Ergosphere

Singularity

Figure 30. The equatorial plane of a Kerr solution with m' > a*. The circles represent the position a short time later of flashes of light emitted by the points represented by heavy dots.

Ergosphere

Singularity

Event horizon

Figure 30. The equatorial plane of a Kerr solution with m' > a*. The circles represent the position a short time later of flashes of light emitted by the points represented by heavy dots.

It is therefore not the event horizon for In fact the event horizon is the surface r = r+ = m + (m2 — a2)i. Figure 30 shows why this is. It shows the equatorial plane 6 = \n\ each point in this figure represents an orbit of the Killing vector Ka, i.e. it is stationary with respect to The small circles represent the position a short time later of flashes of light emitted from the points represented by the heavy black dots. Outside the stationary limit the Killing vector Ka is timelike and so lies within the light cone. This means that the point in figure 30 representing the orbit of emission lies within the wavefront of the light.

On the stationary limit surface, Ka is null and so the point representing the orbit of emission lies on the wavefront. However the wave-front lies partly within and partly outside the stationary limit surface; it is therefore possible for a particle travelling along a timelike curve to escape to infinity from this surface. In the ergosphere between the stationary limit surface and r = r+, the Killing vector Ka is spacelike and so the point representing the orbit of emission lies outside the wavefront. In this region it is impossible for a particle moving on a timelike or null curve to travel along an orbit of the Killing vector and so to remain at rest with respect to infinity. However the positions of the wavefronts are such that the particles can still escape across the stationary limit surface and so out to infinity. On the surface r = r+, the Killing vector Ka is still spacelike. However the wavefront corresponding to a point on this surface lies entirely within the surface. This means that a particle travelling on a timelike curve from a point on or inside the surface cannot get outside the surface and so cannot get out to infinity. The surface r = r+ is therefore the event horizon for J+ and ¡6 a nun surface.

Although the Killing vector Ka is spacelike in the ergosphere, the magnitude KaRbK(aRb) of the Killing bivector K(aRb] is negative everywhere outside r — r+, except on the axis Ra = 0 where it vanishes. Therefore Ka and Ra span a timelike two-surface and so at each point outside r = r+ off the axis there is a linear combination of Ka and Ra which is timelike. In a sense, therefore, the solution in the ergosphere is locally stationary, although it is not stationary with respect to infinity. In fact there is no one linear combination of Ka and Ra which is timelike everywhere outside r = r+. The magnitude of the Killing bivector vanishes on r = r+, and is positive just inside this surface. On r = r+, both Ka and Ra are spacelike but there is a linear combination which is null everywhere on r = r+ (Carter (1969)).

The behaviour of the ergo sphere and the horizon we have discussed will play an important part in our discussion of black holes in § 9.2 and §9.3.

Just as the Reissner-Nordstrom solution can be thought of as a charged version of the Schwarzschild solution, so there is a family of charged Kerr solutions (Carter (1968a)). Their global properties are very similar to those of the uncharged Kerr solutions.

5.7 Gddel's universe

In 1949, Kurt Godel published a paper (Godel (1949)) which provided a considerable stimulus to investigation of exact solutions more complex than those examined so far. He gave an exact solution of Einstein's field equations in which the matter takes the form of a pressure-free perfect fluid (T^ = puaub where p is the matter density and ua the normalized four-velocity vector). The manifold is R* and the metric can be given in the form ds2 = — di2 + da:2 —Jexp (2(^2) ojx) dy2 + dz2-2 exp ((^2) cox) dt dy, where to > 0 is a constant; the field equations are satisfied if u = dj8x° (i.e. ua = S"0) and 4np = ^ =

The constant oj is in fact the magnitude of the vorticity of the flow vector ua.

This space-time has a five-dimensional group of isometries which is transitive, i.e. it is a completely homogeneous space-time. (An action of a group is transitive on Ui if it can map any point of Ut into any other point of .) The metric is the direct sum of the metric g, given by ds,2 = - dt2 + da;2 -1 exp (2(^2) ojx) dy* - 2 exp ((^2) ojx) dt dy on the manifold Ut, = F? defined by the coordinates (t,x,y), and the metric g2 given by ds22 = dz2

on the manifold = R1 defined by the coordinate z. In order to describe the properties of the solution it is sufficient to consider only il).

Defining new coordinates (<', r, <p) on by exp ((j 2) ojx) = cosh 2r + cos <j> sinh 2r, toy exp ((<J2) ojx) = sin <p sinh 2r, tan \(<j) + ojt- (jJ2) f) = exp (- 2 r) tan \4>, the metric gj takes the form dsj2 = 2&r2( - dt'2 + dr2 - (sinh4 r - sinh2 r) d^2 + 2(J2) sinh2 rd^di), where — oo<i<oo, 0^r<oo, and 0 < < 2tt, <f> = 0 being identified with <j> = 2tt\ the flow vector in these coordinates is u = (w/ (J2)) 8/81'. This form exhibits the rotational symmetry of the solution about the axis r = 0. By a different choice of coordinates the axis could be chosen to lie on any flow line of the matter.

(coordinate axis)

(coordinate axis)

Figure 31. G6del's universe with the irrelevant coordinate z suppressed. The space is rotationally symmetric about any point; the diagram represents correctly the rotational symmetry about the axis r — 0, and the time invariance. The light cone opens out and tips over as r increases (see line L) resulting in closed timelike curves. The diagram does not correctly represent the fact that all points are in fact equivalent.

Figure 31. G6del's universe with the irrelevant coordinate z suppressed. The space is rotationally symmetric about any point; the diagram represents correctly the rotational symmetry about the axis r — 0, and the time invariance. The light cone opens out and tips over as r increases (see line L) resulting in closed timelike curves. The diagram does not correctly represent the fact that all points are in fact equivalent.

The behaviour of gi) is illustrated in figure 31. The light cones on the axis r = 0 contain the direction 8\8t' (the vertical direction on the diagram) but not the horizontal directions 8\8r and 8[8<j). As one moves away from the axis, the light cones open out and tilt in the

^-direction so that at a radius r = log (1+^/2), 8/8<p is a null vector and the circle of this radius about the origin is a closed null curve. At greater values of r, 8/8<f> is a timelike vector and circles of constant r, t' are closed timelike curves. As gj) has a four-dimensional group of isometries which is transitive, there are closed timelike curves through every point of gj), and hence through every point of the Godel solution g). __

This suggests that the solution is not very physical. The existence of closed timelike curves in this solution implies that there are no imbedded three-dimensional surfaces without boundary in which are spacelike everywhere. For a closed timelike curve which crossed such a surface would cross it an odd number of times. This would mean that the curve could not be continuously deformed to zero, since a continuous deformation can change the number of crossings only by an even number. This would contradict the fact that is simply connected, being homeomorphic to -R4. The existence of closed timelike lines also shows that there can be no cosmic time coordinate t in which increases along every future-directed timelike or null curve.

The Godel solution is geodesically complete. The behaviour of the geodesies can be described in terms of the decomposition into gi) and g2). Since the metric g2 of is flat, the component of the geodesic tangent vector in is constant, i.e. the z-coordinate varies linearly with the affine parameter on the geodesic. It is sufficient therefore to describe the behaviour of geodesies in gj). The null geodesies from a pointy on the axis of coordinates (figure 31) diverge from the axis initially, reach a caustic at r = log (1 + (V2))» and then reconverge to a point p' on the axis. The behaviour of timelike geodesies is similar: they reach some maximum value of r less than log (1 + (V2)) and then recon verge to p'. A point q at a radius r greater than log (1 + (V2)) can be joined to p by a timelike curve but not by a timelike or null geodesic.

Further details of Godel's solution can be found in Godel (1949), Kundt (1956).

5-8 Taub-NUT space

In 1951, Taub discovered a spatially homogeneous empty space solution of Einstein's equations with topology RxS3 and metric given by

where

U(t) = — 1 + ^^ > m and I are positive constants.

Here 6, <j>, i¡r are Euler coordinates on Ss, so 0 ^ %jr ^ <ln, 0 < 6 < it, 0 ^ 4> < 2n. This metric is singular at t = t± =m± (m2 + l2)i, where [/ = 0. It can in fact be extended across these surfaces to give a space found hj Newman, Tamburino and Unti (1963), but before discussing the extension we shall consider a simple two-dimensional example given by Misner (1967) which has many similar properties.

This space has the topology Sl x R1 and the metric g given by ds2 = —t~lát2 + tái¡/z where 0 ^ íjr < 2n. This metric is singular when t = 0. However if one takes the manifold Jl defined by ijr and by 0 < t < oo, (JK, g) can be extended by defining i¡r' = ^ — log¿. The metric then takes the form g' given by ^ = + 2 d< +

This is analytic on the manifold with topology Sl x R1 defined by \¡r' and by — oo < t < oo. The region í > 0 of (Jl', g') is isometric with g). The behaviour of (JK', g') is shown in figure 32. There are closed timelike lines in the region t < 0, but there are none when t > 0. One family of null geodesies is represented by the vertical lines in figure 32; these cross the surface t = 0. The other family spiral round and round as they approach t = 0, but never actually cross this surface, and these geodesies have only finite affine length. Thus the extension (Jf', g') is not symmetric between the two families of null geodesies, although the original space (JÍ, g) was. However one can define another extension W, g") in which the behaviour of the two families of null geodesies is interchanged. To do so define t¡r" by i¡r" = rjr + logt. The metric takes the form g" given by ds2 = -2df"dt+t{df")2.

This is analytic on the manifold with topology S1 x R1 defined by ijf" and - oo < t < oo. The region í > 0 of (Jf", g") is isometric with (JÍ, g). In a sense, what we have done by defining i]r" is to untwist the second family of null geodesies so that they become vertical lines, and can be continued beyond t = 0. However this twisting winds up the first family of null geodesies so that they spiral around and cannot be continued beyond t = 0. One has therefore two inequivalent locally inextendible analytic extensions of g), both of which are geodesic-

Figure 32. Misner's two-dimensional example.

(i) Extension of region I across the boundary t = 0 into II. The vertical null geodesies are complete, but the twisted null geodesies are incomplete.

(ii) The universal covering space is two-dimensional Minkowski space. Under the discrete subgroup O of the Lorentz group, points s are equivalent; similarly points r, q and t are equivalent, (i) is obtained by identifying equivalent points in regions I and II.

Figure 32. Misner's two-dimensional example.

(i) Extension of region I across the boundary t = 0 into II. The vertical null geodesies are complete, but the twisted null geodesies are incomplete.

(ii) The universal covering space is two-dimensional Minkowski space. Under the discrete subgroup O of the Lorentz group, points s are equivalent; similarly points r, q and t are equivalent, (i) is obtained by identifying equivalent points in regions I and II.

ally incomplete. The relation between these two extensions can be seen clearly by going to the covering space of g).

This is in fact the region I of two-dimensional Minkowski space fj) contained within the future null cone of a point p (figure 32(H)). The isometries of (^tf, r)) which leave p fixed form a one-dimensional group (the Lorentz group of fj) whose orbits are the hyperbolae {cr = constant} where cr = I2 —x2 and I, x are the usual Minkowski coordinates. The space (J?, g) is the quotient of (I, fj) by the discrete subgroup G of the Lorentz group consisting of An (n integer) where A maps (I, x) to

(I cosh Tr + x sinh n, x cosh n + l sinh n), i.e. one identifies the points

{I cosh nn + x sinh nn, x cosh nn + l sinh nn) for all integer values of n, and these correspond to the point t = \(l2 -x2), f = 2 arc tanh (xft) in

The action of the isometry group G in the region I is properly discontinuous. The action of a group H on a manifold JT is said to be properly discontinuous if:

(1) each point qejV has a neighbourhood Ql such that A(%)C\%= 0 for each AeH which is not the identity element, and

(2) if q, rejV are such that there is no A e H with Aq = r, then there are neighbourhoods % and of q and r respectively such that there is no BeH with B{<%) n W * 0.

Condition (1) implies tlint. thn quotient .A^/H in n. manifold, and condition (2) implies that it is Hausdorff. Thus the quotient (I,fj)/(? is the Hausdorff space g). The action of G is also properly discontinuous in the regions I+II (? > —x). Thus (I + II,Tj)/Cr is also a Hausdorff space; in fact it is {J(\ g'). Similarly (I + III, r^jG is the Hausdorff space (ui"', g") where I + III is the region I > x. From this it can be seen how it is that one family of null geodesies can be completed in the extension g') while the other family can be completed in the extension g"). This suggests that one might perform both extensions at the same time. However the action of the group on the region (I + II + III) (i.e. I > —1£|) satisfies condition (1) but condition (2) is not satisfied for points q on the boundary between I and II and points r on the boundary between I and III. Therefore the quotient (I + II + III, r\)!G is not Hausdorff although it is still a manifold.

This kind of non-Hausdorff behaviour is different from that in the example given in §2.1. In that example, one could have continuous curves which bifurcate, one branch going into one region and another branch going into another region. Such behaviour of an observer's world-line would be very uncomfortable. However the manifold (I + II + III)/Cr does not have any such bifurcating curves; curves in I can be extended into II or III but not into both simultaneously. Thus one might be prepared to relax the Hausdorff requirement on a spacetime model to allow this sort of situation but not the sort in which one gets bifurcating curves. Further work on non-Hausdorff space-times can be found in the papers of Hajicek (1971).

Condition (1) is in fact satisfied by the action of G on UK — {p}. Thus the space (UK — [p], r¡)/G is in some sense the maximal non-Hausdorff extension of (UK, g). However it is still not geodesically complete because there are geodesies which pass through the point p which has been left out. If p is included the action of the group does not satisfy condition (1), and so the quotient UK ¡G is not even a non-Hausdorff manifold. However consider the bundle of linear frames L(UK), i.e. the collection of all pairs (X, Y), X, Y e Tg, of linearly independent vectors at all points q eUK. The action of an element A of the isometry group G on UK induces an action A m on L(UK) which takes the frame (X, Y) at q to the frame (A^X,A^Y) at A(q). This action satisfies condition (1) because even for (X, Y)eTp, A+X =f= X and A+Y =f= Y unless A = identity, and satisfies condition (2) even if X and Y lie on the null cone of p. Thus the quotient L(UK)fG is a Hausdorff manifold. It is a fibre bundle over the non-Hausdorff non-manifold UK¡G. One could in a sense regard it as the bundle of linear frames for this space. The fact that the bundle of frames can be well behaved even though the space is not, suggests that it is useful to look at singularities by using the bundle of linear frames. A general procedure for doing this will be given in § 8.3.

We shall now return to the four-dimensional Taub space (UK, g) where UfiaR1x S3 and g is given by (5.32). As UK is simply connected, one cannot take a covering space as we did in the two-dimensional example. However one can achieve a similar result by considering UK as a fibre bundle over S2 with fibre R1 x S1; the bundle projection n: UK-^-S1 is defined by (t, x¡/, 6, <f>)-*-(8, <j>). This is in fact the product with the i-axis of the Hopf fibering S3-*-S2 (Steenrod (1951)) which has fibre S1. The space (UK, g) admits a four-dimensional group of isometries whose surfaces of transitivity are the three-spheres

{i = constant}. This group of isometries maps fibres of the bundle 7r: JC-+&into fibres, and so the pairs , g) are all isometric, where 8? is a fibre (SP « R1 x S1) and g is the metric induced on the fibre by the four-dimensional metric g on Jl. The fibre & can be regarded as the plane, and the metric g on ¡F is obtained from (5.32) by dropping the terms in dd and d<j>\ thus g is given by

The tangent space Tg at the point q^JC can be decomposed into a vertical subspace Vg which is tangent to the fibre and is spanned by the vectors djdt and d/dft, and a horizontal subspace Hg which is spanned by the vectors 8/86 and 8/ckf> —cos dd/dijr. Any vector XeTe can be split into a part XF lying in Vg and a part XH lying in Hg. The metric g on Te can then be expressed as where gv = g and gH is the standard metric on the two-sphere given by ds2 = d#2 + sin2<?d$i2. Thus although the metric g is not the direct sum of gr and (<2+i2)g# (because R1 x S3 is not the direct product of jR1 x S1 with Sz) it can nevertheless be regarded as such a sum locally.

The interesting part of the metric g is contained in gF and we shall therefore consider analytic extensions of the pair gF). When combined with the metric gH of the two-sphere as in (5.34), these give analytic extensions of g).

The metric gF, given by (5.33), has singularities at t = i± where U = 0. However if one takes the manifold defined by i/r and by < t < t+, gF) can be extended by defining

The metric then takes the form gF' given by ds2 = Udf'(lU(t)df -dt).

This is analytic on the manifold ¡F' with topology S1xR defined by and by — oo < t < oo. The region < t < t+of(&r', gF') is isometric with gF). There are no closed timelike curves in the region < t < t+ but there are for t <t_ and for t > t+. The behaviour is very much as for the space g') we considered before, except that there are now two horizons (at t = t_ and t = t+) instead of the one horizon (at t = 0). One family of null geodesies crosses both horizons t = t_ and ds2 = - ¿7-1 di2 + 4i2 ¿7(d^)2.

t = t+ but the other family spirals round near these surfaces and is incomplete.

As before, one can make another extension by defining the coordinate . , r^-hiwr

The metric then takes the form gF" given by ds2 = 4f dijr"(W{t) dijr" + dt)

which is analytic on the manifold 3F" defined by and by — oo < t < oo, and is again isometric to (cF"0, gr) on t_ < t < t+.

Once again one can show the relation between the different extensions by going to the covering space. The covering space of is the manifold defined by the coordinates — oo < ft < oo and by t_ < t < t+. On the metric gr can be written in the double null form ds2 = 4f2i7(i) dijr' dijr", (5.35)

where —co<ijr'<oo,—oo<i/f"< oo. One can extend this in a manner similar to that used in the Reissner-Nordstrom solution. Define new coordinates (u+, v+) and {u_, v_) on by u± = arc tan (exp ^r'/a±)> v± — arc tan (— (exp — v^"/a±)).

n is some integer greater than (mi, + l2)l(mt_ -I I2). Thou ( ho mcLric g,, obtained by applying this transformation to (5.35) is analytic on the manifold & shown in figure 33, where the coordinates (u+,v+) are analytic coordinates except at t = t_ where they are at least C3, and the coordinates (u_, v_) are analytic coordinates except at t = t+ where they are at least C3. This is rather similar to the extension of the (t, r) plane of the Reissner-Nordstrom solution.

The space gv) has a one-dimensional group of isometries, the orbits of which are shown in figure 33. Near the points p+, p_ the action of this group is similar to that of the Lorentz group in two-dimensional Minkowski space (figure 32 (ii)). Let G be the discrete subgroup of the isometry group generated by a non-trivial element A of the isometry group. The space gF) is the quotient of one of the regions (II+, gF) by G. The space (3F', gr') is the quotient (I_ + II+ + III_, gV)\G, and g/) is the quotient

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