X

'sin^; if JsT = +l, X if K = °> sinh^; if K = — 1.

The coordinate x runs from 0 to oo if K = 0 or — 1, but runs from 0 to 2rr if K = + 1. Whon K = 0 or — 1, the thrce-spaces are diffoomorphic to Rs and so are 'infinite', but when K = +1 they are diffeomorphic to a three-sphere S3 and so are compact (' closed' or' finite'). One could identify suitable points in these three-spaces to obtain other global topologies; it is even possible to do this, in the case of negative or zero curvature, in such a way that the resulting three-space is compact (Lobell (1931)). However such a compact surface of constant negative curvature would have no continuous groups of isometries (Yano and Bochner (1953))-although Killing vectors exist at each point, they would not determine any global Killing vector fields and the local groups of isometries they generate would not link up to form global groups. In the case of zero curvature, a compact space could only have a three-parameter group of isometries. In neither case would the resulting space-time be isotropic. We shall not make such identifications, as our original reason for considering these spaces was that they were isotropic (and so had a six-parameter group of isometries). In fact the only identifications which would not result in an anisotropic space would be to identify antipodal points on S3 in the case of constant positive curvature.

The symmetry of the Robertson-Walker solutions requires that the energy-momentum tensor has the form of a perfect fluid whose density ¡i and pressure p are functions of the time coordinate t only, and whose flow lines are the curves (x> 4>) constant (so the coordinates are comoving coordinates). This fluid can be thought of as a smoothed out approximation to the matter in the universe; then the function S(t) represents the separation of neighbouring flow lines, that is, of 'nearby' galaxies.

The equation of conservation of energy (3.9) in these spaces takes the form . „. .

The Raychaudhuri equation (4.26) takes the form

The remaining field equation (which is essentially (2.35)) can be written

Whenever S' #0,(5.12) can in fact be derived, with an arbitrary value of the constant K, as a first integral of (5.10), (5.11); so the real effect of this field equation is to identify the integration constant as the curvature of the metric do-2 of the three-spaces {t = constant}.

It is reasonable to assume (cf. the energy conditions, § 4.3) that ft is positive and p is non-ncgativc. (In fact, present estimates are 10 »"gin cm 3 si fiQ lo gin cm ®, /i0 p p0 ^ 0). Then, il' A is zero, (5.11) shows that S cannot be constant; in other words the field equations then imply the universe is either expanding or contracting. Observations of other galaxies show, as first found by Slipher and Hubble, that they are moving away from us, and so indicate that the matter in the universe is expanding at the present time. Current observations give the value of S'/S at the present time as

H s (S-/S)\0 x 10~10year_1, believed correct to within a factor 2. From this, (5.11) shows that if A is zero, S must have been zero a finite time t0 ago (that is, a time t0 measured along the world-line of our galaxy) where t0 < H-1 « 1010 years.

From (5.10) it follows that the density decreases as the universe expands, and conversely that the density was higher in the past, increasing without bound as S->0. This is therefore not merely a coordinate singularity (as for example, in anti-de Sitter universe expressed in coordinates (5.9)); the fact that the density is infinite there shows that some scalar defined by the curvature tensor is also infinite. It is this that makes the singularity so much worse than in the corresponding Newtonian situation; in both cases the world-lines of all the particles intersect in a point and the density becomes infinite, but here space-time itself becomes singular at the point S = 0. We must therefore exclude this point from the space-time manifold, as no known physical laws could be valid there.

This singularity is the most striking feature of the RobertsonWalker solutions. It occurs in all models in which ft + Zp is positive and A is negative, zero, or with not too large a positive value. It would imply that the universe (or at least that part of which we can have any physical knowledge) had a beginning a finite time ago. However this result has here been deduced from the assumptions of exact spatial homogeneity and spherical symmetry. While these may be reasonable approximations on a large enough scale at the present time, they certainly do not hold locally. One might think that, as one traced the evolution of the universe back in time, the local irregularities would grow and could prevent the occurrence of a singularity, causing the universe to' bounce' instead. Whether this could happen, and whether physically realistic solutions with inhomogeneities would contain singularities, is a central question of cosmology and constitutes the principal problem dealt with in this book; it will turn out that there is good evidence to believe that the physical universe does in fact become singular in the past.

If some suitable relation between p and fi is specified, (5.10) can be integrated to give fi as a function of S. In fact the pressure is very small at the present epoch. If one takes it and A to be zero, one finds from (5.10)

where M is a constant, and (5.12) becomes

The first equation expresses the conservation of mass when the pressure is zero, while the second (the Friedmann equation) is an energy conservation equation for a comoving volume of matter; the constant E represents the sum of the kinetic and potential energies. If E is negative (i.e. K is positive), S will increase to some maximum value and then decrease to zero; if E is positive or zero (i.e. K is negative or zero), S will increase indefinitely.

The explicit solutions of (5.13) have a simple form if given in terms of a rescaled time parameter r(t), defined by dr/d t = S-^t); (5.14)

they take the form

S= (£73)(coshT-l), t = (^/3)(sinhT-T), if K = -1; S = t\ < = £r3, if K = 0;

(The case K — 0 is the Einstein-de Sitter universe; clearly S oc tl.)

lip is non-zero but positive, the qualitative behaviour is the same. In particular if p = (y— 1)^ where y is a constant, 1 < y < 2, one finds

|77/i = MIS3?, and the solution of (5.12) near the singularity takes the form

If A is negative, the solution expands from an initial singularity, reaches a maximum and then recollapses to a second singularity. If A is positive, then for K = 0 or — 1 the solution expands forever and asymptotically approaches the steady state model. For K = +1 there are several possibilities. If A is greater than some value Acrlt (Acrlt =(-El3Mfl(3M)i if p = 0) the solution will start from an initial singularity and will expand forever asymptotically approaching the steady state model. If A = Acrlt there is a static solution, the Einstein static universe. (The metric form (5.7) is that of the particular Einstein static solution for which fi+p = (4rr)—A = 1 + 87753.) There is also a solution which starts from an initial singularity and asymptotically approaches the Einstein universe, and one which starts from the Einstein universe in the infinite past and expands forever. If A < Acrlt there are two solutions—one expands from an initial singularity and then recollapses to a second singularity; the other contracts from an infinite radius in the infinite past, reaches a minimum radius, and then re-expands. This and the universe asymptotic to the static universe in the infinite past are the only solutions which could represent the observed universe and which do not have a singularity. In these models, S" is always positive, and this seems to be in conflict with observations of redshifts of distant galaxies (Sandage (1961, 1968)). Also, the maximum density in these models would not have been very much larger than the present density. This would make it difficult to understand phenomena such as the microwave background radiation and the cosmic abundance of helium, which seem to point to a very hot dense phase in the history of the universe.

Just as in the previous cases we have studied, one can find conformal mappings of the Robertson-Walker spaces into the Einstein static space. We use the coordinate t defined by (5.14) as a time coordinate; then the metric takes the form d«2= /»^{-d^ + d^+Z^fd^+sin^d^2)}. (5.15)

In the case K = +1, this is already conformal to the Einstein static space (put t = t', x = / to agree with the notation of (5.7)). Thus these spaces are mapped into precisely that part of the Einstein static space determined by the values taken by t. When p = A = 0, t lies in the range 0 < t < it, so the whole space is mapped into this region in the Einstein static universe while its boundary is mapped into the three-spheres t = 0, t = it. (If p > 0, it is mapped into a region for which t takes values 0 < t < a < it, for some number a.) In the case K = 0, the same coordinates represent the space as conformal to flat space (see (5.15)), so on using the conformal transformations of §5.1, one obtains these spaces mapped into some part of the diamond representing Minkowski space-time in the Einstein static universe (see figure 14); the actual region is again determined by the values taken by r. When A = 0, 0 < t < oo, so this space (which is the Einstein-de Sitter space when p = 0) is conformal to the half t' > 0 of the diamond which represents Minkowski space-time. In the case K = — 1, one obtains the metric conformal to part of the region of the Einstein static space for which \n > t' + r' > — \tt, > t' — r' > —\n, on defining t' = arc tan (tanh + x)) + arc tan (tanh £(t ~x)),

The part of this diamond-shaped region covered depends on the range of t; when A = 0, the space is mapped into the upper half.

One thus obtains these spaces and their boundaries conformal to some (generally finite) region of the Einstein static space, see figure 21 (i). However there is an important difference from the previous cases: part of the boundary is not 'infinity' in the sense it was previously, but represents the singularity when S = 0. (The conformal factor can be thought of as making infinity finite by giving an infinite compression, but making the singular point S = 0 finite by an infinite expansion.) In fact this makes little difference to the conformal diagrams; one can give the Penrose diagrams as before (see figures 21 (ii) and 21 (iii)). In each case when p > 0 the singularity at t = 0 is represented by a spacelike surface; this corresponds to the existence of particle horizons (defined precisely as in §5.2) in these spaces. Also when K = +1 the future boundary is spacelike, implying the existence of event horizons for the fundamental observers; when K = 0 or — 1 and A = 0, future infinity is null and there are no future event horizon^ for the fundamental observers in these spaces.

At this stage, one should examine the following question: anti-de Sitter space could be expressed in the Robertson-Walker form (5.9) and then expressed conformally as part of the Einstein static universe. When one did so, one found that the Robertson-Walker coordinates

(i)

(coordinate singularity)

(i) The Robertson-Walker spaces (p = A = 0) are conformal to the regions of the Einstein static universe shown, in the three cases K = + 1, 0 and — 1.

(ii) Penrose diagram of a Robertson—Walker space with K = + 1 and p = A = 0.

(iii) Penrose diagram of a Robertson-Walker space with if = 0 or — 1 and p = A = 0.

covered only a small part of the full space-time. That is to say, the space-time described by the Robertson-Walker coordinates could be extended. One should therefore show that the Robertson-Walker universes in which there is matter are in fact inextendible. This follows because one can show that if fi > 0, p > 0 and X is any vector at any point q, the geodesic y(v) through q = y(0) in the direction of X is such that either f*lr — -n\

(coordinate singularity)

(coordinate (coordinate singularity) singularity)

(i) y{v) can be extended to arbitrary positive values of v, or

(ii) there is some v0 > 0 such that the scalar invariant

(R{j - \ESij) (RV - \RgU) = (/i + A)2 + 3(p — A)2

It is now clear that the surfaces {t = constant} are Cauchy surfaces in these spaces. Further one sees that the singularity is universal in the following sense: all timelike and null geodesies through any point in the space approach it for some finite value of their affine parameter.

5.4 Spatially homogeneous cosmological models

We have seen that there are singularities in any Robertson-Walker space-time in which fi > 0, p > 0 and A is not too large. However one could not conclude from this that there would be singularities in more realistic world models which allow for the fact that the universe is not homogeneous and isotropic. In fact, one does not expect to find that the universe can be very accurately described by any attainable exact solution. However one can find exact solutions, less restricted than the Robertson-Walker solutions, which may be reasonable models of the universe, and see if singularities occur in them or not; the fact that singularities do occur in such models gives an indication that the existence of singularities may be a general property of all space-times which can be regarded as reasonable models of the universe.

A simple class of suoh solutions are those in which the requirement of isotropy is dropped but the requirement of spatial homogeneity (the strict Copernican principle) is retained (although the universe seems approximately isotropic at the present time, there might have been large anisotropics at an earlier epoch). Thus in these models one assumes there exists a group of isometries Or whose orbits in some part of the model are spacelike hypersurfaces. (The orbit of a pointy under the group 6r is the set of points into which p is moved by the action of all elements of the group.) These models may be constructed locally by well-known methods; see Heckmann and Schiicking (1962) for the case r = 3, and Kantowski and Sachs (1067) for the case r = 4 (if r > 4, the space-time is necessarily a Robertson-Walker space).

The simplest spatially homogeneous space-times are those in which the group of isometries is Abelian; the group is then of type I in the classification given by Bianchi (1918), so we call these Bianchi 1 spaces. We discuss Bianchi I spaces in some detail, and then give a theorem showing singularities will occur in all non-empty spatially homogeneous models in which the timelike convergence condition (§4.3) is satisfied.

Suppose the spatially homogeneous space-time has an Abelian isometry group; for simplicity we assume A = 0 and that the matter content is a pressure-free perfect fluid ('dust'). Then there exist comoving coordinates (i, x, y, z) such that the metric takes the form d«2 = -dt* + X*(t)dx2 + Y2(t)dy2 + Z2(t)dz2. (5.16)

Defining the function S(t) by S3 = XYZ, the conservation equations show that the density of matter is given by = M/Sa, where M is a suitably chosen constant. The general solution of the field equations can be written

Z = S(it/iS)S!Bta <«+*">, where S is given by ¿js = p^ + 2).

£ (> 0) is a constant determining the magnitude of the anisotropy (we exclude the isotropic case (2 = 0), which is the Einstein-de Sitter universe (§5.3)), and a( — fan < a < fan) is a constant determining the direction in which the most rapid expansion takes place. The average rate of expansion is given by

the expansion in the a;-direction is

and the expansions Y'\Y, Z'jZ in the y, z directions are given by similar expressions in which a is replaced by a + cc + %n respectively.

The solution expands from a highly anisotropic singular state at t = 0, reaching a nearly isotropic phase for large t when it is nearly the same as the Einstein-de Sitter universe. The average length S increases monotonically as t increases, its initial high rate of change (S oc ti for small t) decreasing steadily (S oc it for large t). Thus the universe evolves more rapidly, at early times, than its isotropic equivalent. Suppose one considers the time-reverse of the model, and follows this forward in time towards the singularity. The initially almost isotropic contraction will become very anisotropic at late times. For general values of a, i.e. a 4= \tt, the term 1 + 2 sin (a + will be negative. Thus the collapse in the z-direction would halt, and, for sufficiently early times, be replaced by an expansion, the rate of expansion becoming indefinitely large for early enough times. In the x- and ^-directions, on the other hand, the collapse would continue mono-tonically towards the singularity. Thus if one considers the forward direction of time in the original model, one has a 'cigar' singularity: matter collapses in along the z-axis from infinity, halts, and then starts re-expanding, while in the x- and ^-directions the matter expands monotonically at all times. If one could receive signals from early enough times in such a model, one would see a maximum red-shift in the z-direction, at earlier times matter in this direction being observed with progressively smaller redshifts and then with indefinitely increasing Wwe-shifts.

The behaviour in the exceptional case a = \n is rather different. In this case, the terms 1 + 2 sin (a+\n) and 1 + 2 sin (a + \n) both vanish. Thus the expansions in the axis directions are

If one follows the time-reversed model, the rate of collapse in the y- and z-directions slows asymptotically down to zero, while the rate of collapse in the a;-direction increases indefinitely. In the original model, one has a 'pancake' singularity: matter expands monotonically in all directions, starting from an indefinitely high expansion ratd in the a;-direction but from zero expansion rates in the y- and z-directions^ Indefinitely high redshifts would be seen in the a;-direction, but there would be limiting redshifts in the y- and z-directions.

Further examination shows that in the general ('cigar') case, there is a particle horizon in every direction despite the anisotropic expansion. However in the exceptional ('pancake') case, no horizon occurs in the a;-direction; in fact the particles that can be seen by an observer at the origin at time t0 are characterized by coordinate values (x, y, z) lying within the infinite cylinder x*+y2 < (P

While we have here considered these models for vanishing pressure and A term only, properties of these spaces with more realistic matter contents can easily be obtained; for example if one has either a perfect fluid with p = (y — \)/i, y a constant (1 < y < 2), or a mixture of a photon gas and matter with pressure p < \[i, the behaviour near the singularity is the same as in the dust case.

An interesting consequence of the non-existence of a particle horizon in the a;-direction in the exceptional ('pancake') case, is that one can extend the solution continuously across the singularity. We shall show this explicitly in the case of the dust solution. The metric takes the form (5.16) where now

*(<) = <«+ 2))-*, Y(t) = Z(t) = $M(t + X))i. (5.17)

We now choose new coordinates t, ri which satisfy the equations tanh (to/MfZ) - ,/r, exp . ^

One then finds that the space with metric (5.16), (5.17) is given in the new coordinates by d s* = A2(t)(-dT* + dvi) + Bi(t){dys + dzi) (5.18)

where

A(t) = exp (fitf (< + £))-*, B(t) = (f2f(< + S))§, (5.19)

the whole space (for t > 0) being mapped into the region defined by t > 0, t® — 7j2 > 0. The function t(r, rj) is now defined implicitly as the solution of the equation t2 — 7j2 = IMP exp + (5.20)

¿j for which t > 0. The (r, rj) plane is given in conformally flat coordinates. The region "f in this plane, bounded by the surface t = 0, is shown in figure 22. In this diagram, the world-lines of the particles are straight lines diverging from the origin.

The functions A(t), B(t) are continuous as 0 from above. One can therefore extend the solution continuously to the whole (t,tj) plane by specifying that (5.19) holds everywhere, (5.20) holds inside "V, and holds outside "f. Then (5.18) is a C° metric which is a solution of the

Galaxies not seen by O at p
(ii)

Figubk 22. Dust-filled Bianchi I space with a pancake singularity.

(ii) A half-section of the space in (r, r¡, y) coordinates (the z-coordinate is suppressed), showing the past light cone of the point p = (r0, 0, 0). There is a particle horizon in the ^-direction but not in the x- (i.e: ij) direction.

field equations equivalent to (5.16), (5.17) inside "V, and is a flat space-time outside "V. However the solution is not C1 across the boundary of "V, and in fact the density of matter becomes infinite on this boundary (as (S->0 there). Since the first derivatives are not square integrable, the Einstein field equations cannot be interpreted on the boundary even in a distributional sense (see § 8.4). While the extension onto the boundary is unique, it is in no way unique beyond the boundary. We have carried out the extension in the case of dust; a similar extension could be carried out if one had a mixture of matter and radiation.

Let us now return to considering general non-empty spatially homogeneous models. The existence of a singularity in these models will follow directly from Raychaudhuri's equation if the motion of the matter is geodesic and without rotation (as must be the case, for example, if the world-lines are orthogonal to the surfaces of homogeneity) and the timelike convergence condition is satisfied; however there exist such spaces in which the matter accelerates and rotates, and either of these factors could possibly prevent the existence of a singularity. The following result, which is an improved version of a theorem of Hawking and Ellis (1965), shows that in fact neither acceleration nor rotation can prevent the existence of singularities in these models.

Theorem g) cannot be timelike geodesically complete if:

(1) RabKaKb > 0 for all timelike and null vectors K (this is true if the energy-momentum tensor is type I (§4.3) and fi+pt > 0,

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