<< стр. 13(всего 24)СОДЕРЖАНИЕ >>
for the matter is of the form

= diag c2 0 0 в€’ 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 в€’ dr 2 в€’ B r d в€’ dz2
2

where B r satisfies
B2
B
в€’ =в€’ c2
4B2
2B
Show further that b r = B r satisfies b = в€’ 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 в‰¤ r0
0
r=
0 for r > r0

By demanding that g в†’ в€’r 2 as r в†’ 0, so that the spacetime geometry is regular
on the axis of the string, show that the line element for r в‰¤ r0 is
2
sin r
ds = c dt в€’ dr в€’ в€’ 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 в€’ dr в€’ + r в€’ r0 cos r0 в€’ 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 в€’ dr 2 в€’ 1 в€’ в€’ dz2
r2 d 2
c2
308 Further spherically symmetric geometries

where = r0 0 is the вЂ˜mass per unit lengthвЂ™ 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
вЋ› вЋћ
0 в€’1 0 0
вЋњ1 0 0 0вЋџ
вЋњ вЋџ
F =E r вЋњ вЋџ
вЋќ0 0 0 0вЋ
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 в€’ B r dr 2 в€’ r 2 d + sin2 d
2 2

the energyвЂ“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 в€’
2
T
2 B A AB AB

12.14 Calculate the invariant curvature scalar R for the ReissnerвЂ“NordstrГ¶m
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 ReissnerвЂ“NordstrГ¶m
geometry are given by
2
2
r+
rв€’ r r
ct = r в€’ в€’1 + в€’ 1 + constant
ln ln outgoing
r + в€’ rв€’ r+ в€’ r в€’
rв€’ r+
2
2
r+
rв€’ r r
ct = в€’r + в€’1 в€’ в€’ 1 + constant
ln ln ingoing
r+ в€’ r в€’ r+ в€’ rв€’
rв€’ r+

12.16 Show that, by introducing the advanced EddingtonвЂ“Finkelstein timelike coordinate
2
2
r+
rв€’ r r
ct = ct в€’ в€’1 + в€’1
ln ln
r+ в€’ r в€’ r + в€’ rв€’
rв€’ r+

the ReissnerвЂ“NordstrГ¶m line element takes the form
2
ds2 = c2 dt в€’ 2 1 в€’ dt dr в€’ 2 в€’ dr 2 в€’ r 2 d + sin2 d
2 2

where в‰Ў r = 1 в€’ 2 /r + q 2 /r 2 . Hence show that the worldlines of radially
moving photons in advanced EddingtonвЂ“Finkelstein coordinates are given by
2в€’
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
309
Exercises

12.17 For a particle of mass m and charge e in geodesic motion in the ReissnerвЂ“
NordstrГ¶m geometry, show that the quantity
q2
2 dt eq
k = m 1в€’ +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 ReissnerвЂ“
NordstrГ¶m 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 в†’ в€’t в†’в€’
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 В±1 В±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 в€’ B d в€’ dt 2 в€’ 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 в€’ B d в€’ dt 2 в€’ C dr 2 в€’ 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 в€’ B gt = B = в€’B grr = в€’C = в€’D
2
g g
where, from now on, we use coordinate names rather than numbers to denote the
components. Note that = в€’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 в†’ в€’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 = в€’1/C = в€’1/D
g
To find the remaining contravariant components, we must invert the matrix

в€’gt
1
gtt gt g
Gв€’1 =
G= в‡’
G в€’gt
gt g gtt

where the determinant G = gtt g в€’ gt = в€’AB. Thus
2

B 2 в€’A
g gt
1 gtt
g= = =в€’ = = =
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 particleвЂ™s 4-momentum is still conserved along its geodesic. Indeed
313
13.2 The dragging of inertial frames

p = в€’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 particleвЂ™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 в€ќ pв€ќ
and
d d
where is an affine parameter along the geodesic and the constants of propor-
tionality in each case are equal. Thus the particleвЂ™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 вЂ˜straight inвЂ™ from infinity p = 0 is
вЂ˜draggedвЂ™ 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 в€ј1/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 вЂ˜draggedвЂ™ by the rotating source. A schematic
314 The Kerr geometry

П†

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 В± 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
вЋЎ вЋ¤1/2
2
gt gt
d gtt вЋ¦
В±вЋЈ
=в€’ в€’
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
=в€’ =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 sourceвЂ™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 В· 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 в†’ 0 in the limit gtt в†’ 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 в€’ B d в€’ dt 2 в€’ C dr 2 в€’ 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 EinsteinвЂ™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 вЂ˜compactвЂ™ rotating body, such as a star or planet, the general
form of the metric (13.10) would also be valid outside a rotating вЂ˜extendedвЂ™
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 в†’ 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 вЂ˜Schwarzschild-likeвЂ™ coordinates t r , the line
element for the Kerr geometry takes the form

4 acr sin2 2
2r
ds = c 1в€’ dt + dt d в€’ dr 2 в€’
2 2 2 2 2
d
2 2
(13.11)
ra2 sin2
2
в€’ 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 в€’ 2 r + a2
2

This standard expression for ds2 is known as the BoyerвЂ“Lindquist form and
as BoyerвЂ“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 в€’ a2 sin2
2

and write the metric as

в€’ a2 sin2 4 ar sin2
ds = c dt +
2 2 2
c dt d
2 2
(13.12)
2 sin2
2
в€’ dr 2 в€’ в€’
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 в€’ d в€’ w dt в€’ dr 2 в€’
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 BoyerвЂ“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 = в€’ =в€’
g
2 2

whereas the remaining contravariant components are given by
a2 sin2 в€’
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 в†’ 0,
2
в†’ r2 1 в€’
r
в†’ r2
2

в†’ r4
2

and so any of the forms for the Kerr metric above tends to the Schwarzschild
form,
в€’1
2 2
ds в†’ c 1в€’ dt в€’ 1 в€’ dr 2 в€’ r 2 d в€’ 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 в†’ 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 BoyerвЂ“Lindquist coordinates is elucidated further by
considering the Kerr metric in the limit в†’ 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 в€’ dr 2 в€’ в€’ 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 в€’dx2 в€’dy2 в€’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 в‰Ґ 0, 0 в‰¤ в‰¤ and 0 в‰¤ < 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
в€’2 =1
a2 sin2 a cos2
Оё=0

z
ПЂ/6
Оё=

r=4
4
ПЂ/
Оё=

r=3
ПЂ /3
Оё=
r=2

r=1
Оё = ПЂ/2
r=0
x
вЂ“a a

Оё=
2ПЂ
/3
Оё=
3ПЂ
/4
Оё=
5ПЂ
/6

= 0 plane in Euclidean space.
Figure 13.2 BoyerвЂ“Lindquist coordinates in the

by r = a sinh
2
The coordinates r are related to the standard oblate spheroidal coordinates
and = в€’ /2; see, for example, M. Abramowitz & I. Stegun, Handbook of Mathematical Functions,
Dover (1972).
321
13.7 The KerrвЂ“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 в†’ 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 KerrвЂ“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 в€’ 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 KerrвЂ“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 в€’ ax z
=c
l
a2 + y2 a2 + y2 r
ВЇ
= t x y z and r is defined implicitly in terms of x y and z by
where x

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

The corresponding form for the line element is given by

2 r3
ВЇ2
ds = c d t в€’ dx в€’ dy в€’ dz в€’ 4
2 2 2 2 2
r + a2 z2
2
r a z
ВЇ
Г— c dt в€’ 2 x dx + y dy в€’ 2 x dy в€’ y dx в€’ 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
ВЇ
c d t = c dt в€’ (13.16)
dr

x = r cos + a sin sin (13.17)
y = r sin в€’ a cos sin (13.18)
z = r cos (13.19)

= d в€’ 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 BoyerвЂ“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 BoyerвЂ“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
ВЇ
see that, in terms of the вЂ˜CartesianвЂ™ 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
rВ± = В± в€’ 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 = в€’

from which we see that the surfaces r = r+ and r = rв€’ , for which = 0, are
in fact event horizons. Thus, the Kerr metric has two event horizons. In the
Schwarzschild limit a в†’ 0, these reduce to r = 2 and r = 0. The surfaces r = rВ±
are axially symmetric, but their intrinsic geometries are not spherically symmetric.
Setting r = rВ± and t = constant in the Kerr metric and noting from (13.20) that
rВ± + a2 = 2 rВ± , we obtain two-dimensional surfaces with the line elements
2

2
2 rВ±
= В±d + sin2 d
2 2 2 2
(13.21)
d
В±

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, rв€’ < r < r+ ; and region III,
0 < r < rв€’ .
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 rв€’ 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 в‰€ 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, PenroseвЂ™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 в€’2 r + a2 cos2
2r 2r
gtt = c 1в€’ =c
2
2 2

these surfaces, S + and S в€’ , occur at
so that (for a2 в‰¤ 2)

1/2
rSВ± = В± в€’ a2 cos2
2

The two surfaces are axisymmetric, but setting r = rSВ± and t = constant in the
Kerr metric, and noting from (13.20) that rSВ± + a2 = 2 rSВ± + a2 sin2 , we obtain
2

two-dimensional surfaces with line elements
2 rSВ± 2 rSВ± + 2a2 sin2
= + sin2 d
2 2 2 2
(13.23)
d d
SВ± 2
SВ±

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 в†’ 0, the surface S + reduces to
r = 2 and S в€’ 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 в€’ coincides with the ring singularity in the equatorial plane.
Moreover, S в€’ lies completely within the inner horizon r = rв€’ (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 = rвЂ“ redshift surface) S+

y

Ergosphere
Infinite redshift
surface S вЂ“

Symmetry axis (Оё = 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 В· 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 holeвЂ™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 В· 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 в€’ < < + , where

вЋЎ вЋ¤1/2
2 1/2
gt gt g g
В±вЋЈ в€’ tt вЋ¦ в€’ tt
=в€’ = В± 2
(13.28)
В±
g g g g

There are clearly two special cases to be considered. First, when gtt = 0 we
have в€’ = 0 and + = 2 . This occurs on the stationary limit surface r = rS+ ,
which is the outer defining surface of the ergoregion. The lower limit в€’ = 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
В± = . 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 в€’ <
< + 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
Hв‰Ў = (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 = в€’ 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 В· 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 в€’
3
< r < rв€’ .
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 particleвЂ™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 particleвЂ™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 в€’ 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 В· p B , where et is the t-coordinate
basis vector, whose squared вЂ˜lengthвЂ™ is given by
et В· 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 в†’ M + pt /c2 (13.33)
B
J в†’ J в€’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 observerвЂ™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 = в€’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 1в€’ dt + dt d в€’ dr в€’ 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 = в€’ =в€’ 1в€’
g
r2 r
From (13.37) one can immediately write down the corresponding вЂ˜LagrangianвЂ™
= g xЛ™ xЛ™ . 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
Л™
EulerвЂ“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 Л™
pt = gtt t + gt Л™ = c2 1 в€’
Л™ Л™
t+ = kc2 (13.38)
r r
2 a2
2 ac
Л™= Л™ = в€’h
Л™ Л™
p = g tt + g tв€’ 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 в†’ 0 they coincide with the constants introduced in Chapter 9. This pair of
simultaneous equations for t and Л™ is straightforwardly solved to give
Л™

2 a2
1 2a
Л™
t= r +a + kв€’
2 2
h
r cr
(13.40)
1 2 ac 2
Л™= k+ 1в€’ h
r r

Instead of using the complicated EulerвЂ“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 = в€’h into (13.41) and
remembering that pr = grr r and g rr = 1/grr , we may then obtain the вЂ˜energyвЂ™
Л™
Л™
equation for equatorial trajectories, which gives r in terms of only r and a set of
constants. This yields

r 2 = g rr
Л™ в€’ g tt c4 k2 + 2g t c2 kh в€’ 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 в€’ в€’ h2 h в€’ ack
2 2 2
2 2
r =c k в€’
Л™ + + +
2 22 2
(13.43)
r2 r3
r

In the limit a в†’ 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 EulerвЂ“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 вЂ˜badвЂ™ coordinates near the horizons. Expressed in terms of these coor-
dinates, a trajectory approaching an horizon (at r+ or rв€’ ) 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 вЂ˜energyвЂ™ equation (13.43) with 2 = c2 , which reads

2 c2 a2 c2 k2 в€’ 1 в€’ h2 2 h в€’ ack 2
r = c k в€’1 +
Л™ + +
2 2 2
(13.44)
r2 r3
r

The interpretation of the constants k and h may be obtained by considering the
limit r в†’ , 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

 << стр. 13(всего 24)СОДЕРЖАНИЕ >>