T T exp rr w

where n is an integer > 2(r+)2 (r_)~2. In these coordinates, the metric is analytic everywhere except at r = r, where it is degenerate. Ttio coordinates v'" and w'" are analytic functions of v" and w" for r 4= r+ or r_. Thus the manifold can be covered by an analytic atlas, consisting of local coordinate neighbourhoods defined by coordinates v" and w" for r+r, and by local coordinate neighbourhoods defined by if and w'" for r 4= r+. The metric is analytic in this atlas.

The case e2 = m2 can be extended similarly; the case e2 > m2 is already inextendible in the original coordinates. The Penrose diagrams of these two cases are given in figure 26.

In all these cases, the singularity is timelike. This means that, unlike in the Schwarzschild solution, timelike and null curves can always avoid hitting the singularities. In fact the singularities appear to be repulsive: no timelike geodesic hits them, though non-geodesic timelike curves and radial null geodesies can. The spaces are thus timelike (though not null) geodesically complete. The timelike character of the singularity also means that there are no Cauchy surfaces in these spaces: given any spacelike surface, one can find timelike or null curves which run into the singularity and do not cross the surface. For example in the case e2 < m2, one can find a spacelike surface which crosses two asymptotically flat regions I (figure 25). This is a Cauchy surface for the two regions I and the two neighbouring regions II. However in the neighbouring regions III to the future there are past-directed inextendible timelike and null curves which approach the singularity and do not cross the surface r = r_. This surface is therefore said to be the future Cauchy horizon for if. The continuation of the solution beyond r = r_ is not determined by the Cauchy data on if. The continuation we have given is the only locally inextendible analytic one, but there will be other non-analytic C® continuations which satisfy the Einstein-Maxwell equations.

(singularity)

(singularity)

Homogeneous surfacés

Homogeneous surfacés

Figure 26. Penrose diagrams for the maximally extended Reissner-Nordstr6m solutions:

In the first case there is an infinite chain or regions I (co > r > m) connected by regions III (m > r> 0). The points p are not part of the singularity at r = 0, but are really exceptional points at infinity.

A particle P crossing the surface r = r+ would appear to have infinite redshift to an observer 0 whose world-line remains outside r = r+ and approaches the future infinity i+ (figure 25). In the region II between r = r+ and r = r_, the surfaces of constant r are spacelike and so each point of the figure represents a two-sphere which is a closed trapped surface. An observer P crossing the surface r = r_ would see the whole of the history of one of the asymptotically flat regions I in a finite time. Objects in this region would therefore appear to be infinitely blue-shifted as they approached i+. This suggests that the surface r = r_ would be unstable against small perturbations in the initial data on the spacelike surface and that such perturbations would in general lead to singularities on r = r_.

5.6 The Kerr solution

In general, astronomical bodies are rotating and so one would not expect the solution outside them to be exactly spherically symmetric. The Kerr solutions are the only known family of exact solutions which could represent the stationary axisymmetric asymptotically flat field outside a rotating massive object. They will be the exterior solutions only for massive rotating bodies with a particular combination of multipole moments; bodies with different combinations of moments will have other exterior solutions. The Kerr solutions do however appear to be the only possible exterior solutions for black holes (see §9.2 and §9.3).

The solutions can be given in Boyer and Lindquist coordinates (r, 6, <j>, t) in which the metric takes the form ds2 = p2 + (r2 + a2) sin2 6 d02 - d<2 + — (a sin2 6 d0 - <fc)2, where p2{r, 6) = r2 + a2 cos2 6 and A(r) = r2-2mr + a2.

m and a are constants, m representing the mass and ma the angular momentum as measured from infinity (Boyer and Price (1965)); when o = 0 the solution reduces to the Schwarzschild solution. This metric form is clearly invariant under simultaneous inversion of t and <j>, i.e. under the transformation <-> — <, — <j>, although it is not invariant under inversion of t alone (except when o = 0). This is what one would expect, since time inversion of a rotating object produces an object rotating in the opposite direction.

2mr p2

When a2 > m2, A > 0 and the above metric is singular only when r = 0. The singularity at r = 0 is not in fact a point but a ring, as can be seen by transforming to Kerr-Schild coordinates (x, y, z, t), where x + iy = (r + ia) sin 0exp i J* (d<f> + aA-1 dr), z = rcostf, < = J (d< + (r2 + a2) A"1 dr) - r.

In these coordinates, the metric takes the form ds2 = cLr2 + dy2 +- dz2 — df2

2mr3 Mxdx + ydy) — a(xdy — ydx) zdz + rl + a2z2\ r^+a2 +~ +

where r is determined implicitly, up to a sign, in terms of x, y, z by r4 - [x2 + y2 + z2- a2) r2 - a2z2 = 0.

For r 4= 0, the surfaces {r = constant} are confocal ellipsoids in the (x, y, z) plane, which degenerate for r = 0 to the disc z2 + y2 < a2,2 = 0. The ring x* + yz = a2, z = 0 which is the boundary of this disc, is a real curvature singularity as the scalar polynomial RabcdRabcd diverges there. However no scalar polynomial diverges on the disc except at the boundary ring. The function r can in fact be analytically continued from positive to negative values through the interior of the disc x2+y2 < a2, z = 0, to obtain a maximal analytic extension of the solution.

To do this, one attaches another plane defined by coordinates (x',y',z') where a point on the top side of the disc xz + y* < a2, z = 0 in the (x, y, z) plane is identified with a point with the same x and y coordinates on the bottom side of the corresponding disc in the (x't y', z') plane. Similarly a point on the bottom side of the disc in the (x, y, z) plane is identified with a point on the top side of the disc in the (x', y\ z') plane (see figure 27). The metric (5.30) extends in the obvious way to this larger manifold. The metric on the (x't y', z') region is again of the form (5.29), but with negative rather than positive values of r. At large negative values of r, the space is again asymptotically flat but this time \^th negative mass. For small negative values of r near the ring singularity, the vector 8/8<p is timelike, so the circles (t = constant, r = constant, 6 = constant) are closed timelike curves. These closed timelike curves can be deformed to pass through any point of the extended space (Carter (1968a)). This solution is geodesic-

ally incomplete at the ring singularity. However the only timelike and null geodesies which reach this singularity are those in the equatorial plane on the positive r side (Carter (1968a)).

Svmmetrv

Symmetry Rxig

Svmmetrv

Symmetry Rxig

Figure 27. The maximal extension of the Kerr solution for a2 > m2 is obtained by identifying the top of the disc x1 + ys < a2, z — 0 in the (x,y,z) plane with the bottom of the corresponding disc in the (x', y', z') plane, and vice versa. The figure shows the sections y = 0, y' = 0 of these planes. On circling twice round the singularity at + = a*, z = 0 one passes from the (x, y, z) plane to the (x', y', z') plane (where r is negative) and back to the (x, y, z) plane (where r is positive).

Figure 27. The maximal extension of the Kerr solution for a2 > m2 is obtained by identifying the top of the disc x1 + ys < a2, z — 0 in the (x,y,z) plane with the bottom of the corresponding disc in the (x', y', z') plane, and vice versa. The figure shows the sections y = 0, y' = 0 of these planes. On circling twice round the singularity at + = a*, z = 0 one passes from the (x, y, z) plane to the (x', y', z') plane (where r is negative) and back to the (x, y, z) plane (where r is positive).

The extension in the case a2 < wi2 is rather more complicated, because of the existence of the two values = m + (m2 — a2)i and r_ = m — (m2—a2)i of r at which A(r) vanishes. These surfaces are similar to the surfaces r = r+, r = r_ in the Reissner-Nordstrom solution. To extend the. metric across these surfaces, one transforms to the Kerr coordinates (r,6,<j)+,u+), where d u+ = d t + (r2 + a2) A"1 dr, = + aA"1 dr. The metric then takes the form ds2 = p2 dd2 - 2a sin2 6 dr d0+ + 2 dr du+

+p-2[(r2 + a2)2 - Aa2 sin2 sin2 0 d<f>+2 - 4ap~hnr sin20d0+ dw+ - (1 - 2 mrp~2) dw+2 (5.31)

on the manifold defined by these coordinates, and is analytic at r = r+ and r = r_. One again has a singularity at r = 0, which has the same ring form and geodesic structure as that described above. The metric can also be extended on the manifold defined by the coordinates (r, 6, u_) where du_ = df - (r2+a2) A-1 dr, d$£_ = d0 - aA'1 dr;

the metric again takes the form (5.31), with <f>+, u+ replaced by — <f>_, — u_. The maximal analytic extension can be built up by a combination of these extensions, as in the Reissner-Nordstrom case (Boyer and Lindquist (1967), Carter (1968a)). The global structure is very similar to that of the Reissner-Nordstrom solution except that one can now continue through the ring to negative values of r. Figure 28 (i) shows the conformal structure of the solution along the symmetry axis. Tho regions 1 represent the asymptotically flat regions in which r > r+. The regions II (r_ < r < r+) contain closed trapped surfaces. The regions III ( — oo < r < r_) contain the ring singularity; there are closod timoliko curvos through every point in a region III, hut no causality violation occurs in the ot.hor two regions.

In the case a" «■ m", r+ and r_ coincide and there is no region II. The maximal extension is similar to that of the Reissner-Nordstrom solution when e2 = m%. The conformal structure along the symmetry axis In tills tmse 1» shown in ilgure 28 (11).

The Kerr solutions, being stationary and axisymmetric, have a two-parameter group of isometries. This group is necessarily Abelian (Carter (1970)). There are thus two independent Killing vector fields which commute. There is a unique linear combination Ka of these Killing vector fields which is timelike at arbitrarily large positive and negative values of r. There is another unique linear combination Ra of the Killing vector fields which is zero on the axis of symmetry. The orbits of the Killing vector Ka define the stationary frame, that is, an object moving along one of these orbits appears to be stationary with respect to infinity. The orbits of the Killing vector Ra are closed curves, and correspond to the rotational symmetry of the solution.

In the Schwarzschild and Reissner-Nordstrom solutions, the Killing vector Ka which is timelike at large values of r is timelike everywhere in the region I, becoming null on the surfaces r = 2m and r = r+ respectively. These surfaces are null. This means that a particle which crosses one of these surfaces in the future direction cannot return again to the same region. They are the boundary of the region

Figure 28. The conformal structure of the Kerr solutions along the axis of symmetry, (i) in the cn.ro 0 < a* < ms, (ii) in the caso o* = m*. The dottrel linns ttrtj 1Mit H ul i'iiitttl ttitI r; i.hu roglouc* 1,11 hi nl Hi in ituiti (I) are iliviiltid by / — r+

and r = r_, and the regions I and III in case (ii) by r — m. In both cases, the structure of the space near the ring singularity is as in figure 27.

Figure 28. The conformal structure of the Kerr solutions along the axis of symmetry, (i) in the cn.ro 0 < a* < ms, (ii) in the caso o* = m*. The dottrel linns ttrtj 1Mit H ul i'iiitttl ttitI r; i.hu roglouc* 1,11 hi nl Hi in ituiti (I) are iliviiltid by / — r+

and r = r_, and the regions I and III in case (ii) by r — m. In both cases, the structure of the space near the ring singularity is as in figure 27.

of the solution from which particles can escape to the infinity «/+ of a particular region I, and are called the event horizons of that (They are in fact the event horizon in the sense of § 5.2 for an observer moving on any of the orbits of the Killing vector Ka in the region I.)

In the Kerr solution on the other hand, the Killing vector Ka is spacelike in a region outside r = r+, called the ergosphere (figure 29). The outer boundary of this region is the surface r = m + (m2 — a2 cos2 6)1 on which Ka is null. This is called the stationary limit surface since it is the boundary of the region in which particles travelling on a timelike curve can travel on an orbit of the Killing vector Ka, and so remain at rest with respect to infinity. The stationary limit surface is a timelike surface except at the two points on the axis, where it is null (at these points it coincides with the surface r = r+). Where it is timelike it can be crossed by particles in either the ingoing or the outgoing direction.

Figure 29. In the Kerr solution with 0 < o* < ms, the ergosphere lies between the stationary limit surface and the horizon at r = r+. Particles can escape to infinity from region I (outside the event horizon r = r+) but not from region II (between r = r+ and r = r_) and region III (r < r_; this region contains the ring singularity).

Figure 29. In the Kerr solution with 0 < o* < ms, the ergosphere lies between the stationary limit surface and the horizon at r = r+. Particles can escape to infinity from region I (outside the event horizon r = r+) but not from region II (between r = r+ and r = r_) and region III (r < r_; this region contains the ring singularity).

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