= diag c2 0 0 ’ c2

T

such that there is a negative pressure (or tension) along the string, and the line

element is of the form

ds2 = c2 dt2 ’ dr 2 ’ B r d ’ dz2

2

where B r satisfies

B2

B

’ =’ c2

4B2

2B

Show further that b r = B r satisfies b = ’ c2 b.

Hint: You may find your answers to Exercises 8.9, 9.28 and 9.29 useful.

12.12 Suppose that the matter distribution in a cosmic string has a uniform density across

the string, such that

for r ¤ r0

0

r=

0 for r > r0

By demanding that g ’ ’r 2 as r ’ 0, so that the spacetime geometry is regular

on the axis of the string, show that the line element for r ¤ r0 is

2

sin r

ds = c dt ’ dr ’ ’ dz2

2 2 2 2 2

d

r

where = 2

0 c . By demanding that g and its derivative with respect to r

are both continuous at r = r0 , show that the line element for r > r0 is

2

sin r0

ds = c dt ’ dr ’ + r ’ r0 cos r0 ’ dz2

2 2 2 2 2

d

r

For the interesting case in which r0 1, show that for r r0 the line element

takes the form

8G

ds2 = c2 dt2 ’ dr 2 ’ 1 ’ ’ dz2

r2 d 2

c2

308 Further spherically symmetric geometries

where = r0 0 is the ˜mass per unit length™ of the string. Interpret this line

2

element physically.

12.13 Show that the electromagnetic field tensor outside a static spherically symmetric

charged matter distribution has the form

⎛ ⎞

0 ’1 0 0

⎜1 0 0 0⎟

⎜ ⎟

F =E r ⎜ ⎟

⎝0 0 0 0⎠

0 000

where E r is some arbitrary function. Hence show that, if the line element outside

the matter distribution has the form

ds2 = A r dt2 ’ B r dr 2 ’ r 2 d + sin2 d

2 2

the energy“momentum tensor of the electromagnetic field in this region is given by

1 r 2 E 2 r 2 E 2 sin2

12 1

= c 0 E diag ’

2

T

2 B A AB AB

12.14 Calculate the invariant curvature scalar R for the Reissner“Nordström

R

geometry and hence show that the only intrinsic singularity occurs at r = 0.

12.15 Show that the worldlines of radially moving photons in the Reissner“Nordström

geometry are given by

2

2

r+

r’ r r

ct = r ’ ’1 + ’ 1 + constant

ln ln outgoing

r + ’ r’ r+ ’ r ’

r’ r+

2

2

r+

r’ r r

ct = ’r + ’1 ’ ’ 1 + constant

ln ln ingoing

r+ ’ r ’ r+ ’ r’

r’ r+

12.16 Show that, by introducing the advanced Eddington“Finkelstein timelike coordinate

2

2

r+

r’ r r

ct = ct ’ ’1 + ’1

ln ln

r+ ’ r ’ r + ’ r’

r’ r+

the Reissner“Nordström line element takes the form

2

ds2 = c2 dt ’ 2 1 ’ dt dr ’ 2 ’ dr 2 ’ r 2 d + sin2 d

2 2

where ≡ r = 1 ’ 2 /r + q 2 /r 2 . Hence show that the worldlines of radially

moving photons in advanced Eddington“Finkelstein coordinates are given by

2’

dt

ct + r = constant =

incoming outgoing

c

dr

What is the significance, if any, of the fact that c dt /dr = 0 at r = 2 for

outgoing radially moving photons?

309

Exercises

12.17 For a particle of mass m and charge e in geodesic motion in the Reissner“

Nordström geometry, show that the quantity

q2

2 dt eq

k = m 1’ +2 +

r r d r

is conserved, and interpret this result physically.

12.18 An observer is in a circular orbit of coordinate radius r = R in the Reissner“

Nordström geometry. Find the components of the magnetic field measured by the

observer.

13

The Kerr geometry

The Schwarzschild solution describes the spacetime geometry outside a spheri-

cally symmetric massive object, characterised only by its mass M. In the previous

chapter we derived further spherically symmetric solutions. Most real astrophys-

ical objects, however, are rotating. In this case, a spherically symmetric solution

cannot apply because the rotation axis of the object defines a special direction, so

destroying the isotropy of the solution. For this reason, in general relativity it is not

possible to find a coordinate system that reduces the spacetime geometry outside

a rotating (uncharged) body to the Schwarzschild geometry. The non-linear field

equations couple the source to the exterior geometry. Moreover, a rotating body

is characterised not only by its mass M but also by its angular momentum J , and

so we would expect the corresponding spacetime metric to depend upon these

two parameters.

We now consider how to derive the metric describing the spacetime geometry

outside a rotating body. Since the mathematical complexity in this case is far

greater than that encountered in deriving the Schwarzschild metric (or the other

spherically symmetric geometries discussed in the previous chapter), we shall

content ourselves with just an outline of how the solution may be obtained.

13.1 The general stationary axisymmetric metric

In our derivation of the Schwarzschild solution, we began by constructing the

general form of the static isotropic metric. We are now interested in deriving the

spacetime geometry outside a steadily rotating massive body. Thus we begin by

constructing the general form of the stationary axisymmetric metric.

For the description of such a spacetime, it is convenient to introduce the

timelike coordinate t = x0 and the azimuthal angle = x3 about the axis of

symmetry. The stationary and axisymmetric character of the spacetime requires

310

311

13.1 The general stationary axisymmetric metric

that the metric coefficients g be independent of t and , so that

=g x1 x2

g

where x1 and x2 are the two remaining spacelike coordinates.

Besides stationarity and axisymmetry, we shall also require that the line element

is invariant to simultaneous inversion of the coordinates t and , i.e. the trans-

formations

t ’ ’t ’’

and

The physical meaning of this additional requirement is that the source of the

gravitational field, whatever it may be, has motions that are purely rotational

about the axis of symmetry, i.e. we are considering the spacetime associated with

a rotating body. This assumed invariance requires that

g01 = g02 = g13 = g23 = 0

since the corresponding terms in the line element would change sign under the

simultaneous inversion of t and . Therefore, under the assumptions made thus

far, the line element must have the form

ds2 = g00 dt2 + 2g03 dt d + g33 d + g11 dx1 2 + 2g12 dx1 dx2 + g22 dx2

2 2

(13.1)

We note that, since the metric coefficients g are functions only of x1 and

x2 , the expression in square brackets in (13.1) can be considered as a separate

two-dimensional submanifold. A further reduction in the form of the metric can

thus be achieved by using the fact that any two-dimensional (pseudo-)Riemannian

manifold is conformally flat, i.e. it is always possible to find a coordinate system

in which the metric takes the form

gab = 2

(13.2)

x ab

where 2 x is an arbitrary function of the coordinates and ab = diag ±1 ±1 ;

the signs depend on the signature of the manifold. We proved this result in

Appendix 11C. Thus, taking advantage of this fact, and writing the result in way

suggestive of a rotating body, we can express the line element (13.1) in the form

ds2 = A dt2 ’ B d ’ dt 2 ’ C dx1 2 + dx2 2

(13.3)

where A B C and are arbitrary functions of the spacelike coordinates x1 and x2 .

For definiteness, let us denote the coordinates x1 and x2 by r and respectively.

For our axisymmetric metric, these coordinates are not so readily associated with

any geometrical meaning. Nevertheless, in order that they can be chosen later to

be as similar as possible to the spherically symmetric r and , it is useful to allow

312 The Kerr geometry

some extra freedom in the metric by not demanding that the metric coefficients

g22 and g33 be identical. Thus, from now on we will work with the metric

ds2 = A dt2 ’ B d ’ dt 2 ’ C dr 2 ’ D d 2

(13.4)

where A B C D and are arbitrary functions of the spacelike coordinates r

and but we have the freedom to relate C and D in such a way so that the

physical meanings of r and are as close as possible to the spherically symmetric

case. The functions in (13.4) are related to the metric coefficients g by

gtt = A ’ B gt = B = ’B grr = ’C = ’D

2

g g

where, from now on, we use coordinate names rather than numbers to denote the

components. Note that = ’gt /g and, if the body is not rotating, we can set

= 0 since in this case we would require that the metric is invariant under the

single transformation t ’ ’t and consequently gt = 0.

For later convenience, let us also calculate the contravariant components g of

the metric corresponding to the line element (13.4). The only off-diagonal terms

involve t and , and so immediately we have

g rr = ’1/C = ’1/D

g

To find the remaining contravariant components, we must invert the matrix

’gt

1

gtt gt g

G’1 =

G= ’

G ’gt

gt g gtt

where the determinant G = gtt g ’ gt = ’AB. Thus

2

B 2 ’A

g gt

1 gtt

g= = =’ = = =

tt t

(13.5)

g g

G G G

A A AB

Shortly we will show that a metric of the form (13.4) can indeed be made to

satisfy the empty-space field equations R = 0 by suitable choice of the functions

A B C D and . Before specialising to any particular solution, however, we

investigate three particularly interesting generic properties of such spacetimes:

the dragging of inertial frames and the existence of stationary limit surfaces and

event horizons.

13.2 The dragging of inertial frames

The presence of gt = 0 in the metric (13.4) introduces qualitatively new effects

into particle trajectories. Since g is independent of , the covariant component

p of a particle™s 4-momentum is still conserved along its geodesic. Indeed

313

13.2 The dragging of inertial frames

p = ’L, where L is the component of angular momentum of the particle along

the rotation axis, which is conserved (note the minus sign, which also occurred in

the Schwarzschild case discussed in Chapter 9). This conservation law is a direct

consequence of the axisymmetry of the spacetime. Note, however, that the total

angular momentum of a particle is not a conserved quantity, since the spacetime

is not spherically symmetric about any point.

The corresponding contravariant component p of the particle™s 4-momentum

is given by

p =g p = g t pt + g p

and similarly the contravariant time component of the 4-momentum is

pt = g t p = g tt pt + g t p

Let is now consider a particle (or photon) with zero angular momentum, so that

p = 0 along its geodesic. Using the definition of the 4-momentum, for either a

massive particle or a photon we have

dt d

pt ∝ p∝

and

d d

where is an affine parameter along the geodesic and the constants of propor-

tionality in each case are equal. Thus the particle™s trajectory is such that

gt

d p

= t = tt = r

dt p g

This equation defines what we mean by : it is the coordinate angular velocity

of a zero-angular-momentum particle.

We shall find the explicit form for for the Kerr geometry later, but it is

clear that this effect is present in any metric for which gt = 0, which in turn

happens whenever the source of the gravitational field is rotating. So we have the

remarkable result that a particle dropped ˜straight in™ from infinity p = 0 is

˜dragged™ just by the influence of gravity so that it acquires an angular velocity

in the same sense as that of the source of the metric. This effect weakens with

distance (roughly as ∼1/r 3 for the Kerr metric) and makes the angular momentum

of the source measurable in practice.

The effect is called the dragging of inertial frames. Remember that inertial

frames are defined as those in which free-falling test bodies are stationary or move

along straight lines at constant speed. Consider the freely falling particle discussed

above. At any spatial point r , in order for the particle to be at rest in some

(inertial) frame the frame must be moving with an angular speed r . Any other

inertial frame is then related to this instantaneous rest frame by a Lorentz transfor-

mation. Thus the inertial frames are ˜dragged™ by the rotating source. A schematic

314 The Kerr geometry

φ

t=0 t>0

Figure 13.1 A schematic illustration of the dragging of inertial frames around

a rotating source.

illustration of this effect in a plane = constant is shown in Figure 13.1, where

the spacetime around the source is viewed along the rotation axis.

13.3 Stationary limit surfaces

A second generic property of spacetimes outside a rotating source is the existence

of stationary limit surfaces; this is related to the dragging of inertial frames. This

effect may be illustrated by considering, for example, photons emitted from a

position with fixed spatial coordinates r in the spacetime. In particular,

consider those photons emitted in the ± directions so that, at first, only dt and

d are non-zero along the path. Since ds2 = 0 for a photon trajectory, we have

gtt dt2 + 2gt dt d + g =0

2

d

from which we obtain

⎡ ¤1/2

2

gt gt

d gtt ¦

±⎣

=’ ’

dt g g g

Now, provided that gtt r > 0 at the point of emission, we see that d /dt is

positive (negative) for a photon emitted in the positive (negative) -direction, as

we would expect, although the value of d /dt is different for the two directions.

= 0, however, a remarkable thing happens.

On any surface defined by gtt r

The two solutions of the above equation in this case are

2gt

d d

=’ =2 =0

and

dt g dt

The first solution represents the photon sent off in the same direction as the source

rotation, and the second solution corresponds to the photon sent in the opposite

direction. For this second case, we see that when gtt = 0 the dragging of orbits is so

315

13.4 Event horizons

severe that the photon initially does not move at all! Clearly, any massive particle,

which must move more slowly than a photon, will therefore have to rotate with

the source, even if it has an angular momentum arbitrarily large in the opposite

= 0 is called a stationary limit surface.

sense. Any surface defined by gtt r

Inside the surface, where gtt < 0, no particle can remain at fixed r but must

instead rotate around the source in the same sense as the source™s rotation. This

is consistent with our discussion of the Schwarzschild metric, for which gtt = 0

occurs at r = 2 , within which no particle can remain at fixed spatial coordinates.

The fact that a particle (or observer) cannot remain at a fixed r inside a

stationary limit surface, where gtt < 0, may also be shown directly by considering

the 4-velocity of an observer at fixed r , which is given by

= ut 0 0 0 (13.6)

u

We require, however, that u · u = gtt ut 2 = c2 , but this cannot be satisfied if

gtt < 0, hence showing that a 4-velocity of the form (13.6) is not possible in such

a region.

Any surface defined by gtt = 0 is also physically interesting in another way. In

Appendix 9A, we presented a general approach to the calculation of gravitational

redshifts. In particular, we showed that, for an emitter E and receiver R with fixed

spatial coordinates in a stationary spacetime (i.e. one for which t g = 0), the

gravitational frequency shift of a photon is, quite generally,

1/2

gA

R

= tt

gtt B

E

where A is the event at which the photon is emitted and B the event at which

it is received. Thus, we see that if the photon is emitted from a point with fixed

spatial coordinates, then R ’ 0 in the limit gtt ’ 0, so that the photon suffers

= 0 is also often called

an infinite redshift. Thus a surface defined by gtt r

an infinite redshift surface. This is again consistent with our discussion of the

Schwarzschild metric, for which the surface r = 2 (where gtt = 0) is indeed an

infinite redshift surface.

13.4 Event horizons

In the Schwarzschild metric, the surface r = 2 is both a surface of infinite

redshift and an event horizon, but in our more general axisymmetric spacetime

these surfaces need not coincide. In general, as we shall see below, the defining

property of an event horizon is that it is a null 3-surface, i.e. a surface whose

normal at every point is a null vector.

316 The Kerr geometry

Before discussing the particular case of a stationary axisymmetric spacetime,

let us briefly consider null 3-surfaces in general. Suppose that such a surface is

defined by the equation

=0

fx

The normal to the surface is directed along the 4-gradient n = f = f

(remembering that f is a scalar quantity), and for a null surface we have

g n n =0 (13.7)

This last property means that the direction of the normal lies in the surface

itself; along the surface df = n dx = 0, and this equation is satisfied when the

directions of the 4-vectors dx and n coincide. In this same direction, from the

property (13.7) we see that the element of length in the 3-surface is ds = 0. In

other words, along this direction the 3-surface is tangent, at any given point, to

the lightcone at that point. Thus, the lightcone at each point of a null 3-surface

(say, in the future direction) lies entirely on one side of the surface and is tangent

to the 3-surface at that point. This means that the (future-directed) worldline of a

particle or photon can cross a null 3-surface in only one direction, and hence the

latter forms an event horizon.

In a stationary axisymmetric spacetime the equation of the surface must take

the form

=0

fr

Moreover, the condition that the surface is null means that

f =0

g f

which, for a metric of the form (13.4), reduces to

+g =0

2 2

g rr (13.8)

rf f

This is therefore the general condition for a surface f r to be an event horizon.

We may, however, choose our coordinates r and in such a way that we can

write the equation of the surface as f r = 0, i.e. as a function of r alone. In this

case, the condition (13.8) reduces to

=0

2

g rr rf

from which we see that an event horizon occurs when g rr = 0, or equivalently

grr = . This is consistent with our analysis of the Schwarzschild metric, for

which grr = at r = 2 .

317

13.5 The Kerr metric

13.5 The Kerr metric

So far our discussion has been limited to using symmetry arguments to restrict

the possible form of the stationary axisymmetric line element, which we assumed

to be

ds2 = gtt dt2 + 2gt dt d + g + grr dr 2 + g d

2 2

(13.9)

d

or, equivalently,

ds2 = A dt2 ’ B d ’ dt 2 ’ C dr 2 ’ D d 2

(13.10)

where the arbitrary functions in either form depend only on r and . As we have

seen, the general form of this line element leads to some interesting new physical

phenomena in such spacetimes. Nevertheless, we must now verify that such a

line element does indeed satisfy Einstein™s gravitational field equations and thus

obtain explicit forms for the metric functions appearing in ds2 .

The general approach to performing this calculation is the same as that used in

deriving the Schwarzschild metric. We first calculate the connection coefficients

for the metric (13.9) or (13.10) and then use these coefficients to obtain

expressions for the components R of the Ricci tensor in terms of the unknown

functions in the line element. Since we are again interested in the spacetime

geometry outside the rotating matter distribution, we must then solve the empty-

space field equations

=0

R

Although this process is conceptually straightforward, it is algebraically very

complicated, and the full calculation is extremely lengthy.1

In fact, one finds that the Einstein equations alone are insufficient to deter-

mine all the unknown functions uniquely. This should not come as a surprise

since the requirement of axisymmetry is far less restrictive than that of spherical

symmetry, used in the derivation of the Schwarzschild geometry. Although we

are envisaging a ˜compact™ rotating body, such as a star or planet, the general

form of the metric (13.10) would also be valid outside a rotating ˜extended™

axisymmetric body, such as a rotating cosmic string. To obtain the Kerr metric,

we must therefore impose some additional conditions on the solution. It tran-

spires that if we demand that the spacetime geometry tends to the Minkowski

form as r ’ and that somewhere there exists a smooth closed convex event

horizon outside which the geometry is non-singular, then the solution is unique.

1

For a full derivation, see (for example) S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford

University Press, 1983.

318 The Kerr geometry

In this case, in terms of our ˜Schwarzschild-like™ coordinates t r , the line

element for the Kerr geometry takes the form

4 acr sin2 2

2r

ds = c 1’ dt + dt d ’ dr 2 ’

2 2 2 2 2

d

2 2

(13.11)

ra2 sin2

2

’ r 2 + a2 + sin2 d 2

2

2

where and a are constants and we have introduced the functions and ,

defined by

= r 2 + a2 cos2 = r 2 ’ 2 r + a2

2

This standard expression for ds2 is known as the Boyer“Lindquist form and

as Boyer“Lindquist coordinates. The dedicated student may wish to

tr

verify that this metric does indeed satisfy the empty-space field equations.

We can write the metric (13.11) in several other useful forms. In particular, it

is common also to define the function

= r 2 + a2 2 ’ a2 sin2

2

and write the metric as

’ a2 sin2 4 ar sin2

ds = c dt +

2 2 2

c dt d

2 2

(13.12)

2 sin2

2

’ dr 2 ’ ’

2 2 2

d d

2

This form can be rearranged in a manner that is more suggestive of a rotating

object, to give

2 sin2

2 2

ds = c dt ’ d ’ w dt ’ dr 2 ’

2 2 2 2 2 2

(13.13)

d

2 2

where the physically meaningful function is given by = 2 cra/ 2 .

For later convenience, it is useful to calculate the covariant components g

of the Kerr metric in Boyer“Lindquist coordinates. Using our earlier calculations

319

13.6 Limits of the Kerr metric

for the general stationary axisymmetric metric, we find that g rr and g are simply

the reciprocals of grr and g respectively,

1

g rr = ’ =’

g

2 2

whereas the remaining contravariant components are given by

a2 sin2 ’

2 2 ar

g= =2 =

tt t

g g

sin2

c2 2 2

c

13.6 Limits of the Kerr metric

We see that the Kerr metric depends on two parameters and a, as we might

expect for a rotating body. Moreover, in the limit a ’ 0,

2

’ r2 1 ’

r

’ r2

2

’ r4

2

and so any of the forms for the Kerr metric above tends to the Schwarzschild

form,

’1

2 2

ds ’ c 1’ dt ’ 1 ’ dr 2 ’ r 2 d ’ r 2 sin2 d

2 2 2 2 2

r r

Thus suggests that we should make the identification = GM/c2 , where M is

the mass of the body, and also that a corresponds in some way to the angular

velocity of the body. In fact, by investigating the slow-rotation weak-field limit

(see Section 13.20), one can show that the angular momentum J of the body

about its rotation axis is given by J = Mac.

The fact that the Kerr metric tends to the Schwarzschild metric as a ’ 0 allows

us to give some geometrical meaning to the coordinates r and in the limit of a

slowly rotating body. In the general case, however, r and are not the standard

Schwarzschild polar coordinates. In particular, from (13.11) we see that surfaces

t = constant, r = constant do not have the metric of 2-spheres.

The geometrical nature of Boyer“Lindquist coordinates is elucidated further by

considering the Kerr metric in the limit ’ 0, i.e. in the absence of a gravitating

mass, in which case the spacetime should be Minkowski. One quickly finds that,

in this limit, the line element becomes

2

ds = c dt ’ dr 2 ’ ’ r 2 + a2 sin2 d

2 2 2 2 2 2

d

r 2 + a2

320 The Kerr geometry

This is indeed the Minkowski metric ds2 = c2 dt2 ’dx2 ’dy2 ’dz2 , but written in

that are related to Cartesian coordinates by2

terms of spatial coordinates r

x= r 2 + a2 sin cos

y= r 2 + a2 sin sin

z = r cos

where r ≥ 0, 0 ¤ ¤ and 0 ¤ < 2 (see Figure 13.2).

In this case (with = 0), the surfaces r = constant are oblate ellipsoids of

rotation about the z-axis, given by

x2 + y2 z2

+ =1

r 2 + a2 r 2

The special case r = 0 corresponds to the disc of radius a in the equatorial plane,

centred on the origin of the Cartesian coordinates. The surfaces = constant

correspond to hyperbolae of revolution about the z-axis given by

x2 + y 2 z2

’2 =1

a2 sin2 a cos2

θ=0

z

π/6

θ=

r=4

4

π/

θ=

r=3

π /3

θ=

r=2

r=1

θ = π/2

r=0

x

“a a

θ=

2π

/3

θ=

3π

/4

θ=

5π

/6

= 0 plane in Euclidean space.

Figure 13.2 Boyer“Lindquist coordinates in the

by r = a sinh

2

The coordinates r are related to the standard oblate spheroidal coordinates

and = ’ /2; see, for example, M. Abramowitz & I. Stegun, Handbook of Mathematical Functions,

Dover (1972).

321

13.7 The Kerr“Schild form of the metric

The asymptote for large values of r is a cone, with its vertex at the origin, that

subtends a half-angle . The angle is the standard azimuthal angle. Clearly, in

the limit a ’ 0 the coordinates r correspond to standard spherical polar

coordinates. It should be remembered, however, that the simple interpretation of

the coordinates given above no longer holds in the general case of the Kerr metric,

when = 0.

13.7 The Kerr“Schild form of the metric

The form (13.11) for the line element is not in fact the form originally discovered

by Roy Kerr in 1963. Indeed, Kerr himself followed an approach to the derivation

very different from that presented here. His original interest was in line elements

of the general form

ds2 = dx dx ’ l l dx dx

where the vector l is null with respect to the Minkowski metric , i.e.

l l =0

This form for a line element is now known as the Kerr“Schild form. Kerr showed

that a line element of this form satisfied the empty-space field equations (together

with our additional conditions on the solution mentioned above), provided that

2 r3

=2

r + a2 z2

rx + ay ry ’ ax z

=c

l

a2 + y2 a2 + y2 r

¯

= t x y z and r is defined implicitly in terms of x y and z by

where x

r 4 ’ r 2 x2 + y2 + z2 ’ a2 ’ a2 z2 = 0 (13.14)

The corresponding form for the line element is given by

2 r3

¯2

ds = c d t ’ dx ’ dy ’ dz ’ 4

2 2 2 2 2

r + a2 z2

2

r a z

¯

— c dt ’ 2 x dx + y dy ’ 2 x dy ’ y dx ’ dz

r + a2 r + a2 r

(13.15)

322 The Kerr geometry

It is straightforward, but lengthy, to show that the two forms (13.15) and (13.11)

for the line element are identical if the two sets of coordinates are related by

2r

¯

c d t = c dt ’ (13.16)

dr

x = r cos + a sin sin (13.17)

y = r sin ’ a cos sin (13.18)

z = r cos (13.19)

= d ’ a/

where d dr.

13.8 The structure of a Kerr black hole

The Kerr metric is the solution to the empty-space field equations outside a

rotating massive object and so is only valid down to the surface of the object.

As in our discussion of the Schwarzschild solution, however, it is of interest to

consider the structure of the full Kerr geometry as a vacuum solution to the field

equations.

Singularities and horizons

The Kerr metric in Boyer“Lindquist coordinates is singular when = 0 and when

= 0. Calculation of the invariant curvature scalar R reveals that only

R

= 0 is an intrinsic singularity. Since

= r 2 + a2 cos2 = 0

2

it follows that this occurs when

r =0 = /2

From our earlier discussion of Boyer“Lindquist coordinates, we recall that r = 0

represents a disc of coordinate radius a in the equatorial plane. Moreover, the

collection of points with r = 0 and = /2 constitutes the outer edge of this disc.

Thus, rather surprisingly, the singularity has the form of a ring, of coordinate

radius a, lying in the equatorial plane. Similarly, using (13.14) and (13.19), we

¯

see that, in terms of the ˜Cartesian™ coordinates t x y z , the singularity occurs

2 + y 2 = a2 and z = 0.

when x

The points where = 0 are coordinate singularities, which occur on the surfaces

1/2

r± = ± ’ a2

2

(13.20)

323

13.8 The structure of a Kerr black hole

As discussed above, event horizons in the Kerr metric will occur where r =

constant is a null 3-surface, and this is given by the condition g rr = 0 or, equiva-

lently, grr = . From (13.11), we have

2

grr = ’

from which we see that the surfaces r = r+ and r = r’ , for which = 0, are

in fact event horizons. Thus, the Kerr metric has two event horizons. In the

Schwarzschild limit a ’ 0, these reduce to r = 2 and r = 0. The surfaces r = r±

are axially symmetric, but their intrinsic geometries are not spherically symmetric.

Setting r = r± and t = constant in the Kerr metric and noting from (13.20) that

r± + a2 = 2 r± , we obtain two-dimensional surfaces with the line elements

2

2

2 r±

= ±d + sin2 d

2 2 2 2

(13.21)

d

±

which do not describe the geometry of a sphere. If one embeds a 2-surface with

geometry given by (13.21) in three-dimensional Euclidean space, one obtains a

surface resembling an axisymmetric ellipsoid, flattened along the rotation axis.

The existence of the outer horizon r = r+ , in particular, shows that the Kerr

geometry represents a (rotating) black hole. It is a one-way surface, like r = 2

in the Schwarzschild geometry. Particles and photons can cross it once, from the

outside, but not in the opposite direction. It is common practice to define three

distinct regions of a Kerr black hole, bounded by the event horizons, in which the

solution is regular: region I, r+ < r < ; region II, r’ < r < r+ ; and region III,

0 < r < r’ .

Not all values of and a correspond to a black hole, however. From (13.20),

we see that horizons (at real values of r) exist only for

a2 < 2

(13.22)

Thus the magnitude of the angular momentum J = Mac of a rotating black hole

is limited by its squared mass. Moreover, if the condition (13.22) is satisfied

then the intrinsic singularity at = 0 is contained safely within the outer horizon

r = r+ . An extreme Kerr black hole is one that has the limiting value a2 = 2 .

In this case, the event horizons r+ and r’ coincide at r = . It may be that

near-extreme Kerr black holes develop naturally in many astrophysical situations.

Matter falling towards a rotating black hole forms an accretion disc that rotates

in the same sense as the hole. As matter from the disc spirals inwards and falls

into the black hole, it carries angular momentum with it and hence increases the

angular momentum of the hole. The process is limited by the fact that radiation

from the infalling matter carries away angular momentum. Detailed calculations

324 The Kerr geometry

suggest that the limiting value is a ≈ 0 998 , which is very close to the extreme

value.

For a2 > 2 we find that > 0 throughout, and so the Kerr metric is regular

everywhere except = 0, where there is a ring singularity. Since the horizons

have disappeared, this means that the ring singularity is visible to the outside

world. In fact, one can show explicitly that timelike and null geodesics in the

equatorial plane can start at the singularity and reach infinity, thereby making

the singularity visible to the outside world. Such a singularity is called a naked

singularity (as mentioned in Section 12.6) and opens up an enormous realm for

some truly wild speculation. However, Penrose™s cosmic censorship hypothesis

only allows singularities that are hidden behind an event horizon.

Stationary limit surfaces

As we showed earlier, in a general stationary axisymmetric spacetime the condi-

tion gtt = 0 defines a surface that is both a stationary limit surface and a surface

of infinite redshift. For the Kerr metric, we have

2 ’2 r + a2 cos2

2r 2r

gtt = c 1’ =c

2

2 2

these surfaces, S + and S ’ , occur at

so that (for a2 ¤ 2)

1/2

rS± = ± ’ a2 cos2

2

The two surfaces are axisymmetric, but setting r = rS± and t = constant in the

Kerr metric, and noting from (13.20) that rS± + a2 = 2 rS± + a2 sin2 , we obtain

2

two-dimensional surfaces with line elements

2 rS± 2 rS± + 2a2 sin2

= + sin2 d

2 2 2 2

(13.23)

d d

S± 2

S±

which again do not describe the geometry of a sphere. If one embeds a 2-surface

with geometry given by (13.23) in three-dimensional Euclidean space then a

surface resembling an axisymmetric ellipsoid, flattened along the rotation axis, is

once more obtained. In the Schwarzschild limit a ’ 0, the surface S + reduces to

r = 2 and S ’ to r = 0. As anticipated we see that, in the Schwarzschild solution,

the surfaces of infinite redshift and the event horizons coincide.

The surface S ’ coincides with the ring singularity in the equatorial plane.

Moreover, S ’ lies completely within the inner horizon r = r’ (except at the poles,

where they touch). The surface S + has coordinate radius 2 at the equator and,

for all , it completely encloses the outer horizon r = r+ (except at the poles,

where they touch), giving rise to a region between the two called the ergoregion.

The structure of a Kerr black hole is illustrated in Figure 13.3.

325

13.8 The structure of a Kerr black hole

z

Event horizon r = r+ Ring singularity

Stationary limit

surface (infinite

Event horizon r = r“ redshift surface) S+

y

Ergosphere

Infinite redshift

surface S “

Symmetry axis (θ = 0)

Figure 13.3 The structure of a Kerr black hole.

The ergoregion

The ergoregion gets its name from the Greek word ergo meaning work. The

key property of an ergoregion (which can occur in other spacetime geometries)

is that it is a region for which gtt < 0 and from which particles can escape.

Clearly, the Schwarzschild geometry does not possess an ergoregion, since gtt < 0

is only satisfied within its event horizon. As we will discuss in Section 13.9,

Roger Penrose has shown that it is possible to extract the rotational energy of

a Kerr black hole from within the ergoregion. To assist in that discussion, it is

useful here to consider the constraints induced by the spacetime geometry on the

motion of observers within the ergoregion.

Since gtt < 0 at all points within the ergoregion, an immediate consequence (as

already discussed in Section 13.3) is that an observer (even in a spaceship with

an arbitrarily powerful rocket) cannot remain at a fixed r position. The

4-velocity of such an observer would be given by

= ut 0 0 0 (13.24)

u

but the requirement that u · u = gtt ut 2 = c2 cannot be satisfied if gtt < 0, showing

that a 4-velocity of the form (13.24) is not possible.

It is possible, however, for a rocket-powered observer to remain at fixed r

and coordinates by rotating around the black hole (with respect to an observer

at infinity) in the same sense as the hole™s rotation; this is an illustration of the

frame-dragging phenomenon discussed in Section 13.3. The 4-velocity of such an

observer is

= ut 1 0 0 (13.25)

u

326 The Kerr geometry

where = d /dt is his angular velocity with respect to the observer at infinity.

For any particular values of r and , there exists a range of allowed values for ,

which we now derive. We again require u · u = g u u = c2 and, using (13.25),

this condition becomes

+ 2gt ut u + g = ut gtt + 2gt +g = c2

2 2 2 2

gtt ut (13.26)

u

Thus, for ut to be real we require that

+ 2gt + gtt > 0

2

(13.27)

g

Since g < 0 everywhere, the left-hand side of (13.27) as a function of gives

rise to an upward pointing parabola. Thus, the allowed range of angular velocities

is given by ’ < < + , where

⎡ ¤1/2

2 1/2

gt gt g g

±⎣ ’ tt ¦ ’ tt

=’ = ± 2

(13.28)

±

g g g g

There are clearly two special cases to be considered. First, when gtt = 0 we

have ’ = 0 and + = 2 . This occurs on the stationary limit surface r = rS+ ,

which is the outer defining surface of the ergoregion. The lower limit ’ = 0 is

precisely the physical meaning of a stationary limit surface: within it an observer

must rotate in the same direction as the black hole and so must be positive.

For larger values of r, however, can be negative. The second special case to

2 = g /g , in which case

± = . Thus, at points where this

consider is when tt

condition holds, every observer on a circular orbit is forced to rotate with angular

velocity = . Where (if anywhere) does this condition hold? Upon inserting

the appropriate expressions for , gtt and g from the Kerr metric (13.13) into

(13.28), one finds, after some careful algebra, that our condition holds where

= 0, i.e. at the outer event horizon r = r+ , which is the inner defining surface

of the ergoregion.

Putting our results together we find that, for an observer at fixed r and

coordinates within the ergoregion, the allowed range of angular velocities ’ <

< + becomes progressively narrower as the observer is located closer and

closer to the horizon r = r+ , and at the horizon itself the angular velocity is

limited to the single value

ac

H≡ = (13.29)

r+

2 r+

327

13.9 The Penrose process

which is, in fact, independent of . We also note that H is the maximum

allowed value of the angular velocity for any observer at fixed r and within the

ergoregion.

Extension of the Kerr metric

So far we have not discussed the disc region interior to the ring singularity.

Although beyond the scope of our discussion, it may be shown that if a particle

passes through the interior of the ring singularity then it emerges into another

asymptotically flat spacetime, but not a copy of the original one. The new space-

time is described by the Kerr metric with r < 0 and hence never vanishes, so

there are no event horizons.3

In the new spacetime, the region in the vicinity of the ring singularity has the

very strange property that it allows the existence of closed timelike curves. For

example, consider a trajectory in the equatorial plane that winds around in

while keeping t and r constant. The line element along such a path is

2 a2

ds = ’ r + a +

2 2 2 2

d

r

which is positive if r is negative and small. These are then closed timelike curves,

which violate causality and would seem highly unphysical. If they represent

worldlines of observers, then these observers would travel back and meet them-

selves in the past! It must be remembered, however, that the analytic extension

of the Kerr metric to negative values of r is subject a number of caveats and

may not be physically meaningful. It seems highly improbable that in practice

the gravitational collapse of a real rotating object would lead to such a strange

spacetime.

13.9 The Penrose process

We now discuss the Penrose process, by which energy may be extracted from

the rotation of a Kerr black hole (or, indeed, from any spacetime possessing

an ergoregion). Suppose that an observer, with a fixed position at infinity, for

simplicity, fires a particle A into the ergoregion of a Kerr black hole. The energy

of particle A, as measured by the observer at the emission event , is given by

A

=pA · uobs = pt

A

(13.30)

E

where p A is the 4-momentum of the particle at this event and uobs is the

4-velocity of the observer, which has components uobs = 1 0 0 0 .

In the extended Kerr solution it is common to define region III to cover the coordinate range ’

3

< r < r’ .

328 The Kerr geometry

Suppose now that, at some point in the ergoregion, particle A decays into two

particles B and C. By the conservation of momentum, at the decay event

one has

=pB +p C

pA (13.31)

If the decay occurs in such a way that particle C (say) eventually reaches infinity,

a stationary observer there would measure the particle™s energy at the reception

event to be

C C

= pt = pt

C

E

where, in the second equality, we have made use of the fact that the covariant time

component of a particle™s 4-momentum is conserved along geodesics in the Kerr

geometry, since the metric is stationary t g = 0 . Similarly, for the original

A A

= pt

particle we have pt . Thus, the time component of the momentum

conservation condition (13.31) may be written in the form

B

=E ’ pt

C A

(13.32)

E

B

where pt is also conserved along the geodesic followed by particle B.

B

The key step is now to note that pt = et · p B , where et is the t-coordinate

basis vector, whose squared ˜length™ is given by

et · et = gtt

If particle B were ever to escape beyond the outer surface of the ergoregion,

B

i.e. to a region where gtt > 0, then et would be timelike. Thus, pt would be

proportional to the particle energy as measured by an observer with 4-velocity

B

along the et -direction. In this case pt must therefore be positive, and so (13.32)

shows that E C < E A , i.e. we get less energy out than we put in. However, if

the particle B were never to escape the ergoregion but instead fall into the black

hole, then it would remain in a region where gtt < 0 and so et is spacelike. In this

B

case pt would be a component of spatial momentum, which might be positive

or negative. For decays where it is negative, from (13.32) we see that E C > E A

and so we have extracted energy from the black hole. This is the Penrose process.

What are the consequences of the Penrose process for the black hole? Once the

particle has fallen inside the event horizon, the mass M and angular momentum

J = Mac of the black hole are changed:

B

M ’ M + pt /c2 (13.33)

B

J ’ J ’p (13.34)

where in the last equation we must remember that, for particle orbits in general,

p is minus the component of angular momentum of the particle along the rotation

329

13.9 The Penrose process

B

axis of the black hole. From (13.33), we see that the negative value of pt for

the infalling particle in the Penrose process reduces the total mass of the black

hole. As we now show, however, the Penrose process also reduces the angular

momentum of the black hole. This is what is meant by saying that the Penrose

process extracts rotational energy from the black hole.

To show that the angular momentum of the black hole is reduced by the

infalling particle, it is useful to consider an observer in the ergoregion at fixed

r and coordinates, who observes the particle B as it passes him. As shown in

Section 13.8, the 4-velocity of such an observer is

= ut 1 0 0 (13.35)

u

where = d /dt is the observer™s angular velocity with respect to infinity. This

observer would measure the energy of particle B to be

B B

= p B u = ut p t + p

B

E

Since this energy must be positive, we require

B

pt

(13.36)

L<

B

where L = ’p is the component of the angular momentum of the particle along

B

the rotation axis of the hole. Since pt is negative in the Penrose process and

must be positive for an observer in the ergoregion, we see that L < 0. Thus the

infalling particle must have negative angular momentum, which therefore reduces

the net angular momentum of the black hole. Rotational energy can continue to

be extracted until the angular momentum of the black hole is reduced to zero and

it becomes a Schwarzschild black hole.

We can, in fact, go slightly further and set a strict upper limit on L (which, since

L is negative, is equivalent to a lower limit on its magnitude). We actually require

(13.36) to hold for any observer at fixed r and in the ergoregion. From our

earlier discussion of the ergoregion, the maximum value of the angular velocity

occurs for an observer at the horizon r = r+ , in which case = H , (13.29).

Thus, denoting the changes in the mass and angular momentum of the black hole

by M and J respectively, the condition (13.36) becomes

c2 M

J<

H

where it should be remembered that both M and J are negative.

330 The Kerr geometry

13.10 Geodesics in the equatorial plane

As one might expect, the general equations for non-null and null geodesics in the

Kerr geometry are much less tractable than in the Schwarzschild case, and particle

trajectories exhibit complicated behaviour. For example, in general the trajectory

of a massive particle or photon is not constrained to lie in a plane. This is a direct

consequence of the fact that the spacetime is not spherically symmetric and so, in

general, the angular momentum of a test particle is not conserved. Since the Kerr

geometry is stationary and axisymmetric, the conserved quantities along particle

trajectories are pt and p . The latter corresponds to the conservation of only the

component of angular momentum along the rotation axis. Nevertheless, since the

metric is reflection-symmetric through the equatorial plane, particles for which

p = 0, i.e. which are initially in the equatorial plane, will always have p = 0 and

so the trajectory remains in this plane. We shall therefore confine our attention to

this simpler special case.

Setting = /2 in the Kerr metric (13.11), we obtain

r2 2 2 a2

2 4 ac

ds = c 1’ dt + dt d ’ dr ’ r + a +

2 2 2 2 2 2

d

r r r

(13.37)

from which the covariant metric components g in the equatorial plane can

be read off. Following the method described in Section 13.1, the corresponding

contravariant metric components are found to be

2 a2

1 2a

g= r +a + gt =

2 2

tt

c2 r cr

1 2

g rr = ’ =’ 1’

g

r2 r

From (13.37) one can immediately write down the corresponding ˜Lagrangian™

= g x™ x™ . In the interests of notational simplicity, for a massive particle we

shall take the particle to have unit rest mass and for a photon we shall choose

an appropriate affine parameter along the null geodesic such that, in both cases,

p = x . One may obtain the geodesic equations by writing down the appropriate

™

Euler“Lagrange equations. It is quicker, however, simply to use the fact that pt and

p are conserved along geodesics (since the metric does not depend explicitly on

t and ), which leads immediately to the first integrals of the t- and - equations.

These are given by

2 2 ac ™

pt = gtt t + gt ™ = c2 1 ’

™ ™

t+ = kc2 (13.38)

r r

2 a2

2 ac

™= ™ = ’h

™ ™

p = g tt + g t’ r +a +

2 2

(13.39)

r r

331

13.10 Geodesics in the equatorial plane

where we have defined the constants k and h so that in the Schwarzschild limit

a ’ 0 they coincide with the constants introduced in Chapter 9. This pair of

simultaneous equations for t and ™ is straightforwardly solved to give

™

2 a2

1 2a

™

t= r +a + k’

2 2

h

r cr

(13.40)

1 2 ac 2

™= k+ 1’ h

r r

Instead of using the complicated Euler“Lagrange equation for r, we may use

the first integral provided by the invariant length of the 4-momentum p. Since

the covariant components of p are particularly simple, the most convenient form

to use is g p p = 2 , where 2 = c2 for a massive particle and 2 = 0 for a

photon.4 Since p = 0 this gives

+ 2g t pt p + g + g rr pr =

2 2 2 2

g tt pt (13.41)

p

where, for the moment, it is simpler not to write out the contravariant metric

components in full. By substituting pt = kc2 and p = ’h into (13.41) and

remembering that pr = grr r and g rr = 1/grr , we may then obtain the ˜energy™

™

™

equation for equatorial trajectories, which gives r in terms of only r and a set of

constants. This yields

r 2 = g rr

™ ’ g tt c4 k2 + 2g t c2 kh ’ g

2

h2 (13.42)

At this stage, we may (if we wish) substitute the explicit forms for the contravariant

metric coefficients to obtain

a2 c2 k2 ’ ’ h2 h ’ ack

2 2 2

2 2

r =c k ’

™ + + +

2 22 2

(13.43)

r2 r3

r

In the limit a ’ 0, the energy equation reduces to those derived in Chapter 9

for massive-particle 2 = c2 and photon 2 = 0 orbits in the Schwarzschild

geometry.

Since we are restricting our attention to the equatorial plane, we need not

consider the Euler“Lagrange equation for , since it will not yield an independent

equation of motion. Thus, equations (13.40) and (13.43) completely determine

the null and non-null geodesics in the equatorial plane for given values of the

constants k and h.

4

The device of working in terms of 2 allows one to calculate the null and non-null geodesic equations

simultaneously; one simply sets 2 to the appropriate value at the end of the calculation.

332 The Kerr geometry

The null and non-null geodesics in the equatorial plane can be delineated in

much the same way as for the Schwarzschild case in Chapter 9, albeit requiring

significantly more complicated algebra. Before moving on to discuss particu-

lar examples, however, it is worth noting two essential differences from the

Schwarzschild case. First, in the Kerr equatorial geometry trajectories will depend

upon whether the particle or photon is in a co-rotating (prograde) or counter-

rotating (retrograde) orbit, i.e. rotating about the symmetry axis in the same sense

or the opposite sense to that of the rotating gravitational source. Second, both t

and are ˜bad™ coordinates near the horizons. Expressed in terms of these coor-

dinates, a trajectory approaching an horizon (at r+ or r’ ) will spiral around the

black hole an infinite number of times, just as it takes an infinite coordinate time

t to cross the horizon; neither behaviour is experienced by an observer comoving

with the particle.

13.11 Equatorial trajectories of massive particles

For a massive particle, the timelike geodesics in the equatorial plane are governed

by (13.40), and the ˜energy™ equation (13.43) with 2 = c2 , which reads

2 c2 a2 c2 k2 ’ 1 ’ h2 2 h ’ ack 2

r = c k ’1 +

™ + +

2 2 2

(13.44)

r2 r3

r

The interpretation of the constants k and h may be obtained by considering the

limit r ’ , in the same way as for the Schwarzschild geometry. One thus finds

that kc2 and h are, respectively, the energy and angular momentum per unit rest

mass of the particle describing the trajectory.

One may rewrite the energy equation (13.44) in the form