.6)
where i = ( - 1) 1/2. Then using the definition (2.4) we get ~oo = gOO,
~a~ = _ g a ~ , (2.7)
goo = goo,
ga~ = - g , ~ .
It is of interest to point out that the two tetrads (2.2), (2.6), are adequate to describe fields of the same strength: namely, fields of the type FOGIII (cf. Mikhail and Wanas, 1981; Wanas, 1986). We are going to use the tetrad (2.6) for our applications. This
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facilitates the comparison with the results of relativistic cosmology, since the metric given by (2.7) is the same one familiarly used in Friedmann-Robertson-Walker (FRW) cosmology.
3. The Field Equations and the World Model According to the generalized field theory given by Mikhall and Wanas (1977), the field equations are given by E.~v : 0,
(3.1)
where E Q is a second-order, non-symmetric tensor defined in the AP-space. Using the tetrad (2.6) to evaluate E.~v, then Equation (3.1) will give rise to: E~ : A'2/A2 El i = E22
+
4k/A 2 = 0,
= E33
"
(3.2)
2 A " / A + A ' e / A 2 + 4k/A 2 = 0 ,"
where dots represent differentiation with respect to (t). The first equation can be easily integrated to give (3.3)
A = .4 + 2 ( - k ) l / 2 t ,
where A is the constant of integration, giving the value of A at t = 0. Solution (3.3) satisfies the second equation of the pair (3.2) without any further conditions. If we take k = 0, then we get A = ,4 = constant, i.e., a static model, while k = + 1 gives an imaginary scale factor (and imaginary Hubble's parameter). Thus, for a nonstatic real model we have to take k = - 1.
(3.4a)
Furthermore, we take the + ve sign in the solution (3.3) to get an expanding model, then A =/~ + 2t.
(3.4b)
The material-energy tensor, according to the generalized field theory, is defined by def
T~,~ = tOuv- a ~ + g, vA, A f89
(3.5)
-
where def
a
e
tO~v = 7.,eT.~v + ~7v~7~.o~, def
% ~ --tr
~ ." ela
~ ) . ~ v "= ~i
(3.6)
~.~a v ~
d e f "o-~.
.
i#; v~
the ; denotes covariant differentiation. Evaluating the tensors given by (3.5) using the tetrad (2.6) we get for the material-energy tensor the values (3.7)
TOo = 9k/A2"
Tll
= T 2 2 = T33 =
3k/A2;
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and for the cosmological function the value (3.8)
A = 9k/A z .
For later convenience, one can write the quantities given by (3.7) and (3.8) using Equations (3.2) in the form ~
TOo = - 9A" 2 / 4 A 2 ,
Tll
= T22 = T33 = -3A"
(3.9)
/
2/4A2,
A = - 9A" 2/4A2.
(3.10)
4. Cosmological Quantities To facilitate comparison of the present model with the results of relativistic cosmology, and those of observations, we are going to use the quantities Ha-e--fA "/A ,
(4.1)
z~fH- 1
(4.2)
The values at the present epoch (i.e., when t = to) usually denoted by Ho, %, are the well-known Hubble's parameter and the age parameter, respectively. The acceleration parameter ho and the deceleration parameter qo are, respectively, the present values of the quantities h~CA "'/A,
(4.3)
q~fh/H
(4.4)
a.
The quantities defined so far, in the present section, are those used in relativistic cosmology. The material-energy tensor of the generalized field theory is a purely geometrical object. Therefore, to get an idea about the material contents of the model, we use the quantity (4.5)
F~ aef _ T O o / 3 H 2 ,
where T~ is a component of the mixed form of the tensor defined by (3.5). The value of this quantity at the present epoch (f~o) is called 'the matter parameter'. In the case of FRW-models, the last quantity will be identical with the well-known density parameter (%). Evaluating the previous quantities for the suggested model, we get H = 2/(A + 2t), q
0
f~
~ = 89 + t,
h = 0,
(4.6)
3
It is to be considered that h, q, and f~ are independent of (t) for the present model.
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5. Comparison with Observations In order to c o m p a r e the new m o d e l with cosmological observations, we are going to start by giving a brief review o f the results o f these observations. These are s u m m a r i z e d in Table I. The third value o f H u b b l e ' s p a r a m e t e r given in this table is the value required by time-scale agreements with F R W - m o d e l s for qo = 0. A s clear from Table I, one can accept the value H o ~ 50 k m s - 1 M p c - 1
(5.1) TABLE I Cosmological observations
Quantity
Observed value (units)
Reference
Hubble's parameter
60 > H o > 50 (km s - 1 Mpc- 1) 5 7 > H o > 4 3 ( k m s - I M p c -1) 52 > Ho > 40 (km s - 1 Mpc - 1)
1 2 2
16 • 109 (years) 2.3 X 109 ( y e a r s ) 0.1 > qo > 0
3 3 1
0.1 > aB > 0.03 0.53 > ac >-_0.003 0.7 > aD > 0.1
4 5 4
Globular cluster Age of Formation time Deceleration parameter Density parameter Baryonic Galaxy data Dynamic (1) (2) (3) (4) (5)
Sandage and Tammann (1975). Sandage and Tammann (1982). Sandage and Tammann (1983). Longair (1982). Narlikar (1983).
A n estimate for the age o f the U n i v e r s e can be o b t a i n e d by adding the age o f globular cluster to its formation time given in Table I. This gives t o = 18.3 x 109 y r .
(5.2)
F o r the suggested m o d e l (qo = 0), the age o f the Universe is less than %. I f we c o n s i d e r , 4 ~ A , then to ~ Ho- 1 = Zo = 2 x 10 l~ y r .
(5.3)
A c o m p a r i s o n between the values given by (4.6) with the o b s e r v e d values of Table I shows that the suggested m o d e l is consistent with the o b s e r v e d values except for f~o which is greater than the o b s e r v e d u p p e r limit. This will be d i s c u s s e d later.
6. Comparison with Relativistic Cosmology To c o m p a r e the scheme of general relativity with that o f the generalized field theory concerning the cosmological problem, we are going to s u m m a r i z e some o f he i m p o r t a n t features o f the two theories in Table II.
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A SELF-CONSISTENT WORLD MODEL TABLE II Comparison with relativistic cosmology Criterion 1. Geometry 2. Dimension 3. Field variables 4. Basic assumption 5. Number of models allowed 6. Equations necessary to construct a model 7. Curvature constant for non-static, non-empty model 8. Deceleration parameter 9. Matter (density) parameter 10. Cosmological member 11. Indicator for strong field
General relativity
Generalized theory
Riemannian 4 The metric tensor g.v (10-components) Homogeneity and isotropy Many Field equations + equation of state +1,0, -1
Absolute parallelism 4 The tetrad vectors 2" (16-components) i Homogeneity and isotropy One
Field equations only -1
+ re, O, - ve
0
oO > ao > O
3
Arbitrary (const.) Singularity
Not arbitrary (func.) A
It is of interest to point out that, for the generalized field theory, one is not free to choose the cosmological member as in the case of general relativity. Also, one can not adjust the material-energy tensor, since it is part of the geometrical structure used. The metric common to all FRW-models is the Robertson-Walker metric. Robertson (1932) has constructed two AP-spaces whose associated Riemannian spaces have the same Robertson-Walker metric. These two AP-spaces were found to be of the types FOGIII and FOGI, respectively (Wanas, 1986). In the present work we have used the AP-space FOGIII, and rejected the other. The second one is rejected on the basis that it is not capable of representing a material distribution or a strong field. This situation is not so obvious in general relativity. Strong fields appear as singularities in this theory. Furthermore, the links between geometry and the material distribution are manifested through its field equations. These links are rather artificial, which is not the case in the generalized field theory. In FRW-models of relativistic cosmology (without the cosmological term), we have the relation 2% = %. This relation does not hold in the present work, since we have A#O.
7. General Discussion of the Results
7.1. T H E CASE OF
+ ve
DEFINITE METRIC
Although the generalized field theory, used in the present work, is not a metric one, yet the metric of the Riemannian space associated with the AP-space can always be obtained, as stated before. The results obtained so far in the present work, depend on the tetrad (2.6) whose associated metric is indefinite with the signature ( + - - - ).
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For some reasons, it is sometimes required to derive cosmological results in the case of a + ve definite metric (cf. Hawking, 1982). Thus it may be useful to reconsider the above results in the light of a + ve definite metric. This can be easily done by replacing the tetrad (2.6) by the tetrad (2.2), since the later will give rise to a n associated Riemannian space with a + ve definite metric. For this purpose, repeating calculations, it can be shown that, to get an expanding non-empty real model, we Should take k = + 1.
(7.1)
All other results remain the same especially those concerning the cosmological quantities of Section 4. To sum up, we have obtained the same model with the same parameters (4.6), the same material-energy tensor (3.9), the same scale factor (3.4b), and the same cosmological function (3.10). The only difference is that the case of indefinite metric (case I) is characterized by a ( - re) curvature constant (3.4a), while the case of + ve definite metric (case II) is characterized by a + ve curvature (7.1), as a result of the solution. 7.2.
THE COSMOLOGICAL FUNCTION
The model with + ve definite metric and k = + 1 will expand forever (qo -- 0). This is due to the - ve value of the cosmological function A ( = - A ' Z / A 2) in the two cases I, II. This shows that the cosmological member will cause a repulsion giving rise to the continuous expansion of the model in spite of the fact that k = + 1. A similar situation was found in the case of FRW-models with a non-vanishing cosmological member. It is to be considered that the cosmological term in the generalized field theory has a different sign compared with the corresponding term in general relativity. 7.3. SINGULARITIES
In the literature there are two criteria for judging the presence of singularities in a space-time. In the following we are going to discuss these criteria in the framework of the generalized field theory. (i) The first criterion is that the space-time is incomplete in a certain sense. In general relativity space-time is singular if it possesses, at least one incomplete geodesic (cf. Wald, 1984; Section 9.1). Singularity theorems of general relativity (cf. Hawking and Ellis, 1973; Ch. 8) are used to show whether space-time is geodesically incomplete. These theorems depend on the following assumptions: (a)The space-time is the Riemannian whose metric is characterized by the Lorentz signature ( - + + + ); (b) Einstein's field equations, with a vanishing cosmological term, hold; (c) as a consequence of the two assumptions above, the motion of a particle (with or without rest mass) is along a geodesic (time-like or null). Furthermore, the proof of these theorems depends mainly on the Landau-Raychaudhuri equation (cf. Hawking and Ellis, 1973; p. 84). In the absence of vorticity, this equation shows that the main term affecting the prediction of singularities is the term Rab V a V b, where V a is a non-spacelike vector and Rab is the Ricci tensor.
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In the generalized field theory, the situation is different. (g) Regarding the first assumption, the generalized field theory is constructed in the AP-space. (b) The field equations used in the present work are different from those of Einstein. Although we can put the symmetric part of the field equations in a form similar to that of general relativity, yet we have a non-vanishing cosmological term. This term is not an arbitrary one in contrast with the case in general relativity. (~) From a geometrical point of view, paths in an AP-space are not, in general, geodesics. The equations of such paths in such spaces are different from geodesics (cf. Eisenhart, 1927; p. 12). From the cosmological point of view, in the very early Universe most of the material contents were in form of particles having spin or charge, or both. It is well known that such particles will not move along geodesics (cf. Galvao and Teitelboim, 1980). From the above discussion it is clear that singularity theorems of general relativity are not relevant to the present work. If one tries to study singularities in AP-spaces using the first criterion, one should generalized the Landau-Raychaudhuri equation. Effective terms are expected to be different (at least, covariant differentiation will be replaced by absolute differentiation giving rise to more terms). The detailed study of singularity theorems in the generalized field theory will appear in a separate paper (in preparation). (ii) The second criterion is the blowing up of a polynomial constructed from the invariants of the theory. In general relativity these scalar polynomials are formed in terms of the curvature tensor. For the present work, we are going to rely on this criterion to get an idea about the singularities in the model. It should be noted that the values of A(t) giving rise to negative distances should be ruled out, as A(t) is used in several definitions of distance (cf. Weinberg, 1972; p. 418). It can be shown that the invariants of the theory like A, T"~T~v, ..., etc., converges to finite values at t = 0, and to zero as t ~ 0o. This shows that the model is free from singularities, according to this criterion, for t > 0, but the model may have a singularity for t < 0. 7.4. ENERGY C O N D I T I O N S
AND MATERIAL CONTENTS
It is to be noted that, as we do not rely neither on Einstein's field equations nor on the Landau-Raychaudhuri equation, the energy conditions will not be relevant in predicting the existence of singularities. However, we are going to discuss these conditions for its own sake. If V~ is a time-like vector (V~V~ --- - 1), then using (3.9) we get T%V~V v -
9A-2 4 A2
> 0.
(7.2)
This shows that the material-energy tensor satisfies the weak energy condition (cf. Hawking and Ellis, 1973; p. 89). The cosmological term in the field equations (3.1) is not an arbitrary term as stated before. Then the strong energy conditions (cf. Hawking and Ellis, 1973; p. 95) cannot be examined if it is taken literally, as it required the vanishing of the cosmological term. But, as this term could be included in the definition of the material-energy tensor as shown by (3.5), then we can test the strong condition. The trace of the material-energy
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tensor gives _ i r -- _9 ;A-x/_j 2 '
2
(7.3
4\A/
If we compare (7.2) with (7.3) we get T,
V~,V v =
- ~lT ,
which shows that the strong energy condition is also satisfied. The components of the material energy tensor (3.9) are not affected by the change of the signature as shown before. The change of the signature affects the curvature constant only. However, the model is filled with matter (characterized by the parameter f~o = 0.75) which is distributed with homogeneity and isotropy. In addition, the material-energy tensor and the cosmological function have the following values in the following two cases: (i) At t = 0, TOo =
- 9 / - ,~2 ,
Tll
= T22
= T33
=
-- 3/~z~2 ,
A
=
- 9/~z~2 .
(ii) As t--, 0%
T~'~0,
A~O.
It has been shown by Coley and Tupper (1982) that FRW-models do not possess unique interpretations. They have shown that these models may be interpreted either as:
(i) perfect fluid solution, or (ii) viscous magnetohydrodynamic solution. It may be impossible, as they stated, to decide which interpretation is the correct one. It is to be considered that, for the generalized field theory, and especially for the present model, this type of ambiguity is ruled out at least for the electromagnetic interpretation. This is due to the fact that all skew tensors of the second order for this model vanish identically. This is clear from the type of the AP-spaces used in the present application (F0gIII). 7.5.
CRITICAL
DENSITY
SCHEME
It seems that the standard treatment of the critical density problem is questionable. Coley and Tupper (1982) have found that, using general relativity, not only the quantity of matter in the Universe, but also its qualitative nature is needed to determine whether the Universe is open or closed. A similar conclusion has been reached by Sandage and Tammann (1983). They have stated that closure does not necessarily mean that the expansion will stop and contraction begins. It has been shown in the present work that the sign of k is independent of the material contents of the model (same f2 for both k = + 1, k = - 1). The question which emerges now, from the metric point of view, is whether the Universe is open with an indefinite
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metric, or closed with a + ve definite metric. In both cases the model will expand forever as shown above. 7.6. UNCERTAINTIES OF OBSERVATIONS Due to the uncertainties of observations of the first ranked cluster galaxies concerning Hubble's diagram, most hopes to obtain qo from that diagram have been destroyed (cf. S andage and Tammann, 1983). However, many astronomers believe that qo may be very close to zero. The value qo = 0 obtained from the present work is not in contradiction with the present data on galaxies. It has been shown by Kristian et al. (1978), on observational grounds, that a self-consistent model with qo ~ 0 that satisfies the known data, can be constructed. 7.7. PARTICLE HORIZON By use of the present model and the metric (2.7), the proper distance R is defined by R d--erA(to)r,
(7.4)
where r is the radial coordinate given by (c = 1) to t~
Y
= ~ dt/A(t) = tl
_ lln(1 + Z)
for the present model ;
(7.5)
and Z is the red-shift defined by def
Z = A ( t o ) / A ( t l ) - 1.
(7.6)
The existence of a finite value for R as Z ~ oo means that a given model has a particle horizon (cf. Narlikar, 1983; p. 127). Thus using relation (7.4) and the value (7.5) we get R = %1n(1 + Z ) ,
(7.7)
which shows that, as Z ~ 0% R ~ oo (and, therefore, for the present model) there is no horizon problem: there is no limit on the proper distance up to which one can observe.
8. Conclusions On the basis of the generalized field theory, a unique model of the Universe has been constructed. The model is an expanding but non-empty one. It has been shown that the model constructed is free from particle horizons. Furthermore, it gives a definite value for the deceleration parameter qo ( = 0), and for the matter parameter flo ( = 0.75). If we confine ourselves with an indefinite metric with the signature ( + - - - ) , for the associated Riemannian space, then the model fixes a unique value for the curvature constant k ( = - 1 ) , i.e., an open Universe. But if the metric of the associated Riemannian space is required to be + ve definite, then again the model fixes a unique
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value for k ( = + 1). Hence, as the two cases have identical parameters, one certain conclusion is that - whether the Universe is open with an indefinite metric or closed with a + ve definite metric - it will not stop its expansion. This means that the event of the creation of the Universe is unique. This conclusion has been reached by Sandage and Tammann (1983) on a completely different basis. However, it remains to find out some way to fix some appropriate initial conditions by which one can be able to evaluate.d, the value of A(t) at t = 0. If we know this value, the age of the Universe can be determined exactly. For the time being A is assumed to be very small compared with the present value A(to); and, hence, the age of the Universe will be of the same order as Hubble's time %. According to the model obtained, a fraction of matter in the Universe should exist in a non-baryonic form. That is because the value of the matter parameter f~o (= 3), as given by the model, is far from its baryonic value. The solution obtained is unique. This is because the geometry is specified (before solving the field equations) much more than in the case of general relativity. Furthermore, the study of the strength of the field equations (Mikhail and Wanas, 1977), shows that Z o = 0 (for a definition of Z o see Einstein, 1955; Appendix II). This means that the theory does not leave free any function of four variables, and, there is no scope left to impose an independent equation of state. Finally, we think that the difference between the results obtained in the present work and the corresponding results of relativistic cosmology is of fundamental nature, which may be clarified as follows. The field equations of the general theory of relativity: viz., 1
R~,,~ - g g u , , R
(8.1)
= - tOT, v ,
involves some kind of duality between the geometrical structure and the physics it may represent. Quite often this may cause some difficulties, as the geometrical structure used, and represented by the left-hand side of the field equations (8.1), may not be suitable for the kind of physics represented by the form chosen for Tuv on the right-hand side. In the present procedure this situation does not exist. Here the geometrical structure used designates, to some extent, the physical content, it represents from purely geometrical considerations.
Acknowledgement The author wishes to express his deep gratitude to Professor F. I. Mikhail for many discussions and comments.
References Coley,A. A. and Tupper, B. O. J.: 1982,'An Essay Submittedto the GravityResearch Foundation Selected for Honorable Mention'. Einstein, A.: 1955, The Meaning of Relativity (5th ed.), Princeton Univ. Press. Eisenhart, L. P.: 1927,Non-Riemannian Geometry, Amer. Math. Soc., New York. Galvao, C. A. P. and Teitelboim,C.: 1980,J. Math. Phys. 21, 1863.
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