Exact solutions

Any space-time metric can in a sense be regarded as satisfying Einstein's field equations

(where we use the units of chapter 3), because, having determined the left-hand side of (5.1) from the metric tensor of the space-time (^.g), one can define T^ as the right-hand side of (5.1). The matter tensor so defined will in general have unreasonable physical properties; the solution will be reasonable only if the matter content is reasonable.

We shall mean by an exact solution of Einstein's equations, a spacetime in which the field equations are satisfied with Tab the energy-momentum tensor of some specified form of matter which obeys postulate (a) ('local causality') of chapter 3, and one of the energy conditions of §4.3. In particular, one may look for exact solutions for empty space (T^ = 0), for an electromagnetic field (T^ has the form (3.7)), for a perfect fluid (T^ has the form (3.8)), or for a space containing an electromagnetic field and a perfect fluid. Because of the complexity of the field equations, one cannot find exact solutions except in spaces of rather high symmetry. Exact solutions are also idealized in that any region of space-time is likely to contain many forms of matter, while one can obtain exact solutions only for rather simple matter content. Nevertheless, exact solutions give an idea of the qualitative features that can arise in General Relativity, and so of possible properties of realistic solutions of the field equations. The examples we give will show many types of behaviour which will be of interest in later chapters. We shall discuss solutions with particular reference to their global properties. Many of these global properties have only recently been discovered, although the solutions have heen known in a local form for some time.

In § 5.1 and § 5.2 we consider the simplest Lorentz metrics: those of constant curvature. The spatially isotropic and homogeneous cosmo-logical models are described in § 5.3, and their simplest anisotropic

generalizations are discussed in § 5.4. It is shown that all such simple models will have a singular origin provided that A does not take large positive values. The spherically symmetric metrics which describe the field outside a massive charged or neutral body are examined in §5.5, and the axially symmetric metrics describing the field outside a special class of massive rotating bodies are described in §5.6. It is shown that some of the apparent singularities are simply due to a bad choice of coordinates. In §5.7 we describe the Godel universe and in § 5.8 the Taub-NUT solutions. These probably do not represent the actual universe but they are of interest because of their pathological global properties. Finally some other exact solutions of interest are mentioned in § 5.9.

5.1 Minkowski space-time

Minkowski space-time is the simplest empty space-time in

General Relativity, and is in fact the space-time of Special Relativity. Mathematically, it is the manifold R* with a flat Lorentz metric rj. In terms of the natural coordinates (x1, x2, x3, x4) on Ri, the metric rj can be expressed in the form dsa = — (da:4)2 + (da:1)2 + (da:2)2 + (da:3)2. (5.2)

If one uses spherical polar coordinates {t,r,6,<j>) where x* = t, x3 = r cos 6, x2 = rsintfcos^, x1 = rsinflsin^, the metric takes the form ds2 = - di2 + dr2 + r2 (d02 + sin2 6 d$6a). (5.3)

This metric is apparently singular for r = 0 and sin 6=0; however this is because the coordinates used are not admissible coordinates at these points. To obtain regular coordinate neighbourhoods one has to restrict the coordinates, e.g. to the ranges 0 < / < oo, 0 < 6 < tt, 0 < ^ < 277. One needs two such coordinate neighbourhoods to cover the whole of Minkowski space.

An alternative coordinate system is given by choosing advanced and retarded null coordinates v, w defined by v = t + r, w = t — r (=> v > w). The metric becomes ds2 = -di;du> + j{i;-u>)2{d02 + sin20d02), (5.4)

where —co<v<cc, —cc<w<cc. The absence in the metric of terms in da2, dw;2 corresponds to the fact that the surfaces {w = constant}, {v = constant} are null (i.e. w.aiv;bgab = 0 = v.av.bgab); see figure 12.

(i) (») Figure 12. Minkowski space. The null coordinate v(w) may be thought of as incoming (outgoing) spherical waves travelling at the speed of light; they are advanced (retarded) time coordinates. The intersection of a surface {v = constant} with a surface {w — constant} is a two-sphere.

(i) The v, w coordinate surfaces (one coordinate is suppressed).

(ii) The (t, r) plane; each point represents a two-sphere of radius r.

In a coordinate system in which the metric takes the form (5.2), the geodesies have the form xa(v) = bav + c° where b° and c° are constants. Thus the exponential map expp: is given by x° (expp X) = X" + x°(p), where X" are the components of X with respect to the coordinate basis {8/8xa} of Tp. Since exp is one-one and onto, it is a diffeomorphism between Tp and Jl. Thus any two points of J( can be joined by a unique geodesic curve. As exp is defined everywhere on Tp for all p, is geodesically complete.

For a spacelike three-surface Sf, the future (past) Cauchy development D+(Sf) (D~(Sf)) is defined as the set of all points q&Ji such that each past-directed (future-directed) inextendible non-spacelike curve through q intersects if, cf. §6.5. If IID"^) = J(, i.e. if every inextendible non-spacelike curve in Jl intersects ¿f, then ¡P is said to be a Cauchy surface. In Minkowski space-time, the surfaces {a;4 = constant} are a family of Cauchy surfaces which cover the whole of^. One can however find inextendible spacelike surfaces which are not Cauchy surfaces; for example the surfaces

Sfa\ {- (a;4)2 + (x1)2 + (x2)2 + (x3)2 = cr = constant}, where a < 0, xt < 0, are spacelike surfaces which lie entirely inside the past null cone of the origin 0, and so are not Cauchy surfaces (see figure 13). In fact the future Cauchy development of is the region bounded by SPa and the past light cone of the origin. By lemma 4.5.2, the timelike geodesies through the origin 0 are orthogonal to the surfaces Sfa. If U then the timelike geodesic through r and 0 is the longest timelike curve between r and S?a. If

Timelike Spacelike Null
Figure 13. A Cauchy surface {x4 = constant} in Minkowski space-time, and spacelike surfaces ¿f^, which are not Cauchy surfaces. The normal geodesies to the surfaces £fa, all intersect at O.

however r does not lie in u there is no longest timelike curve between r and £fa\ either r lies in the region cr ^ 0, in which case there is no timelike geodesic through r orthogonal to or r lies in the region a < 0, x* > 0, in which case there is a timelike geodesic through r orthogonal to but this geodesic is not the longest curve-between r and <5^. as it contains a conjugate point to at O (cf. figure 13).

To study the structure of infinity in Minkowski space-time, we shall use the interesting representation of this space-time given by Penrose. From the null coordinates v, w, we define new null coordinates in which the infinities of v, w have been transformed to finite values; thus we define p, q by tanp = v, tang = w where —\n<p< \it, — \n < q < \n (and js > q). Then the metric of takes the form ds2 = sec2psec2q(-dpdq+isin2 (p- q) (d<92 4- sin2 6d<f>2)).

The physical metric rj is therefore conformal to the metric g given by dsa = -4dpdq + sin2 (p - q) (d0* + sin2 6 d^2). (5.5)

This metric can be reduced to a more usual form by defining t'=p + q, r'=p-q, where — n < t' +r' < n, -n < t' — r' < n, r' ^ 0; (5.6)

(5.5) is then ds2 = -(di')2+(dr')2 + sin2r'(dea + sin2edi62). (5.7)

Thus the whole of Minkowski space-time is given by the region (5.6) of the metric ^ = isec2(j{('+r'))sec2{j{f_/))dg2

where ds2 is determined by (5.7); the coordinates t, r of (5.3) are related to t', r' by

Now the metric (5.7) is locally identical to that of the Einstein static universe (see §5.3), which is a completely homogeneous space-time. One can analytically extend (5.7) to the whole of the Einstein static universe, that is one can extend the coordinates to cover the manifold R1 x S3 where — oo < t' < oo and r', 6, $ are regarded as coordinates on S3 (with coordinate singularities at r' = 0, r' = n and 6 = 0, 6 = n similar to the coordinate singularities in (5.3); these singularities can be removed by transforming to other local coordinates in a neighbourhood of points where (5.7) is singular). On suppressing two dimensions, one can represent the Einstein static universe as the cylinder x* + y2 = 1 imbedded in a three-dimensional Minkowski space with metric ds2 = — dt2 + dx2 + dy2 (the full Einstein static universe can be imbedded as the cylinder x2 + y2 + z* + w2 = 1 in a five-dimensional Euclidean space with metric ds2 = — d<2 + da;2 + dy2 + dz2 + dw;2, cf. Robertson (1933)).

One therefore has the situation: the whole of Minkowski space-time is conformal to the region (5.6) of the Einstein static universe, that is, to the shaded area in figure 14. The boundary of this region may therefore be thought of as representing the conformal structure of infinity of Minkowski space-time. It consists of the null surfaces p = (labelled J+) and q = -\n (labelled J~) together with points p = £77, q = \n (labelled i+), p = \n, q = - \n (labelled i°) and p = -g = —\n (labelled Any future-directed timelike geodesic in

Figure 14. The Einstein static universe represented by an imbedded cylinder; the coordinates 6, tf> have been suppressed. Each point represents one half of a two-sphere of area 4ffsin*r'. The shaded region is conformal to the whole of Minkowski space-time; its boundary(part of the null cones of i" and i~)may be regarded as the conformal infinity of Minkowski space-time.

Figure 14. The Einstein static universe represented by an imbedded cylinder; the coordinates 6, tf> have been suppressed. Each point represents one half of a two-sphere of area 4ffsin*r'. The shaded region is conformal to the whole of Minkowski space-time; its boundary(part of the null cones of i" and i~)may be regarded as the conformal infinity of Minkowski space-time.

Minkowski space approaches i+ (i~) for indefinitely large positive (negative) values of its affine parameter, so one can regard any timelike geodesic as originating at and finishing at i+ (cf. figure 15(¿)). Similarly one can regard null geodesies as originating at J" and ending atwhile spacelike geodesies both originate and end at i°. Thus one may regard i+ and i~ as representing future and past timelike infinity, and as representing future and past null infinity, and i° as representing spacelike infinity. (However non-geodesic curves do not obey these rules; e.g. non-geodesic timelike curves may start on and end on •/+.) Since any Cauchy surface intersects all timelike and null geodesies, it is clear that it will appear as a cross-section of the space everywhere reaching the boundary at i°.

Spacelike geodesic

Timelike _ geodesies

Null'

Figure 16

(i) The shaded region of figure 14, with only one coordinate suppressed, representing Minkowski space-time and its conformal infinity.

(ii) The Penrose diagram of Minkowski space-time; each point represents a two-sphere, except for i+, i" and each of which is a single point, and points on the line r = 0 (where the polar coordinates are singular).

i° (regard as one point) Surface

Surfaces [t = constant}

i° (regard as one point) Surface

^Two-spheres {r = constant}

^Two-spheres {r = constant}

Figure 16

(i) The shaded region of figure 14, with only one coordinate suppressed, representing Minkowski space-time and its conformal infinity.

(ii) The Penrose diagram of Minkowski space-time; each point represents a two-sphere, except for i+, i" and each of which is a single point, and points on the line r = 0 (where the polar coordinates are singular).

One can also represent the conformal structure of infinity by drawing a diagram of the (t',r') plane, see figure 15 (ii). As in figure 12 (ii), each point of this diagram represents a sphere S2, and radial null geodesies are represented by straight lines at + 45°. In fact, the structure of infinity in any spherically symmetric space-time can be represented by a diagram of this sort, which we shall call a Penrose diagram. On such diagrams we shall represent infinity by single lines, the origin of polar coordinates by dotted lines, and irremovable singularities of the metric by double lines.

The conformal structure of Minkowski space we have described is what one would regard as the 'normal' behaviour of a space-time at infinity; we shall encounter different types of behaviour in later sections.

Finally, we mention that one can obtain spaces locally identical to (^.rj) but with different (large scale) topological properties by identi-

fying points in Jl which are equivalent under a discrete isometry without a fixed point (e.g. identifying the point (x1, x2, x3, x*) with the point (x1,xi,x3,x? + c), where c is a constant, changes the topological structure from R* to Ra x S1, and introduces closed timelike lines into the space-time). Clearly, (-Ji, rj) is the universal covering space for all such derived spaces, which have been studied in detail by Auslander and Markus (1958).

5.2 De Sitter and anti-de Sitter space-times The space-time metrics of constant curvature are locally characterized by the condition Rabcd = ^R(gac9bd-9ad9bc)- This equation is equivalent to Cabcd = 0 = Rab — \Rgab, thus the Riemann tensor is determined by the Ricci scalar R alone. It follows at once from the contracted Bianchi identities that R is constant throughout space-time; in fact these space-times are homogeneous. The Einstein tensor is

One can therefore regard these spaces as solutions of the field equations for an empty space with A = \R, or for a perfect fluid with a constant density R/32n and a constant pressure — Rj32n. However the latter choice does not seem reasonable, as in this case one cannot have both the density and the pressure positive; in addition, the equation of motion (3.10) is indeterminate for such a fluid.

The space of constant curvature with R = 0 is Minkowski spacetime. The space for R > 0 is de Sitter space-time, which has the topology R1 x S3 (see Schrödinger (1956) for an interesting account of this space). It is easiest visualized as the hyperboloid

in flat five-dimensional space ii5 with metric

(see figure 16). One can introduce coordinates (t, 6, <f>) on the hyperboloid by the relations a sinh (arH) = v, a cosh (arH) cos X — w, a. cosh (arH) sin % cos 6 — x, a. cosh (arH) sin x sin 6 cos <p = y, a cosh (arH) sin x sin 6 sin <p = z.

t increases

I = 0; minimum distance between geodesic normals X increases

X increases X = 0

are boundaries of coordinate patch £ increases x = 0

Geodesic normals

Surfaces of constant time t

X increases X = 0

Geodesic normals

Surfaces of constant time t

are boundaries of coordinate patch £ increases x = 0

Surfaces of constant time t

Timelike geodesic which does not cross surfaces {t = constant}

Surfaces of constant time t

Timelike geodesic which does not cross surfaces {t = constant}

Figure 16. De Sitter space-time represented by a hyperboloid imbedded in a five-dimensional flat space (two dimensions are suppressed in the figure).

(i) Coordinates (t, x< 8, <j>) cover the whole hyperboloid; the sections {t = constant} are surfaces of curvature A; = +1.

(ii) Coordinates (£, y,z) cover half the hyperboloid; the surfaces {i = constant} are flat three-spaces, their geodesic normals diverging from a pcint in the infinite past.

In these coordinates, the metric has the form ds2 = - di2 + a2. cosh2 [a~H). {d/ + sin2 x(d<?2 + sin2 8 d{42)}.

The singularities in the metric at x = 0, x — 71 and at 8 = 0, 8 = n, are simply those that occur with polar coordinates. Apart from these trivial singularities, the coordinates cover the whole space for — oo < < < oo, 0 ^ x ^ n> 27r. The spatial sections of constant t are spheres S3 of constant positive curvature and are Cauchy surfaces. Their geodesic normals are lines which contract monotonically to a minimum spatial separation and then re-expand to infinity (see figure 16 (i)).

One can also introduce coordinates f , w + v . ax . ay . az t = a log-, £ =-, Q = —2 =-

a w + v w + v w + v on the hyperboloid. In these coordinates, the metric takes the form ds2 = - di2 + exp (2a~lt) (d£2 + d£2 + dS2).

However these coordinates cover only half the hyperboloid as i is not defined for w + v < 0 (see figure 16 (ii)).

The region of de Sitter space for which v + w> 0 forms the spacetime for the steady state model of the universe proposed by Bondi and Gold (1948) and Hoyle (1948). In this model, the matter is supposed to move along the geodesic normals to the surfaces {t = constant}. As the matter moves further apart, it is assumed that more matter is continuously created to maintain the density at a constant value. Bondi and Gold did not seek to provide field equations for this model, but Pirani (1955), and Hoyle and Narlikar (1964) have pointed out that the metric can be considered as a solution of the Einstein equations (with A = 0) if in addition to the ordinary matter one introduces a scalar field of negative energy density. This 'C'-field would also be responsible for the continual creation of matter.

The steady state theory has the advantage of making simple and definite predictions. However from our point of view there are two unsatisfactory features. The first is the existence of negative energy, which was discussed in § 4.3. The other is the fact that the space-time is extendible, being only half of de Sitter space. Despite these aesthetic objections, the real test of the steady state theory is whether its predictions agree with observations or not. At the moment it seems that they do not, though the observations are not yet quite conclusive.

de Sitter space is geodesically complete; however, there are points in the space which cannot be joined to each other by any geodesic. This is in contrast to spaces with a positive definite metric, when geodesic completeness guarantees that any two points of a space can be joined by at least one geodesic. The half of de Sitter space which represents the steady state universe is not complete in the past (there are geodesies which are complete in the full space, and cross the boundary of the steady state region; they are therefore incomplete in that region).

To study infinity in de Sitter space-time, we define a time coordinate where ds2 is given by (5.7) on identifying r' = Thus the de Sitter space is conformal to that part of the Einstein static universe defined by (5.8) (see figure 17 (i)). The Penrose diagram of de Sitter space is accordingly as in figure 17 (ii). One half of this figure gives the Penrose

*'by t' = 2 arc tan (exp arx1) — r, — \n<t'< \ir. ds2 = a2 cosh2 {arH'). ds2, where Then

singularity) singularity (x = 0)

(i) De Sitter space-time is conformal to the region — in < t' < Jw of the Einstein static universe. The steady state universe is conformal to the shaded region.

(n) The Penrose diagram of de Sitter space-time.

(iii) The Penrose diagram of the steady state universe.

In (ii), (iii) each point represents a two-sphere of area 2ir Bin* null lines are at 46°. x = 0 and X~ 77 are identified.

singularity) singularity (x = 0)

(i) De Sitter space-time is conformal to the region — in < t' < Jw of the Einstein static universe. The steady state universe is conformal to the shaded region.

(n) The Penrose diagram of de Sitter space-time.

(iii) The Penrose diagram of the steady state universe.

In (ii), (iii) each point represents a two-sphere of area 2ir Bin* null lines are at 46°. x = 0 and X~ 77 are identified.

diagram of the half of de Sitter space-time which constitutes the steady state universe (figure 17 (iii)).

One sees that de Sitter space has, in contrast to Minkowski space, a spacelike infinity for timelike and null lines, both in the future and the past. This difference corresponds to the existence in de Sitter space-time of both particle and event horizons for geodesic families of observers.

In de Sitter space, consider a family of particles whose histories are timelike geodesies; these must originate at the spacelike infinity and end at the spacelike infinity •/+. Let p be some event on the world-

Particle has been

Particle has been

O'b world-line

O'b world-line

Figure 18

(i) The particle horizon defined by a congruence of geodesic curves when past null infinity is spacelike.

(ii) Lack of suoh a horizon is null.

Figure 18

(i) The particle horizon defined by a congruence of geodesic curves when past null infinity is spacelike.

(ii) Lack of suoh a horizon is null.

line of a particle 0 in this family, i.e. some time in its history (proper time measured along 0's world-line). The past null cone of p is the set of events in space-time which can be observed by 0 at that time. The world-lines of some other particles may intersect this null cone; these particles are visible to 0. However, there can exist particles whose world-lines do not intersect this null cone, and so are not yet visible to 0. At a later time 0 can observe more particles, but there still exist particles not visible to 0 at that time. We say that the division of particles into those seen by 0 at p and those not seen by 0 at p, is the particle horizon for the observer 0 at the event p\ it represents the history of those particles lying at the limits of O's vision. Note that it is determined only when the world-lines of all the particles in the family are known. If some particle lies on the horizon, then the event p is the event at which the particle's creation light cone intersects 0's world-line. In Minkowski space, on the other hand, all the other particles are visible at any event p on O'b world-line if they move on timelike geodesies. As long as one considers only families of geodesic observers, one may think of the existence of the particle horizon as a consequence of past null infinity being spacelike (see figure 18).

All events outside the past null cone of p are events which are not, and never have been, observable by 0 up to the time represented by the event p. There is a limit to 0's world-line on In de Sitter spacetime, the past null cone of this point (obtained by a limiting process in the actual space-time, or directly from the conformal space-time) is a boundary between events which will at some time be observable by 0, and those that will never be observable by 0. We call this surface the future event horizon of the world-line. It is the boundary of the past of the world-line. In Minkowski space-time, on the other hand, the limiting null cone of any geodesic observer includes the whole of space-time, so there are no events which a geodesic observer will never be able to see. However if an observer moves with uniform acceleration his world-line may have a future event horizon. One may think of the existence of a future event horizon for a geodesic observer as being a consequence of J+ being spacelike (see figure 19).

Consider the event horizon for the observer 0 in de Sitter space-time and suppose that at some proper time (event p) on his world-line, his light cone intersects the world-line of the particle Q. Then Q is always visible to 0 at times after p. However there is on Q's world-line an event r which lies on 0's future event horizon; 0 can never see later events on Q's world-line than r. Moreover an infinite proper time elapses on O'b world-line from any given point till he observes r, but a finite proper time elapses along Q's world-line from any given event to r, which is a perfectly ordinary event on his world-line. Thus 0 sees a finite part of Q's history in an infinite time; expressed more physically, as 0 observes Q he sees a redshift which approaches infinity as 0 observes points on Q's world-line which approach r. Correspondingly, Q never sees beyond some point on O's world-line, and sees nearby points on O's world-line only with a very large redshift.

At any point on O's world-line, the future null cone is the boundary of the set of events in space-time which 0 can influence at and after that time. To obtain the maximal set of events in space-time that 0 could at any time influence, we take the future light cone of the limit

O'b world-line

KvmLo wlilflll' will never be seen by O

O't future.

event horizon

O'b world-line

KvmLo wlilflll' will never be seen by O

O't future.

event horizon

Q's world-line O's past null cone at p

O'a future null cone at p

O't past event horizon ('creation light cone')

Events O will never be able to influence

Q's world-line O's past null cone at p

Past light cone of O at p; eventually includes all space-time

Non-geodesic observer M's world-line

Future event horizon for R

Figure 19

(i) The future event horizon for a particle O which exists when futuro infinity •/+ is spacelike; also the past event horizon which exists when past infinity is spacelike.

(ii) If future infinity consists of a null •/+ and there is no future event horizon for a geodesic observer O. However an accelerating observer R may have a future event horizon.

Geodesic observer O's world-line

Non-geodesic observer M's world-line

Future event horizon for R

Figure 19

(i) The future event horizon for a particle O which exists when futuro infinity •/+ is spacelike; also the past event horizon which exists when past infinity is spacelike.

(ii) If future infinity consists of a null •/+ and there is no future event horizon for a geodesic observer O. However an accelerating observer R may have a future event horizon.

point of O'b world-line on past infinity that is, we take the boundary of the future of the world-line (which can be regarded as O's creation light cone). This has a non-trivial existence for a geodesic observer only if the past infinity is spacelike (and is in fact then O's past event horizon). It is clear from the above discussion that in the steady state universe, which has a null past infinity for timelike and null geodesies and a spacelike future infinity, any fundamental observer has a future event horizon but no past particle horizon.

One can obtain other spaces which are locally equivalent to the de Sitter space, by identifying points in de Sitter space. The simplest such identification is to identify antipodal points p, p' (see figure 16) on the hyperboloid. The resulting space is not time orientable; if time increases in the direction of the arrow at p, the antipodal identification implies it must increase in the direction of the arrow at p', but one cannot continuously extend this identification of future and past half null cones over the whole hyperboloid. Calabi and Markus (1962) have studied in detail the spaces resulting from such identifications; they show in particular that an arbitrary point in the resulting space can be joined to any other point by a geodesic if and only if it is not time orientable.

The space of constant curvature with R < 0 is called anti-de Sitter space. It has the topology S1xiP, and can be represented as the hyperboloid -u*-v* + x* + y* + z* = 1

in the flat five-dimensional space ii6 with metric dsa = - (dw)a - (dt>)2 + (da;)a + (dj/)a + (dz)a.

There are closed timelike lines in this space; however it is not simply connected, and if one unwraps the circle S1 (to obtain its covering space R1) one obtains the universal covering space of anti-de Sitter space which does not contain any closed timelike lines. This has the topology of R*. We shall in future mean by 'anti-de Sitter space', this universal covering space.

It can be represented by the metric dsa = -dfa + cosai{dxa + sinhax(d<?a + sina0d0a)}. (5.9)

This coordinate system covers only part of the space, and has apparent singularities at t = ± \n. The whole space can be covered by coordinates {<', r, 6,0} for which the metric has the static form dsa = - cosh2 r dr2 + dr2 + sinh2 r(d^ + sin2 <9 d^2).

In this form, the space is covered by the surfaces {<' = constant) which have non-geodesic normals.

To study the structure at infinity, define the coordinate r' by r' = 2 arctan (exp r) — \n, 0 < r' <

Then one finds dsa = cosh8rdaz, where ds2 is given by (5.7); that is, the whole of anti-de Sitter space is conformalto the region 0 < r' < \tt of the Einstein static cylinder. The Penrose diagram is shown in figure 20; null and spacelike infinity can be thought of as a timelike surface in this case. This surface has the topology R1 x /S2.

Xyz Solution Space

mmm mmm

LineB

Surfaces ~~ {f = constant} {t=+ co} Surfaces {t = constant} {t eo}

Lines

= constant} Null geodesies from infinity to r

Timelike geodesies from p

Null geodesic from p

Timelike geodesies from p

Null geodesic from p

Figtjbe 20

(i) Universal anti-de Sitter space is conformal to one half of the Einstein static universe. While coordinates («', r, 0, <f>) cover the whole space, coordinates (t, X' 4>) cover only one diamond-shaped region as shown. .The geodesies orthogonal to the surfaces {t = constant} all converge at p and q, and then diverge out into similar diamond-shaped regions.

(ii) The Penrose diagram of universal anti-de Sitter space. Infinity consists of the timelike surface J and the disjoint points i+, The projection of some timelike and null geodesies is shown.

One cannot find a conformal transformation which makes timelike infinity finite without pinching off the Einstein static universe to a point (if a conformal transformation makes the time coordinate finite it also scales the space sections by an infinite factor), so we represent timelike infinity by the disjoint points i+, i~.

The lines {x, 6, (j> constant} are the geodesies orthogonal to the surfaces {t = constant}; they all converge to points q (respectively, p) in the future (respectively, past) of the surface, and this convergence is the reason for the apparent (coordinate) singularities in the original metric form. The region covered by these coordinates is the region between the surface t = 0 and the null surfaces on which these normals become degenerate.

The space has two further interesting properties. First, as a consequence of the timelike infinity, there exists no Cauchy surface whatever in the space. While one can find families of spacelike surfaces (such as the surfaces {<' = constant}) which cover the space completely, each surface being a complete cross-section of the spacetime, one can find null geodesies which never intersect any given surface in the family. Given initial data on any such surface, one cannot predict beyond the Cauchy development of the surface; thus from the surface {t = 0}, one can predict only in the region covered by the coordinates t, x, 6, <j>. Any attempt to predict beyond this region is prevented by fresh information coming in from the timelike infinity.

Secondly, corresponding to the fact that the geodesic normals from t = 0 all converge at p and q, all the past timelike geodesies from p expand out (normal to the surfaces {t = constant}) and reconverge at q. In fact, all the timelike geodesies from any point in this space (to either the past or future) reconverge to an image point, diverging again from this image point to refocus at a second image point, and so on. The future timelike geodesies from p therefore never reach S, in contrast to the future null geodesies which goto./ from p and form the boundary of the future of p. This separation of timelike and null geodesies results in the existence of regions in the future of p (i.e. which can be reached from p by a future-directed timelike line) which cannot be reached from p by any geodesic. The set of points which can be reached by future-directed timelike lines from p is the set of points lying beyond the future null cone of p; the set of points which can be reached from p by future-directed timelike geodesies is the interior of the infinite chain of diamond-shaped regions similar to that covered by coordinates (t,x,d,<j>). One notes that all points in the Cauchy development of the surface t = 0 can be reached from this surface by a unique geodesic normal to this surface, but that a general point outside this Cauchy development cannot be reached by any geodesic normal to the surface.

5.3 Robertson-Walker spaces

So far, we have not considered the relation of exact solutions to the physical universe. Following Einstein, we can ask: can one find spacetimes which are exact solutions for some suitable form of matter and which give a good representation of the large scale properties of the observable universe? If so, we can claim to have a reasonable 'cosmo-logical model' or model of the physical universe.

However we are not able to make cosmological models without some admixture of ideology. In the earliest cosmologies, man placed himself in a commanding position at the centre of the universe. Since the time of Copernicus we have been steadily demoted to a medium sized planet going round a medium sized star on the outer edge of a fairly average galaxy, which is itself simply one of a local group of galaxies. Indeed we are now so democratic that we would not claim that our position in space is specially distinguished in any way. We shall, following Bondi (1960), call this assumption the Copernican principle.

A reasonable interpretation of this somewhat vague principle is to understand it as implying that, when viewed on a suitable scale, the universe is approximately spatially homogeneous.

By spatially homogeneous, we mean there is a group of isometries which acts freely on and whose surfaces of transitivity are spacelike three-surfaces; in other words, any point on one of these surfaces is equivalent to any other point on the same surface. Of course, the universe is not exactly spatially homogeneous; there are local irregularities, such as stars and galaxies. Nevertheless it might seem reasonable to suppose that the universe is spatially homogeneous on a large enough scale.

While one can build mathematical models fulfilling this requirement of homogeneity (see next section), it is difficult to test homogeneity directly by observation, as there is no simple way of measuring the separation between us and distant objects. This difficulty is eased by the fact that we can, in principle, fairly easily observe isotropics in extragalactic observations (i.e. we can see if these observations are the same in different directions, or not), and isotropies are closely con nected with homogeneity. Those observational investigations of iso-tropy which have been carried out so far support the conclusion that the universe is approximately spherically symmetric about us.

In particular, it has been shown that extragalactic radio sources are distributed approximately isotropically, and that the recently observed microwave background radiation, where it has been examined, is very highly isotropic (see chapter 10 for further discussion).

It is possible to write down and examine the metrics of all spacetimes which are spherically symmetric; particular examples are the Schwarzschild and Reissner-Nordstrom solutions (see § 5.5); however these are asymptotically flat spaces. In general, there can exist at most two points in a spherically symmetric space from which the space looks spherically symmetric. While these may serve as models of space-time near a massive body, they can only be models of the universe consistent with the isotropy of our observations if we are located near a very special position. The exceptional cases are those in which the universe is isotropic about every point in space time; so we shall interpret the Copernican principle as stating that the universe is approximately spherically symmetric about every point (since it is approximately spherically symmetric about us).

As has been shown by Walker (1944), exact spherical symmetry about every point would imply that the universe is spatially homogeneous and admits a six-parameter group of isometries whose surfaces of transitivity are spacelike three-surfaces of constant curvature. Such a space is called a Robertson-Walker (or Friedmann) space (Minkowski space, de Sitter space and anti-de Sitter space are all special cases of the general Robertson-Walker spaces). Our conclusion, then, is that these spaces are a good approximation to the large scale geometry of space-time in the region that we can observe.

In the Robertson-Walker spaces, one can choose coordinates so that the metric has the form d«2 = —dts + Ss(t)do"2, where dor2 is the metric of a three-space of constant curvature and is independent of time. The geometry of these three-spaces is qualitatively different according to whether they are three-spaces of constant positive, negative or zero curvature; by rescaling the function S, one can normalize this curvature K to be +1 or — 1 in the first two cases. Then the metric do-2 can be written do-2 = dx*+fHx) (d02+sin20d02), where

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