A

integral curves of this vector field are curves in Jr which are mapped by the geodesic normals to one point in the surface s = s0. Thus this surface will be at most two-dimensional. As the geodesies lie in the Cauchy development of \$ for |«| < |«0|, the surface s = s0 will lie in the Cauchy development or on the boundary of the Cauchy development of Jv. By condition (1), the energy-momentum tensor has a unique timelike eigenvector at each point. These eigenvectors will form a C1 timelike vector field whose integral curves may be thought-of as representing the flow lines of the matter. As the surface s = s0 lies in the Cauchy development of J?3 or on its boundary, all the flow lines that pass through it must intersect Jf. But then as Jf is homogeneous, all the flow linek\that pass through & must pass through s = s0. Thus the flow lines define a diffeomorphism between JP and the surface

8 = s0. This is impossible, as if is three-dimensional and s = s0 is two-dimensional. □

In fact, if all the flow lines were to pass through a two-dimensional surface, one would expect the matter density to become infinite. We have now seen that a large scale rotation or acceleration cannot, by itself, prevent the occurrence of singularities in a universe model obeying the strict Copernican principle. In later theorems we shall see that irregularities are in general also unable to prevent the occurrence of singularities in world models.

5.5 The Schwarzschild and Reissner-Nordstrdm solutions While the spatially homogeneous solutions may be good models for the large scale distribution of matter in the universe, they are inadequate for describing, for example, the local geometry of space-time in the solar system. One can describe this geometry to a good approximation by the Schwarzschild solution, which represents the spherically symmetric empty space-time outside a spherically symmetric massive body. In fact, all the experiments which have so far been carried out to test the difference between the General Theory of Relativity and Newtonian theory are based on predictions by this solution.

The metric can be given in the form

where r > 2m. It can be seen that this space-time is static, i.e. 8/8t is a timelike Killing vector which is a gradient, and is spherically symmetric, i.e. is invariant under the group of isometries S0(3) operating on the spacelike two-spheres {t, r constant} (cf. appendix B). The coordinate r in this metric form is intrinsically defined by the requirement that 477T2 is the area of these surfaces of transitivity. The solution is asymptotically flat as the metric has the form gab = i)ab + 0(1/r) for large r. Comparison with Newtonian theory (cf. § 3.4) shows that m should be regarded as the gravitational mass, as measured from infinity, of the body producing the field. It should be emphasized that this solution is unique: if any solution of the vacuum field equations is spherically symmetric, it is locally isometric to the Schwarzschild solution (although it may of course look totally different if it is given in some other coordinate system; see appendix B and Bergmann, Cahen and Komar (1965)).

Normally one would regard the Schwarzschild metric for r greater than some value r0 > 2m as being the solution outside some spherical body, the metric inside the body (r < r0) having a different form determined by the energy-momentum tensor of the matter in the body. However it is interesting to see what happens when the metric is regarded as an empty space solution for all values of r.

The metric is then singular when r = 0 and when r = 2m (there are also the trivial singularities of polar coordinates when 6=0 and 6 = n). One must therefore cut r = 0 and r = 2m out of the manifold defined by the coordinates (i, r, 6, since in § 3.1 we took space-time to be represented by a manifold with a Lorentz metric. Cutting out the surface r = 2m divides the manifold into two disconnected components for which 0 < r < 2m and 2m < r < oo. Since we took the space-time manifold to be connected, we must consider only one of these components and the obvious one to choose is the one for r > 2m, which represents the external field. One must then ask whether this manifold J( with the Schwarzschild metric g is extendible, i.e. whether there is a larger manifold into which can be imbedded and a suitably differentiable Lorentz metric g' on which coincides with g on the image of J(. The obvious place where <J( might be extended is where r tends to 2m. A calculation shows that although the metric is singular at r = 2m in the Schwarzschild coordinates (t, r, 6, <p), no scalar polynomials of the curvature tensor and the metric diverge as r 2m. This suggests that the singularity at r = 2m is not a real physical singularity, but rather one which is a result of a bad choice of coordinates.

To confirm this, and to show that (^tf, g) can be extended, define is a retarded null coordinate. Using coordinates (v, r, 6, the metric takes the Eddington-Finkelstein form g' given by

The manifold JC is the region 2m < r < oo, but the metric (5.22) is non-singular and indeed analytic on the larger manifold for which v s t + r*

ds2 = — i 1 — dv2 + 2dv dr + r2(d<92 + sin26dç£2). (5.22)

0 < r < oo. The region of (J?', g') for which 0 < r < 2m is in fact isometric to the region of the Schwarzschild metric for which 0 < r < 2m. Thus by using different coordinates, i.e. by taking a different manifold, we have extended the Schwarzschild metric so that it is no longer singular at r = 2m. In the manifold JC' the surface r = 2m is a null surface, as can be seen from the Finkelstein diagram (figure 23). This is a section (0,0 constant) of the space-time; each point represents a two-sphere of area 4m2. Some null cones and radial null geodesies are indicated on this diagram. Surfaces {t = constant} are indicated; one sees that t becomes infinite on the surface r = 2m.

This representation of the Schwarzschild solution has the odd feature that it is not time symmetric. One might expect this from the cross term (dvdr)in (6.22); it is qualitatively clear from the Finkelstein diagram. The most obvious asymmetry is that the surface r — 2m acts as a one-way membrane, letting future-directed timelike and null curves cross only from the outside (r > 2m) to the inside (r < 2m). Any past-directed timelike or null curve in the outside region cannot cross into the inside region. No past-directed timelike or null curve within r = 2m can approach r = 0. However any future-directed timelike or null curve which crosses the surface r = 2m approaches r = 0 within a finite affine distance. As r 0, the scalar RabcdRabcd diverges as m2/r®. Therefore r = 0 is a real singularity; the pair (JC, g') cannot be extended in a C2 manner or in fact even in a C° manner across r = 0.

If one uses the coordinate w instead of v, the metric takes the form g" given by

This is analytic on the manifold JC defined by the coordinates (w, r, 6, <j>) for 0 < r < oo. Again the manifold is the region 2m < r < oo and the new region 0 < r < 2m is isometric to the region 0 < r < 2m of the Schwarzschild metric, but the isometry reverses the direction of time. In the manifold JC, the surface r = 2m is again a null surface which acts as a one-way membrane. However this time it acts in the other direction of time, letting only past-directed timelike or null curves cross from the outside (r > 2m) to the inside (r < 2m).

One can in fact make both extensions g') and (Ji", g") simultaneously; that is to say, there is a still larger manifold with metric g* into which both g') and {JC, g") can be isometrically imbedded, so that they coincide on the region r > 2m which is ds2 = - du? - 2 du> dr + r2(d02 + sin2 6 d02).

Figure 23. Section (6, <f>) constant of the Schwarzschild solution.

(i) Apparent singularity at r = 2m when coordinates (i, r) are used.

(ii) Finkelstein diagram obtained by using coordinates (v, r) (lines at 46° are lines of constant v). Surface r = 2m is a null surface on which t = co.

Figure 23. Section (6, <f>) constant of the Schwarzschild solution.

(i) Apparent singularity at r = 2m when coordinates (i, r) are used.

(ii) Finkelstein diagram obtained by using coordinates (v, r) (lines at 46° are lines of constant v). Surface r = 2m is a null surface on which t = co.

isometric to (Ui, g). A construction of this larger manifold has been given by Kruskal (1960). To obtain it, consider (UK, g) in the coordinates (v, w,6, <f>)\ then the metric takes the form ds2 = r*( d0® + sin8 6 d02), where r is determined by

This presents the two-space (6,<f> constant) in null conformally flat coordinates, as the space with metric ds2 = — dv Aw is flat. The most general coordinate transformation which leaves this two-space expressed in such conformally flat double null coordinates is v' = v'(v), w' = w'(w) where v' and w' are arbitrary C1 functions. The resulting metric is

To reduce this to a form corresponding to that obtained earlier for Minkowski space-time, define x' = \$(v'-w'), t' = l(v' + w').

The metric takes the final form ds2 = F2{t\ x') (- dt'8 + da;'8) + r*{t', x') (d0s 4- sin2 6 d<f>2). (5.23)

The choice of the functions v', w' determines the precise form of the metric. Kruskal's choice was v' = exp («/4m), w' = — exp (— w/4m). Then r is determined implicitly by the equation

and F is given by

On the manifold defined by the coordinates (<', x', 6, <f>) for (<')2~ (a;')2 < 2m, the functions r and F (defined by (5.24), (5.25)) are positive and analytic. Defining the metric g* by (5.23), the region I of (JK*, g*) defined by x' > |i'| is isometric to (ui'.g), the region of the Schwarzschild solution for which r > 2m. The region defined by x' > — t' (regions I and II in figure 24) is isometric to the advanced Finkelstein extension (JK', g'). Similarly the region defined by x' > t' (regions I and II' in figure 24) is isometric to the retarded Finkelstein extension (JK", g"). There is also a region I', defined by x' < —1«'|, r = constant <2m

—r = constant > 2m r = constant <2m r = constant > 2 m

= 2m r = constant > 2 m r = 0 = 0 (singularity)!

\ future singularityJ

\ future singularityJ

1 = constant t = constant

/ past singularity \

Figure 24. The maximal analytic Schwarzschild extension. The 6, <j> coordinates are suppressed ; null lines are at ± 46°. Surfaces {r = constant} are homogeneous.

(i) The Kruakal diagram, showing asymptotically flat regions I and I' and regions II, II' for which r < 2m.

(ii) Penrose diagram, showing conformai infinity as well as the two singularities.

Cauchy surface y

/ past singularity \

Figure 24. The maximal analytic Schwarzschild extension. The 6, <j> coordinates are suppressed ; null lines are at ± 46°. Surfaces {r = constant} are homogeneous.

(i) The Kruakal diagram, showing asymptotically flat regions I and I' and regions II, II' for which r < 2m.

(ii) Penrose diagram, showing conformai infinity as well as the two singularities.

which turns out tpJbe-again isometric with the exterior Schwarzschild solution (Jl, g)/This can be regarded as another asymptotically flat universe on the/other side of the Schwarzschild' throat(Consider the section t = 0. The two-spheres {r = constant} behave as in Euclidean space, for large r; however for small r, they have an area which decreases to the minimum value 16rrm2 and then increases again, as the two spheres expand into the other asymptotically flat three-space.) The regions I' and II are isometric with the advanced Finkelstein extension of region I', and similarly I' and II' are isometric with the retarded Finkelstein extension of I', as can be seen from figure 24. There are no timelike or null curves which go from region I to region I'. All future-directed timelike or null curves which cross the part of the surface r = 2m represented here by t' = \x'\ approach the singularity at t' = (2m + (x')2)i, where r = 0. Similarly past-directed timelike or null curves which cross t' = — |a:'| approach another singularity at t' = — (2m+(x')2)i, where again r = 0.

The Kruskal extension g*) is the unique analytic and locally inextendible extension of the Schwarzschild solution. One can construct the Penrose diagram of the Kruskal extension by defining new advanced and retarded null coordinates v" = arc tan (v'(2m)~i), w" = arc tan (w'(2m)~h)

for —n<v" + w"<n and — ^n < v" < \$n, — < w" <

(see figure 24 (ii)). This may be compared with the Penrose diagram for Minkowski space (figure 15 (ii)). One now has future, past and null infinities for each of the asymptotically flat regions I and I'. Unlike Minkowski space, the conformal metric is continuous but not differ-entiable at the points i°.

If we consider the future light cone of any point outside r = 2m, the radial outwards geodesic reaches infinity but the inwards one reaches the future singularity; if the point lies inside r = 2m, both these geodesies hit the singularity, and the entire future of the point is ended by the singularity. Thus the singularity may be avoided by any particle outside r = 2m (so it is not' universal' as it is in the RobertsonWalker spaces), but once a particle has fallen inside r = 2m (in region II) it cannot evade the singularity. This fact will turn out to be closely related to the following property: each point inside region II represents a two-sphere that is a closed trapped surface. This means the following: consider any two-sphere p (represented by a point in figure 24) and two two-spheres q, s. formed by photons emitted radially outwards, inwards at one instant from p. The area of q (which is given by Attt2) will be greater than the area of p, but the area of s will be less than the area of p, if all three lie in a region r > 2m. However if they all lie in the region II where r < 2m, then the areas of both q and s will be less than the area of p (in the figure, r decreases as one moves from the bottom to the top of region II). In that case, we say that p is a closed trapped surface. Each point inside region II' represents a time-reversed closed trapped surface (the existence of trapped surfaces is a necessary consequence of the fact that the surfaces r = constant are spacelike), and correspondingly all particles in region II' must have come from the singularity in the past. We shall see in chapter 8 that the existence of the singularities is closely related to the existence of the closed trapped surfaces.

The Reissner-Nordstrôm solution represents the space-time outside a spherically symmetric charged body carrying an electric charge (but with no spin or magnetic dipole, so this is not a good representation of the field outside an electron). The energy-momentum tensor is therefore that of the electromagnetic field in the space-time which results from the charge on the body. It is the unique spherically symmetric asymptotically flat solution of the Einstein-Maxwell equations and is locally rather similar to the Schwarzschild solution; there exist coordinates in which the metric has the form

(2»n e2\ 1 1m />2\_1 1 - -y + y di2 +11 —- + ;») dr2 + r2{d62 + sin2 6 dçS2),

where m represents the gravitational mass and c the electric charge of the body. This asymptotically flat solution would normally be regarded as the solution outside the body only, the interior being filled in with some other suitable metric; but it is again interesting to see what happens if we regard it as a solution for all r.

If e2 > m2 the metric is non-singular everywhere except for the irremovable singularity at r = 0; this may be thought of as the point charge which produces the field. If e2 < m2, the metric also has singularities at r+ and r_, where r± = m± (m2 —e2)i; it is regular in the regions defined by oo > r > r+, r+ > r > r_ and r_> r > 0 (if e2 = m2, only the first and third regions exist). As in the Schwarzschild case, these singularities may be removed by introducing suitable coordinates and extending the manifold to obtain a maximal analytic extension (Graves and Brill (1960), Carter (1966)). The major differences that arise are due to thé existence of two zeros in the factor in front of di2, rather than one as in the Schwarzschild case. In particular this implies that the first and third regions are both static, whereas the second region (when it exists) is spatially homogeneous but is not static.

To obtain the maximally extended manifold, we proceed in steps analogous to those in the Schwarzschild case. Defining the coordinate

r* = r + -—±—log(r-r+) —--log(r-r_) if e2 < m2,