Vistas in Astronomy, Vol. 29, pp. 281-289, 1986 0083-6656/86 $0.00 + .50 Printed in Great Britain. All rights reserved. Copyright © 1986 Pergamon Journals Ltd.

N E W I D E A S IN C O S M O L O G Y A N D N E W S O L U T I O N S IN G E N E R A L R E L A T I V I T Y

Paul S. Wesson Department of PhysiCs, University of Waterloo, Waterloo, Ontario,

Canada N2L 3G1

ABSTRACT

New ideas in cosmology that have become popular in recent years include matter production, inflation and bubbles in the early Universe. New solutions in general relativity include Friedmann/Robertson/Walker ones which start in Minkowski space, and spherically- symmetric ones with expanding, singular hypersurfaces which are spherical in ordinary space. The ideas in cosmology and the solutions which describe them are reviewed. In particular, Einstein's equations are given in forms suitable for finding other solutions; and certain recent solutions are quoted in forms suitable for working out astrophysical consequences. It is hoped that the present review will serve as a stimulus to further research in this field.

I. INTRODUCTION

Recently, some long-standing problems with standard models have led to the appearance of several new ideas in cosmology. To describe these ideas, new solutions in general relativity with properties not seen before have been introduced. It is the aim of the present work to briefly review these ideas; and to give a mathematical framework for their description, extensive enough that interested workers can find other solutions of Einstein's equations, and examine the astrophysical consequences of recently discovered solutions.

The standard models of cosmolo~ are the Friedmann/Robertson/Walker (FRW) ones which start with a big bang (Robertson and Noonan, 1968). They are solutions of the field equations of general relativity in the form of the Friedmann equations, and have the Robertson/Walker line element (see below). The Friedmann equations presume that the cosmological medium behaves as a perfect fluid, and the Robertson/Walker line element pres~nes that the medium is homogeneous and isotropic. The FRW models are mathematically simple, but those which start with a big bang have some physical problems. There is disagreement about which of these problems are serious enough to warrant an interest in non-standard models, but two problems may be mentioned here because they can be resolved (at least partially) by models of the types discussed below. Firstly, there is the problem of the origin of the matter in the Universe. In the standard FRW models, the density of matter is infinite at time zero. This means that the origin of matter in these models is not open to analysis, a situation that is widely regarded as unsatisfactory. Secondly, there is the horizon problem. In most standard FRW models there are horizons, which imply that in the early stages of a given model there are regions that are causally di~oint, meaning they cannot communicate with each other by signals travelling at the speed of light or less. But in the real Universe, photons of the microwave background have nearly the same temperature irrespective of which direction they come from, which implies that the early stages of the real Universe did not have regions that were causally di~oint. This means that a conflict exists between the standard FRW models and the real Universe, at least insofar as the early stages are concerned.

New ideas to help resolve these and other problems have appeared recently. It was suggested some while ago that the big bang may have been a quantum phenomenon (Tryon, 1973; Albrow, 1973). But the idea was not examined in detail until later, when it was investigated in the context of eonformally-invariant quantum field theories (B rout et al., 1978, 1979). Such theories predicted that particles could be produced from the vacuun, and this process became the basis for several new models of the origin of the matter in the Universe (loc.cit., and Atkatz and Pagels, 1982; Gott, 1982; Henriksen et al., 1983; Henriksen, 1983). Meanwhile, the horizon problem was tackled by using the new idea that the Universe in the early stages may have undergone a period of de Sitter (i.e., exponential) or "inflationary" expansion (Guth, 1981). This idea also implied the associated one of phase changes, and it was suggested that in the early Universe "bubbles" of a new phase may have appeared in an old phase (loc.cit., and Guth and Weinberg, 1981). The inflationary Universe model quickly became a subject of intense interest, and reviews of it are available (Levi, 1983; Guth and Steinhardt, 1984). After a while, the distinction between models to explain the origin of the matter in the Universe and models to resolve the horizon problem became blurred, and models which were combinations of both previous types were suggested (e.g., Vilenkin, 1982, 1983). The present situation is that none of the new ideas recently proposed solves all of

281

282 P.S. Wesson

the problems encountered by the standard FRW models, but that the whole field of cosmology as it applies to the early Universe is now open to re-examination, with a good chance of finding more satisfactory models.

New solutions in general relativity which describe the preceding ideas can be obtained by assuming a certain form for the line element ds 2 = gijdxid~ and solving Einstein's equations Gcj = -8~Tij. Here gi4 is the metric tensor, x i are coordinates (x ° = t = time, xl,2, 3 = r,0,O = spherica~ polars), Gij is the Einstein tensor and Tij is the energy-momentum tensor. The latter is usually assumed to be that corresponHin~ to a perfect fluid, in which case it depends on the metric tensor, the 4-veloclties u i = dxl/ds and only two matter parameters, namely the density p and the pressure p. With the perfect-fluid assumption Einstein's equations take their usual form,

Gij = -87 [(p+p)uiu j - pgij ] . (I.I)

In writing these equations, units have been chosen which render the magnitudes of the gravitational constant and the velocity of light equal to unity, a choice which will prevail in what follows. It can also be remarked here that in solutions of (I.i) to be discussed below, the symbols p and p for a cosmological perfect fluid may include more than the density and pressure intrinsic to the particles of the fluid. For example, it has been realized for some time that these symbols really include contributions from the gravitational and electromagnetic interactions between particles (Robertson and Noonan, 1968, p. 417). And more recently it has become common to include other contributions from what is naively called the vacuum by writing the total density P and total pressure p as sums of a matter part and a vacuum part: thus p = Pm + 0v and p = Pm + Pv, where Pv = -Pv applies as the effective equation of state of the vacuum in Einstein's theory (see, e.g., Henriksen, 1983; Henriksen et al., 1983; and Linde, 1974). This technique also allows the cosmological constant A to be handled in a neat way. For this parsmeter is equivalent to a constant density and pressure for the vacuum, where Pv = -Pv = A/8~. This means that if A does not appear explicitly in the equations, it may be regarded as being present implicitly in the density and pressure. Thus, the symbols p and p allow of several subtly different interpretations, and these should be kept in mind when discussing equations (1.1) and their solutions.

In the following two sections, solutions of the perfect-fluid Einstein equations (I.I) will be discussed which are of the FRW (homogeneous and isotropic) type and the spherically symmetric (in general, inhomogeneous) type, respectively.

2. FRIEDMANN/ROBERTSON/WALKER SOLUTIONS

In this section, a discussion will be given of FRW solutions which do not begin in a big bang of the conventional type but produce matter from empty Minkowski space. Two special models will be discussed, in each of which the density starts from zero, rises sharply to a maximum and then declines. The discussion is based on work by Wesson (1985). It is essentially a re-examination of the work on matter production mentioned in section 1 above (Brout et al., 1978, 1979; Atkatz and Pagels, 1982; Gott, 1982; Henriksen et al., 1983; Henriksen, 1983), but in the context of the FRW solutions.

The line element for the FRW solutions is

ds 2 = dt 2 - S 2 {. dr2 + r 2 (d02 + sin2Od~ 2) }

(i + kr2/4) 2

Here, S = S(t) is the scale factor and k(= ±I,0) is the curvature constant. Derivatives of S with respect to t will be denoted by a dot. For the above metric, the field equations (I.I) reduce to two relations, the Freidmann equations (Robertson and Noonan, 1968, p. 372). These are

3 8~p = ~2 (~2 + k)

=i (2s~+ §2 8~p = ~ + k). (2.1)

The cosmological constant has not been included in these equations, hut if so desired it may be regarded as incorporated in p,p by using the technique outlined in Section i. In that case, however, the symbols p,p will have somewhat different meanings than those used in the rest of this section, where they will be taken to refer to the properties of ordinary matter.

The standard solutions of (2%1) obey certain traditional conditions involving P and p. These conditions include p>O, (p + p)>0 and (3p + p)>0. But it is clear that if equations (2.1) are to give a classical description of the partlele production found in quantum field theories, these conditions have to be relaxed (Brout et al., 1978; Henriksen et al., 1983; Henriksen, 1983). This can he seen directly from (2.1). For these imply

Cosmology and General Relativity 283

(pS3) " + p(S3) " = 0, or m + pV = 0 where m = pS 3 is mess and V = S 3 is volume (Robertson and

Noonan, 1968, p. 373). So m>0 requires p<0 in an expanding model. The pressure p is a phenomenologieal parameter, and p<0 is to be interpreted as a tension. It is quite possible that quantum mechanical processes and the interactions between particles resulted in an enviroranent in the early Universe which can formally be described by p<0 (see above). It should also be appreciated that there are no firm experimental or observational data on the equation of state of matter for temperatures greater than 1011K (Barrow, 1978). For these reasons, it is permissible to study matter production in FRW models by solving (2.1) with p not subject to preconditions.

The standard models are mostly derived from (2.1) by adopting an equation of state p = p(p) and solving for S(t). But this procedure is of course impossible here because the equation of state for the early Universe is unknown. Instead, it is necessary to choose S(t) in accordance with what kind of model is desired, and let the equation of state be determined implieity by (2.1). There are in general an infinite number of choices for S(t), but in the present case they are severely restricted by the wish to explain the origin of the matter in the Universe. In fact, it is possible to be quite specific about this. Minkowski space which is empty (p = 0, p = 0) is the natural starting state for models of the type being considered here, because it is the most basic type of space in general relativity and describes a state without matter. There are only two spaces of this type, as can be seen

from (2.1): M1 has k = 0, S = 0 while M2 has k = -i, S = i. These two Minkowski spaces should be distinguished, even though they can be obtained from each other by coordinate transformations (Robertson and Noonan, 1968, p. 362). In the following two paragraphs, two solutions of (2.1) will be discussed which have desirable properties and which start in M1 and M2 respectively.

(M1) A model which starts in an empty, static state and is spatially flat has been given by Bonnor (1960; see also Israel, 1960). The general form of this model is quite complicated, but a special form may be considered here that is typical. It has a scale factor

S = (1 + ate-b/t) 2/3 (2.2)

and a density and pressure given by (2.1) as

4~ b 2 e -2b/t 8up = (1 + ~) (2.3)

(i + ateb't) 2 - / "

4ab 2 e -b/t 8~p (2.4)

3t 3 (i + ate -b/t )

Here, a and b are positive constants. This model has a density which starts from zero, rises to a maximum and then declines, as may be seen by considerln~ the following properties. For t+0, S+I and all derivatives of S+O, while p + -ab2eLb/t/6~t 3 and @ + a2b2e-2b/t/6~t 2. For t+ =, S ÷ (at) 2/3 while p + -b2/6~t 4 and @ + I/6~t 2, so it is Einstein/de Sitter in this limit. These properties imply that it is possible to have an extended version of this model which exists as the M1 state for t<0, undergoes some kind of perturbative event at t = 0, and evolves smoothly into an expanding, matter-filled state for t>0. However, the model has some drawbacks. For example, while the model in general form has several adjustable parameters, in its most natural form it tends to Einstein/de Sitter at late times with the critical density of matter, whereas the real Universe appears to have a significantly lower density. It is therefore worthwhile to direct attention to the other alternative.

(M2) A model which starts in an empty, expanding state (S = i) and is spatially open has several natural advantages. For example, it can evolve into a relatively low-density state for t÷=; and it has no horizon for t+O at least (Rindler, 1956), and so may avoid the horizon problem which plagues standard FRW models (Guth, 1981). There are several cosmologically acceptable forms for the scale factor for t>O, but one particularly convenient form is

S = t + ~e -B/t (2.5)

where ~ and B are positive constants. The density and pressure are given by (2.1) as

3~B ~Be -B/t e -B/t

8~p = W (2 + t ~ ) (t + ~e-B/t) 2 (2.6)

=B 4~e -B/t 2B 3~Be -B/t e -B/t

8~p = t ~--5- (2 + - - t t t 2 ) (t + ae-B/t) 2 (2.7)

284 P.S . Wesson

This model also has some interesting properties. For t+0, {+i and all higher derivatives of S+O, while p+ -~B 2 e-~/t/4~t 5 ande ~ 3~e-8/t/4~t~" For t+~, S+t while p+ ~/4wt # and

p÷ 3~/4~t 4, so it has the ultra-relativistic or radiation equation of state (p = p/3) in this limit. A numerical investigation of (2.5), (2.6) and (2.7) has been carried out (Wesson, 1985). The main features are illustrated in Fig. i. The detailed results show that

after going negative p becomes positive quite early on, and that the equation of state tends to that of radiation quite quickly (p=p/3 at t=2 for ~=i and 8=1). Therefore this model may be interpreted as one for the early, hot stages of the Universe, and an FRW model of more standard type may succeed it as a description of the later Universe. If it is desired to have a two-stage model of the Bonnor type, then S of (2.5) may be replaced by the form S' = t + ~e-~/(t-l), which leads to the same values for p and p in the limit t +~. Thus it is possible to have an extended model which exists as the M2 state (S = t) for t<l, undergoes some kind of perturbative event at t = I, and evolves smoothly into a matter-filled state (S' as above) for t>l.

The solutions considered in the preceding paragraphs are only two among many which involve matter production. It is possible to choose forms for the scale factor different from either (2.2) or (2.5) which, when substituted into (2.1), show that matter is produced. The cases considered above should be viewed as plausible examples which illustrate two

important points. Firstly, the big bang of standard FRW models can be replaced by an event in which the density starts from zero, rises to a (finite) maximum and then declines. Secondly, there are FRW models in which a matter-filled phase is preceded by an empty phase, the transition between the two being an event which might be identified with a quantum phenomenon.

3. SPHERICALLY-SYMMETRIC SOLUTIONS

In this section, a discussion will be given of spherlcally-symmetric solutions which have matter production and surfaces in ordinary space which may be identified with bubbles. Solutions of two types will be discussed, ones with p ¢ 0 and ones with p = 0. The discussion is based on work by Henriksen et al. (1983) and Wesson (1984), respectively. It is essentially an account of solutions which have properties analogous to those required by

eonformal quantum field theories (Brout et al., 1978, 1979) and the inflationary Universe model (Guth, 1981; Gnth and Weinberg, 1981), but in the context of classical general relativity.

Solutions with p ~ 0 will in general involve a finite cosmological constant A which can be included implicitly in the way outlined in Section i. Thus, the density p and pressure p

will be understood to mean p = Pm + 0v, P = Pm + Pv with Pv = -Pv = A/8~, and the mass m introduced below will be understood to mean m = nhn + m v with m v = 4~R3pv/3 (subscripts refer to matter and vaeuum). To find solutions of Einstein's equations (I.i) relevant to quantum field theories with matter production (Brout et al., 1978, 1979), a good way to proceed is to look for solutions with a conformal Killing vector. The existence of such a vector is connected with the existence of eonformal symmetries, whose presence simplifies the search for solutions from both the mathematical and physical viewpoints. [See Robertson and Noonan, 1968, Chapter 13 and Yano, 1970, for discussions of Killing's equation, Killing vectors and symmetries.] In mathematics, conformal symmetries have long been known to be important, because they cause partial differential equations to be converted to ordinary differential equations. In physics, these symmetries have also long been known to be important, in the guise of self-similar or scale-free solutions to the equations of Newton's theory and Einstein's theory. There is therefore quite a literature in existence on this topic. The main books which deal with it are those by Langhaar (1951), Sedov (1959), Birkhoff (1960) and Bluman and Cole (1974). Some key articles which deal with it are those by Cahill and Taub (1971), Barenblatt and Zel'dovich (1972), Zel'dovich (1977) and Henriksen and Wesson (1978). A particularly interesting class of conformal solutions in general relativity was studied recently (Henriksen et al., 1983), which includes a new exact solution, and some aspects of this class will now be examined.

The line element for spherically-symmetric solutions can be taken to be

ds 2 = e o dt 2 - e ~ dr 2 - R 2 (d02 + sin2Od#2).

Here, r is a Lagrangian coordinate, comoving with the matter. The metric coefficients o,e,R are all functions of t,r. Derivatives with respect to t,r will be denoted by (.),(?)

respectively. For the above metric, the field equations (i.I) reduce to five relations. Of these, the first is really a definition for the mass m, while the others come from the non-vanishing components of (i.I), which are thereby reduced from second order to first order in the derivatives. The five relations are

2m = 1 + e -~ ~2 _ e-toRt2 R

= -4~R2~p

m t = 4~R2Rt 0

t = -2p t o (p+o)

= -2~ 4~ (P+O) R (3.1)

Cosmology and General Relativity 285

40 ' ' ' l i

87rp,

87rp 20

I0

0

- I0

-20

-30

8"lrp

2 S

0

- 4 0 L ..... I I I 0 .0 0.5 1.0 1.5 2.0

t

Fig. I A plot of (2.5), (2.6) and (2.7) for u=l and •=I.

JPVA 29: 3-D

286 P.S. Wesson

These equations were originally derived in this form by Podurets (1964), and have been used in several applications (Henriksen and Wesson, 1978; Wesson, 1978, 1979, 1984; Henriksen et el. , 1983). They are eminently suitable for finding astrophysical solutions in general relativity, because they have fairly immediate physical interpretations. For example, the

second equation of (3.1) shows that the mass of a region increases (m>0) if that region is

expanding (R>0) and has a negative pressure (p<0). This set of circumstances can reflect matter production in the early Universe.

To find solutions of (3.1) with a conformal symmetry, let it be assumed that there are canonical coordinates t' ,r' in terms of which functions depend on the single variable

= t'/r'. This is allowable, provided it is realized that it leads to a splitting of one of the field equations into two (see below), which makes the syste~ exactly determined and so fixes the equation of state p = p(p) automatically. The transformation from the comoving coordinates t,r to the canonical coordinates t',r' has the form

dt = e A°/2 dr' , Ao = A~(t')

dr = e Am/2 dr' , Ae = Am(r'), (3.2)

under which o' = o + Ao and m' = ~ + A~ in the metric. Before proceeding, it is convenient to redefine some parameters in terms of the canonical coordinates t' ,r' and the variable $. The parameters which define the matter properties are Pro, Pm and the mass m m within some radius from the origin of coordinates. The parameters which define the line element and therefore the dynamics are o,~ and R. In terms of t',r' and ~, these parameters can he written

P(~) ~(~) _ r'M($)

Pm = ' Om = ' mm - 2 8~(r') 2 8~(r') 2

o = ~(~) , ~ = ~($) , R = r'S(~) (3.3)

Thus P,n,M,o,e,S are now the unknowns. If d/dE is denoted (*), the field equations (3.1) necessary to find these unknowns can be re-expressed as

1 - M 8~Pv(r')2S2 S - 3 = e-°r'Am(S-~S*)2 - e-°-A°(s*)2

M* = -ps2s *

M-~M* = ~S2(S-~S *)

o* - -2($2P)* ~2(p+~)

~, = -2n* 4S* (P+n) S " (3.4)

These are the basic equations which it is desired to solve.

It is not difficult to see that equations (3.4) do indeed admit solutions with a conformal symmetry. A separate proof is necessary for the cases A>O and A<O, but since they are alike only the former will be given here. For A>0, an inspection of the first equation of (3.4) shows that the only way to maintain the assumed symmetry is to set e -A° = A(t')2/3, e -a~ = I. Then the noted equation splits into 1-M/S = e-m(S-~S*) 2 and S 2 = e-O($S*) 2. These two equations plus the other four of (3.4) provide six relations for the six unknowns P,n,M,o,m,S as functions of E = t'/r'. By (3.2), the relation between the canonical coordinates and the comoving coordinates is found to be

t' = (const) exp [(A/3)i/2t] , r'=r. (3.5)

The variable ~ can therefore be taken to be

= (3)I/2 exp[(A/3) I/2t]r (3.6)

The generator of the corresponding Lie group symmetry may be taken to be

$i = exp [(A/3)I/2t] {(3/A) I/2, r, O, O} , (3.7)

for which the Lie derivative of the metric (see Yano, 1970) is

~ % ~ 2 exp [(u3)l/2t] % . (3.8)

Cosmology and General Relativity 287

The symmetry is thus seen to be eonformal. In fact, (3.7) and (3.8) represent an extension to inhomogeneous spacetimes of the de Sitter space eonformal symmetry.

Equations (3.4) result in physical solutions when A>0 (i.e., Pv>0) and unphysical ones when A<O, in the sense that the former have properties similar to those required by the inflationary Universe model whereas the latter do not. The A>0 solutions will therefore be concentrated on here. Several solutions of this type have been found, including an analytic one that is homogeneous in the density and several numerically determined ones (Henriksen et al., 1983). The analytic solution is typical, in that it and the numerically-determined solutions have like behaviours. The analytic solution is

3C tanh (C £n ~) - 1 p = K2~2[I-C tanh(C ~n ~)]

3

K2~2

M = K ~ sech 3 (C ~n ~)

e ~/2 = i - C tanh (C £n E)

e m/2 = C K $ seth (C £n ~)

S = K ~ sech (C £n ~) (3.9)

Here, C and K are arbitrary constants of integration. This solution is restricted to the domain M<S because of a constraint set by A>0 and the splitting of the first of (3.4) necessary to maintain a eonformal syo~netry. Now by (3.9), as ~+I, M is asymptotically equal to S. Also, there is another analytic solution of (3.4), determined up to a relation between

and ~, that admits homogeneity and has M = S. It is therefore possible to patch these two solutions together at ~ = i, in the manner illustrated in diagram form in Fig. 2. Here, region I is described by (3.9), which has a de Sitter spacetime for ~+% region II is a thin

! I I I I I

r=O

Fig. 2. A patched solution

transition zone near ~ ~ i; and region III is described by the M = S solution of (3.4), which has a spacetlme with S = ~, where now ~<I. This patched global solution is a model for a bubble, which propagates into one region of the Universe while containing another region that is rapidly expanding. This interpretation is in accord with the ideas behind the inflationary Universe model mentioned in Section I. Other such patched solutlonsmay be found that contain an inflationary region and describe two phases.

The particle production mentioned in Section 1 is also accommodated by solutions of the type being discussed here. This can be appreciated by notlng some results from the numerically determined solutions of (3.4). These solutions typically have a behaviour in the P/S plane where P rises from negative values to a positive maximum, and then falls to near

288 P.S. Wesson

zero, as S increases. By (3.3), P is proportional to the matter pressure Pm and S is proportional to the distance measure R. These solutions therefore have a period of negative matter pressure in the early stages of expansion, and so can describe matter production in the early Universe.

Qualitatively, the solutions discussed above have properties in remarkable conformity with those of the inflationay Universe model (Guth, 1981; Guth and Weinberg, 1981) and conformal quantum field theory (Brout et al., 1978, 1979). They are inflationary, can describe two phases, and have a period of particle production. Presumably, if a solution of the kind discussed above can be applied to the real Universe, the surface of the bubble now lles beyond the horizon. Quantitatively, the solutions discussed above imply a nmch larger value of A than that of the real Universe at the present time. But this, in a way, is no surprise: the phase changes which are an integral part of the inflationary model must involve changes in the value of A, and these cannot be included in solutions of the type discussed above.

In this section so far, solutions have been discussed in which p * 0 (however this symbol is interpreteD. As can be seem from the above, such solutions are quite complicated. To end this section, a solution will be discussed in which p = 0, and which is relatively simple. This solution was found by Wesson (1984), and despite its simplicity is of interest because it displays a bubble-like structure.

It may be verified by direct substitution that an exact solution of (3.1) is given by

2k;r3 p = 0 , P 2~f(2k 2 + 3f) , m 9

2k 2 e °/2 = 1 , e~/2 = f2/3 ii + ~- ) , R = f2/3r

f = klt + k2£nr + k 3 (3.10)

Here the k's are constants. This solution has some interesting properties, of which the main ones may be mentioned here. For t+~, f÷klt , e~/2+f 2/3, p+(6zt2) -I and is homogeneous, and the solution tends to Einstein/de Sitter. For t+0, O is inhomogeneous. The density is singular (p+~) on two hypersurfaces, namely f = 0 and (2k 2 + 3f) = 0. On these hypersurfaces, the metric is also singular, since R+0 and e~/2÷0 respectively. This behaviour appears to be typical of solutions like this, since in another solution of this type a like behaviour is found (Henriksen and Wesson, 1978). It is not necessarily unphysical, since by a suitable choice of parameters such solutions can be interpreted as cosmological models valid even for the present Universe (Wesson, 1979). However, the hypersurfaces in (3.10) are most appropriately interpreted as bubbles of the sort found in the inflationary picture, in which case this solution is most relevant to models of the early Universe.

4. CONCLUSION

The standard models of cosmology have some long-standing problems, such as those to do with the origin of matter and the horizon. New ideas have been proposed in response to these problems, notably involving matter production from the vacuum, inflationary expansion, and phase changes which proceed by the formation of bubbles. New solutions in general relativity which describe these ideas are of two main types: Friedmann/Robertson/Walker ones, where matter is produced from the vacuum and where the matter-filled phase can be preceded by an empty phase; and spherically-symmetric ones, where there are hypersurfaces that are spherical in ordinary space and can be identified with surfaces that separate different phases.

The solutions discussed above are mathematically interesting. However, none of them has all the features desired of a model of the early Universe; and all of them have restrictions imposed by the assumptions on which they are based. Of these latter assumptions, that involving the use of a perfect fluid has been made explicit, while others which underlie the use of classical general relativity are implicit (e.g., the absence of quantum gravity effects). But despite their shortcomings, solutions of the types discussed in Sections 2,3 show that important new features are still to be found even in well-studied areas of general relativity. In the above, an attempt has been made to write Einstein's field equations in forms suitable for the finding of other physically relevant solutions. This has been done in the hope that those interested in this kind of research can pursue it with the least possible delay. Another topic for future research concerns working out the astrophysical consequences of solutions, like those discussed above, which have been found recently and about which little is known. There are several major unsolved problems in cosmology, such as the nature of galaxy formation, which should be examined afresh now that new ideas and solutions are available.

ACKNOWLEDGEMENT S

Thanks for previous discussions go to W.B. Bonnor and R.N. Henriksen. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

Cosmology and General Relativity 289

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