Sumita chaudhury, Department of Physics, Lady Keane College, SHILLONG-793001(India)
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Cosmology is the study of the universe as a whole. Matter in the universe is found to be distributed in agglomerations of stars, galaxies and clusters of galaxies. Cosmology treats this distribution as a fine structure, which is ignored in the zeroth approximation, and the universe is described in the continumm approximation, i.e., via a cosmological fluid. The cosmological fluid has the property of being isotropic and homogenous ; in other words, of being isotropic and homogenous about every spatial point. This means that on a sufficiently large scale, there is no privileged direction or a privileged location in the universe. The hypothesis that the universe is homogenous and isotropic is known as the ‘Cosmological Principle’ and is the working assumption in designing a mathematical model of the universe. A comprehensive discussion is presented in this paper as to how such models may be designed on the basis of these simplying assumptions.
Keywords : Cosmology, Cosmological Principle, Homogeity, Isotropy
After several attempts, in the year 1915, Albert Einstein ultimately succeeded in arriving at the desired field equation of general relativity. The following tensor equation is the celebrated equation of gravitation as Einstein deduced:
R ik – ½ g ik R = 8T ik
In the process of derivation, Einstein realised that a tensor Riklm called Riemann curvature tensor, plays a fundamental role in the intrintic geometry of spacetime and contains first and second spacetime derivatives of g ik, the metric tensor and was of rank 4. For obvious reasons of reducing it to equation of rank 2, the method of contraction was made use of by him and a combination of the type
aR ik + b g ik (a,b constants)
was obtained. Einstein’s initial guess was that b = 0 and so the field equation would to the form
R ik = (constant)T ik
However, he, after a prolonged introspection, realised that such an equation would require the additional condition of energy momentum conservation of the form
T ik ; k = 0
The above condition led to the values of a & b as b= – a/2. Accordingly, Einstein arrived at the following supposedly finest of equations ever written that mathematically represents the real world
G ik = R ik – ½ g ik R = kT ik
The coupling constant k , on comparison with experiments, turned out to have value equal to 8?.
In 1917, barely two years after formulating his general relativistic theory of gravitation, Einstein applied the theory to construct a model for the way the universe really works. Shortly after Einstein published his field equations of general relativity, Karl Schwarzschild was the first person to solve them to find the spacetime geometry outside a spherical distribution of mass M. Schwarzschild solution was the first physically significant solution of Einstein’s field equations. It showed how the spacetime is curved around a spherically symmetric distribution of mass. In an attempt to explain the working of the universe, Einstein realised that Schwarzschild solution was basically a local problem as the spacetime would be asymptotically flat. Thus the solution would cease to provide the required spacetime geometry of the universe as a whole, since the universe cannot be approximated by a local distribution of matter only. It was not far too long that Einstein felt the necessity of a new type of solution to describe a universe filled everywhere with matter. Such a picture of the universe was formulated by Einstein in the year 1917.
Now, how to construct a model of the universe making use of the field equations of general relativity, became the primary concern that Einstein got preoccupied with immediately after formulation of his field equations of general relativity. The field equations of general relativity, as they stand , are a set of interlinked non-linear partial differential equations and are virtually not solvable when attempted to formulate a model of the universe. Therefore, in order to make any progress towards solution, Einstein felt the necessity of imposing some simplifying assumptions to the equations. What are these simplifying assumptions and how they would be used in modelling the universe? These two simplifying assumptions used by the cosmologists in order formulate model for the universe are (i) homogeneity & isotropy and (ii) Weyl’s posulates. Einstein too, was no exception in making use of these assumptions in formulating his mathematical model for the universe. Let us look critically into the essence of these assumptions.
MODELLING THE UNIVERSE
Homogeneity & Isotropy
The hypothesis that the universe is homogenous and isotropic is known as ‘Cosmological Principle’.
Homogeneity basically refers to the notion that we do not occupy a privileged position in our universe; that if we were located in a different region of our universe, the basic characteristics of our surroundings would appear the same. Homogeneity implies that a quantity that a physicist measures would be the same wherever he might be located in the spacetime. An observer viewing the universe from any vintage point will find that it presents the same aspect from all such points.
Isotropy implies the notion that there are no preferred directions in space; the observations on sufficiently large scales should yield results which do not depend on which direction we look. The universe from any vantage point would look the same in whichever direction the observer looks. Isotropy implies invariance with respect to the change of direction.
Mathematical definition of homogeneity & isotropy
These two symmetry properties of spacetime, when structured mathematically, would, by their implicit mechanism, bring remarkable simplification when attempted to model the universe. Loosely speaking, homogeneity means that at any given ‘instant of time’, each point of ‘space’ should ‘look like’ any other point. A precise formulation may be given as follows: A spacetime is said to be (spatially) homogenous if there exists a one –parameter family of spacelike hypersurface ?t foliating the spacetime ( see Fig 1) such that for each t and for any point p , q ? ?t there exists an isometry of the spacetime metric gik which takes p into q.
With regard to isotropy, it should be pointed out, in general, that at each point, at the most one observer can see the universe as isotropic. For example, if ordinary matter fills the universe, any observer in motion relative to the matter must see an anisotropic velocity distribution of the matter. With this idea in view, a precise formulation of the notion of isotropy may be given as follows : A spacetime is said to be (spatially) isotropic at each point if there exists a congruence of timelike curves (i.e., observers) , with tangents denoted by u a, filling the spacetime ( Fig2) and satisfying the following property: Given any point p and any two unit “spatial” tangent vectors s a1, s a2 ? Vp (i.e., vectors at p orthogonal to ua ), there exists an isometry of gab which leaves p and ua at p fixed but rotates sa1 into s a2. Thus, in an isotropic universe, it is impossible to construct a geometrically preferred tangent vector orthogonal to u a.
With the help of Killing vectors, it is also possible to express these spacetime properties more precisely and formally. We have defined above that the spacetime Mn is said to be homogenous if there are infinitesimal isometries which carry a typical point p to q in its immediate neighbourhood. This means that that the Killing vectors at p can take all possible values, and we can choose at p, n linearly independent Killing vectors. By suitable choice, we can therefore have a basis of n Killing vector fields ? i (k) (X,P) at a general point X in the neighbourhood of P such that
Lim i (k) (X , P) = i k , ( k= 1,2,3 )
Clearly, by a succession of infinitesimal displacements, we can take p to any distant point q.
Likewise, the spacetime Mn is said to be isotropic at a given point p if there are Killing vectors ? i in the neighbourhood of p such that ? i (P) and ? i;k (P) span the space of antisymmetric second rank tensors at p. Thus we need ½ n(n-1) linearly independent ? i ;k at p. In an isotropic spacetime at p, we can choose coordinates in the neighbourhood of p such that there are 1/2n (n-1) Killing vector fields ? i[ pq] ( X, P ) with the properties
i [pq] (X, P ) = – ? i [qp] (X, P )
i [pq] (P, P ) = 0,
i;k [pq] (P, P ) = [ ? i;k [pq] (X, P ) ] X=P = ? i p ? k q – ? i q ? k p
( p,q = 1,2, . . . . . , n )
It may be proved with the help of Killing vectors that any Mn which is isotropic about every point is also homogenous.
It is worth noticing how the idea of imposing spacetime symmetries, viz., homogeneity and isotropy in the field equations of general relativity led Einstein to model universe in its large-scale structure.
More than two hundred years ago, Newton had attempted to find out a model of a highly symmetrically distributed matter-filled universe of infinite extent. This would lead to a solution in Newtonian gravity as such a condition would incorporate both homogeneity and isotropy in its entirety .Newton found that such a universe would be static for particles being equally attracted on all sides by matter should stay put where they are. But with the symmetry properties introduced, a problem cropped up, because homogeneity precludes any any pressure gradient in the universe and consequently such a universe would tend to shrink under its own gravity. Newton realised that such model would, therefore, be unstable.
Einstein found, as mentioned earlier, that the solution of the field equations of general relativity, as they stand , is apparently beyond the range of techniques available in applied mathematics. It is only by imposing simplifying assumptions regarding spacetime symmetries that any progress towards a solution may be made. In order to get a cosmological solution, Einstein ,therefore, assumed the idea of homogeneity and isotropy to the field equations .One additional condition that he assumed, like Schwarzchild, that the spacetime is static.
Einstein’s Model of the Universe
Let us now try to understand how the conditions of homogeneity and isotropy helped Einstein in deriving a model of the universe. This allowed him to choose a time coordinate t such that the line element of spacetime could be described by
ds 2 = c 2 dt 2 – ? µ? dx µ dx ? ,
where ?µ? are functions of spacetime coordinates x µ ( µ? = 1, 2, 3 only ).
It is worth noticing that the constraint of homogeneity implies that the coefficient of dt 2 can only be constant, which has been normalized to c 2. Similarly, the constraint of isotropy implied that there there should be no term of the form dt dx µ in the line element. One more idea is required – what physical significance can be given to the term ?µ? ? Einstein believed that the universe contained so much matter so as to ‘ close’ the space. This assumption led him to obtain a specific form for ? µ?. But how?
In an attempt to construct the homogenous and isotropic closed space of three dimensions that Einstein wanted for his model of the universe, he recognised it to be S3 , the 3- surface of a four-dimensional hypersphere of radius, say, S. The equation of such a 3-sphere is given in Cartesian coordinates x 1, x 2, x 3 and x 4 by
(x 1)2 + (x 2)2 +(x 3)2 +(x 4)2 = S 2
Expressing x 1 , x 2 ,x 3, and x 4 in terms of the coordinates intrinsic to the surface and defining
X 4 = S cos ? , x 1 = S sin? cos?, x 2 = S sin? cos? , x 3 = S sin? sin? sin?
The spatial line element on the surface S 3 is therefore given by
d? 2 = (dx 1)2 + (dx 2)2 +(dx 3)2 +(dx 4)2
= S 2 [ d? 2 + sin 2 ? ( d ? 2 sin 2 ? d? 2 )]
The range of ? , ? and ? are given by
0 ? ? ? ? , 0 ? ? ? ? , 0 ? ? ? 2?.
As the range of ? is taken as 0 ? ? ? ? , it gives us what is known as spherical space.
Representing d? 2 in terms of r, ? and ?, with r = sin ? ( 0 ? r ? 1) with possibility of both spherical and elliptical space , we get
d?2 = S 2 [ dr 2/ 1- r 2 + r 2( d ? 2+ sin2 ? d? 2 )]
ds2 = c 2 dt 2 – d? 2
= c 2 dt 2 – S 2 [ d? 2 + sin 2 ? ( d ? 2 + sin 2 ? d? 2 )]
= c 2 dt 2 – S 2 [ dr 2 / 1- r 2 + r 2 ( d ? 2 + sin 2 ? d? 2 )]
It may be noted that it is assumption of the symmetry considerations , i.e., homogeneity and isotropy of the spacetime structure that led to the ultimate formulation for the line element of Einstein universe.
When the above line element is used in the left- hand side of Einstein’s equation and non-zero components of Einstein tensor computed, the two deciding equations representing model of the Einstein universe comes out to be of the form
-3/S 2 = -8?G?o /c 2 , -1/S 2 = 0
It is known to students of cosmology that the above two famous equations could not help Einstein in formulating a static homogenous isotropic and dense model of the universe as is obvious from the appearance.
The idea of bringing in the ?-term into the model in order to get a sensible solution by Einstein and the more elaborate idea of expanding universe first by w. de Sitter and all the consequences thereafter fall now within the purview of history of cosmology, a huge literature of which fills shelfs of any prestigious library of Cosmology. Needless to say , whatever attempt might have been made in designing model for the universe either by Einstein or any other cosmologist, symmetry considerations in the structure of the spacetime, viz., homogeneity and isotropy, got implicitly incorporated into the mathematical formulation of the models.
This postulate referes to the regularity of motion of the galaxies at a given cosmic time. Given the idea that most matter in the universe is confined to galaxies , the postulate asserts that the galaxies move in a systemic fashion along well-defined trajectories never colliding with each other in such a way that it is possible to state clearly at what time which galaxy was where in space. This allows the cosmologists to imagine a universal clock that all galaxies follow. The time measured by this clock is what is cosmic time . With this notion, it is possible to make assertions about the state of the universe as a whole at a given cosmic time. This intuitive picture of the motion of the galaxies is often expressed formally as the Weyl postulate following the work of the famous mathematician Hermann Weyl. The formal idea of the postulate is : the world lines of galaxies form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces.
Mathematical definition of Weyl’s postulate
It can be expressed in terms of coordinates and metric of spacetime. Three spacelike coordinates x µ (µ=1 ,2, 3) are used to label a typical world line in the 3-bundle of galaxies . Let the coordinate x 0 label a typical member of the series of spacelike hypersurfaces of the given system. Thus it follows that
X 0 = constant
is a typical spacelike hypersurface orthogonal to the typical world line given by
X µ = constant
Instead of considering galaxies forming a discrete set, a continuum of smooth-fluid approximation is attempted in order to proceed towards a model.
Let the metric in terms of these coordinates be given by the tensor g ik . Now what assertion may be assigned to the metric on the basis of Weyl’s postulate ? The orthogonality condition gives
g 0µ = 0
Again, the line x µ = constant is a geodesic and the geodesic equations
d 2 x i / d x 2 + ? i kl dx k /ds dx l /ds = 0
are satisfied for x i = constant , I = 1,2,3. Therefore
? µ 00 = 0, µ = 1,2,3.
We therefore get,
?g 00 / ?x µ = 0, µ = 1, 2, 3.
Thus we get that g 00 depends on x 0 only . Hence it is possible to replace x 0 by a suitable function of x 0 to make g 00 constant. Hence without loss of generality, we can take
g 00 = 1.
The line element therefore becomes
ds 2 = ( dx 0 ) 2 + g µ? dx µ dx ?
= c 2 dt 2 + g µ? dx µ dx ?
where ct = x 0 has been substituted. This time coordinate is the cosmic time.
It is observed that the spacelike hypersurfaces in Weyl’s postulate are the surfaces of simultaneity with respect to the cosmic time. Here ‘t’ is the proper time kept by any galaxy .
It may be noted that almost all the pioneer model builders of the universe starting from Newton, Einstein to de Sitter, Friedmann and Lamaitre, were using the assumptions of homogeneity and isotropy of spacetime and also the concept of Weyl’s postulate without explicitly stating them. It was in the year 1930s, that H. P. Robertson and A. G .Walker, for the first time, introduced them into cosmology in a formal mathematical framework and obtained the famous Robertson-Walker line element which may be written as
ds2 = c2 dt2 – S2(t)[ dr2 / 1-kr2 + r2(d ? 2 + sin2 ? d?2]
where the 3-spaces t = constant are Euclidean for k=0, closed with positive curvature for k= +1 and open with negative curvature for k= -1. The scale factor S(t) is often called the expansion factor.
The Robertson-Walker line element signifying geometry of a homogenous and isotropic spacetime was obtained by Robertson and Walker (independently) purely on consideration of the symmetry postulates ( homogeneity and isotropy of the spacetime) and it is remarkable how these simplifying assumptions of Cosmology reduced the number of unknowns in the metric tensor from 10 to the single function S(t) and the discrete parameter k that characterize the Robertson-Walker metric. The line element is virtually the stepping stone to obtaining subsequent realistic models of the universe we live in.
The author expresses her profound gratitude to her research guide Dr K. D. Krori, UGC Emeritus Professor, for initiating her into the enchanting world of General Relativity and Cosmology.
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