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Section 1 Introduction
F1
PROJECT F
The Spinning Black Hole
Black holes are macroscopic objects with masses varying from a few solar
masses to millions of solar masses. To the extent they may be considered as
stationary and isolated, to that extent, they are all, every single one of
them, described exactly by the Kerr solution. This is the only instance we
have of an exact description of a macroscopic object. Macroscopic objects, as
we see them all around us, are governed by a variety of forces, derived from
a variety of approximations to a variety of physical theories. In contrast,
the only elements in the construction of black holes are our basic concepts
of space and time. They are, thus, almost by deﬁnition, the most perfect
macroscopic objects there are in the universe. And since the general theory
of relativity provides a single unique twoparameter family of solutions for
their description, they are the simplest objects as well.
—S. Chandrasekhar
1 Introduction
In this project we explore some of the properties of spacetime near a spin
ning black hole. Analogous properties describe spacetime external to the
surface of the spinning Earth, Sun, or other spinning uncharged heavenly
body. For a black hole these properties are truly remarkable. Near enough
to a spinning black hole—even outside its horizon—you cannot resist
being swept along tangentially in the direction of rotation. You can have a
negative total energy. From outside the horizon you can, in principle, har
ness the rotational energy of the black hole.
Do spinning black holes exist? The primary question is: Do black holes
exist? If the answer is yes, then spinning black holes are inevitable, since
astronomical bodies most often rotate. As evidence, consider the most
compact stellar object short of a black hole, the neutron star. Detection of
radio and Xray pulses from some spinning neutron stars (called
pulsars
)
tells us that many neutron stars rotate, some of them very rapidly. These
are impressive structures, with more mass than our Sun, some of them
spinning once every few milliseconds. Conclusion: If black holes exist,
then spinning black holes exist.
General relativity predicts that when an isolated spinning star collapses to
a black hole, gravitational radiation quickly (in a few seconds of faraway
time!) smooths any irregularities in rotation. Thereafter the metric exterior
F2
P
ROJECT
F The Spinning Black Hole
to the horizon of the spinning black hole will be the Kerr metric used in
this project.
However, not all spinning black holes are isolated; many are surrounded
by other matter attracted to them. The inwardswirling mass of a resulting
accretion disk
can affect spacetime in its vicinity, distorting the metric
away from that of the isolated spinning black hole that we analyze here.
2 Angular Momentum of the Black Hole
An isolated spinning uncharged black hole is completely speciﬁed by just
two quantities: its mass
M
and its angular momentum. In Chapter 4 (page
43) we deﬁned the angular momentum per unit mass for a particle orbit
ing a nonspinning black hole as
L/m
=
r
2
d
φ
/
d
τ
. In this expression, the
angle
φ
has no units and proper time
τ
has the unit meter. Therefore
L/m
has the unit meter. To avoid confusion, the angular momentum of a spin
ning black hole of mass
M
is given the symbol
J
and its angular
momentum per unit mass is written
J/M
. The ratio
J/M
appears so often in
the analysis that it is given its own symbol:
a = J/M
. We call the constant
“
a
” the
angular momentum parameter
. Just as the angular momentum
L/m
of a stone orbiting a nonrotating black hole has the unit meter, so
does the angular momentum parameter
a
=
J/M
have the unit meter. In
what follows it is usually sufﬁcient to treat the angular momentum
parameter
a
as a positive scalar quantity.
Newman and others found the metric for a spinning black hole with net
electric charge (see references in Section 14 and also equation [51] for the
metric of a charged
nonspinning
black hole). The most general steadystate
black hole has mass, angular momentum, and electric charge. However,
we have no evidence that astronomical bodies carry sufﬁcient net electric
charge (which would ordinarily be rapidly neutralized) to affect the met
ric. If actual black holes are uncharged, then the Kerr metric describes the
most general stable isolated black hole likely to exist in Nature.
3 The Kerr Metric in the Equatorial Plane
For simplicity we are going to study spacetime and particle motion in the
equatorial plane
of a symmetric spinning black hole of angular momen
tum
J
and mass
M
. The equatorial plane is the plane through the center of
the spinning black hole and perpendicular to the spin axis.
Here is the
Kerr metric
in the equatorial plane, expressed
in what are
called
BoyerLindquist coordinates
. The angular momentum parameter
a
=
J/M
appears in a few unaccustomed places.
[1] dτ212M
r
–
dt24Ma
r
 dtdφdr2
12M
r
–a2
r2
+
–1
a2
r2
 2Ma2
r3
++
r2dφ2
–+=
Section 3 The Kerr Metric in the Equatorial Plane
F3
For the
nonrotating
black hole examined in Chapters 2 through 5, the
Schwarzschild metric describing spacetime on a plane is the same for
any
plane that cuts through the center of the black hole, since the Schwarz
schild black hole is spherically symmetric. The situation is quite different
for the spinning Kerr black hole; the metric [1] is correct
only
for neighbor
ing events that occur in the plane passing through the center of the black
hole and perpendicular to its axis of rotation. We choose the equatorial
plane because it leads to the simplest and most interesting results.
The time
t
in equation [1] is the “faraway time” registered on clocks far
from the center of attraction, just as for the Schwarzschild metric. In con
trast, for
a
> 0 the BoyerLindquist
r
coordinate does
not
have the simple
geometrical meaning that it had for the Schwarzschild metric. More on the
meaning of
r
in Sections 4 and 9. The metric [1] provides a
complete
description of spacetime in the equatorial plane outside the horizon of a
spinning uncharged black hole. No additional information is needed to
answer every possible question about its (nonquantum) properties and
(with the Principle of Extremal Aging) about orbits of free particles and
light pulses in the equatorial plane.
You say that the Kerr metric provides a complete
nonquantum
description of the spin
ning black hole. Why this reservation? What more do we need to know to apply
general relativity to quantum phenomena?
In answer, listen to Stephen Hawking as he discusses the “singularity” of spacetime at
the beginning of the Universe. A similar comment applies to the singularity inside any
black hole.
Suggestion:
As you go along, check the units of all equations, the equations
in the project and also your own derived equations. An equation can be
wrong if the units are right, but the equation cannot be right if the units
are wrong!
The general theory of relativity is what is called a classical theory.
That is, it does not take into account the fact that particles do not
have precisely deﬁned positions and velocities but are “smeared
out” over a small region by the uncertainty principle of quantum
mechanics that does not allow us to measure simultaneously both
the position and the velocity. This does not matter in normal
situations, because the radius of curvature of spacetime is very
large compared to the uncertainty in the position of a particle.
However, the singularity theorems indicate that spacetime will
be highly distorted, with a small radius of curvature at the
beginning of the present expansion phase of the universe [or at the
center of a black hole]. In this situation, the uncertainty principle
will be very important. Thus, general relativity brings about its
own downfall by predicting singularities. In order to discuss the
beginning of the universe [or the center of a black hole], we need a
theory that combines general relativity with quantum mechanics.
—Stephen Hawking
F4
P
ROJECT
F The Spinning Black Hole
The Kerr metric has four central new features that distinguish it from the
Schwarzschild metric.
The ﬁrst new feature of the Kerr metric is a new
r
value for the horizon
.
In the Schwarzschild metric, the coefﬁcient of
dr
2
is 1/(1 –
2M/r
). This coef
ﬁcient increases without limit at the Schwarzschild horizon,
r
H
= 2
M
. For
the Kerr metric, in contrast, the horizon—the point of no return—has an
r

value that depends on the value of the angular momentum parameter
a.
(Note:
A true proof that a horizon exists requires the demonstration that
worldlines can run through it only in the inward direction, not outward.
For the corresponding proof for the nonspinning black hole, see Project B,
pages B14–15. Our choice here of the horizon at the place where the coef
ﬁcient of
dr
2
blows up is an intuitive, but yet correct, selection.)
QUERY 1 Equatorialplane Kerr metric in the limit of zero angular momentum. Show
that for zero angular momentum (a = J/M = 0), the Kerr metric, equation [1],
reduces to the Schwarzschild metric (equation [A] in Selected Formulas at the
end of this book).
QUERY 2 Motion stays in plane. Make an argument from symmetry that a free object
that begins to orbit a spinning black hole in the equatorial plane will stay in
the equatorial plane.
Do Spinning Black Holes Power Quasars?
In contrast to dead solitary black holes, the most powerful
steady source of energy we know or conceive or see in all
the universe may be powered by a spinning black hole of
many millions of solar masses, gulping down enormous
amounts of matter swirling around it. Maarten Schmidt,
working at the Palomar Mountain Observatory in 1956, was
the ﬁrst to uncover evidence for these quasistellar
objects, or quasars, starlike sources of light located not bil
lions of kilometers but billions of lightyears away. Despite
being far smaller than any galaxy, the typical quasar man
ages to put out more than a hundred times as much energy
as our entire Milky Way with its hundred billion stars.
Quasars—unsurpassed in brilliance and remoteness—can
justly be called lighthouses of the heavens.
Observation and theory have come together to explain in
broad outline how a quasar operates. A spinning black hole
of some hundreds of millions of solar masses, itself perhaps
built by accretion, accretes more mass from its surroundings.
The incoming gas, and stars converted to gas, does not fall
in directly, any more than the water rushes directly down the
bathtub drain when the plug is pulled. This gas, as it goes
round and round, slowly makes its way inward to regions of
everstronger gravity. In the process it is compressed and
heated and ﬁnally breaks up into positive ions and electrons,
which emit copious amounts of radiation at many wave
lengths. The infalling matter brings with it some weak
magnetic ﬁelds, which are also compressed and powerfully
strengthened. These magnetic ﬁelds link the swirling elec
trons and ions into a gigantic accretion disk. Matter little by
little makes its way to the inner boundary of this accretion
disk and then, in a great swoop, falls across the horizon into
the black hole. During that last swoop, hold on the particle is
relinquished. Therefore, the chance is lost to extract as
energy the full 100 percent of the mass of each infalling bit
of matter. However, magnetic ﬁelds do hold on to the ions
effectively enough and long enough to extract, as radiant
energy, several percent of the mass. In contrast, neither
nuclear ﬁssion nor nuclear fusion is able to obtain a conver
sion efﬁciency of more than a fraction of 1 percent. No one
has ever seen evidence for a more effective process to con
vert bulk matter into energy than accretion into a spinning
black hole, and no one has ever been able to come up with a
more feasible scheme to explain the action of quasars.
See Section 11 for more details.
Section 4 The Kerr Metric for Extreme Angular Momentum
F5
Unless stated otherwise, when we say “the horizon” we refer to equation
[2] with the plus sign.
Research note:
Choosing the minus sign in equation [2] leads to a second
horizon that is
inside
the outer, plussign horizon. This inner horizon is
called the
Cauchy horizon
. Theoretical research shows that spacetime is
stable (correctly described by the Kerr metric) immediately inside the
outer horizon and most of the way down to the inner (Cauchy) horizon.
However, near the Cauchy horizon, spacetime becomes unstable and
therefore is
not
described by the Kerr metric. At the Cauchy horizon is
located the socalled
massinﬂation singularity
described in the box on page
B5. The presence of the massinﬂation singularity at the Cauchy horizon
bodes ill for a diver wishing to experience in person the region between
the outer horizon and the center of a rotating black hole. It is delightful to
read in a serious theoretical research paper a sentence such as the follow
ing: “Such . . . results strongly suggest (though they do not prove) that
inside a black hole formed in a generic collapse, an observer falling
toward the inner [Cauchy] horizon should be engulfed in a wall of (classi
cally) inﬁnite density immediately after seeing the entire future history of
the outer universe pass before his eyes in a ﬂash.” (Poisson and Israel)
4 The Kerr Metric for Extreme Angular Momentum
In this project we want to uncover the central features of the spinning
black hole with minimum formalism. The equations become simpler for
the case of a black hole that is spinning at the maximum possible rate.
A black hole spinning at the maximum rate derived in Query 4 is called an
extreme Kerr black hole
.
How fast are existing black holes likely to spin;
how “live” are they likely to be? Listen to Misner, Thorne, and Wheeler
QUERY 3 Radial coordinate of the horizon. Show that for the spinning black hole,
the coefﬁcient of dr2 increases without limit at the rvalue:
[2]
Look ﬁrst at the case with the plus sign. What value does rH have when
a = 0? For a spinning black hole, is the value of rH greater or less than the
corresponding rvalue for the Schwarzschild horizon?
rHMM
2a2
–()
12⁄
±=
QUERY 4 Maximum value of the angular momentum. How “live” can a black hole be?
That is, how large is it possible to make its angular momentum parameter
a = J/M? Show that the largest value of the angular momentum parameter, a,
consistent with a real value of rH is a = M. This maximum value of the angular
momentum parameter a is equivalent to angular momentum J = M2. What
happens to the inner (Cauchy) horizon in this case?
F6 PROJECT F The Spinning Black Hole
(page 885): “Most objects (massive stars; galactic nuclei; . . .) that can col
lapse to form black holes have so much angular momentum that the holes
they produce should be ‘very live’ (the angular momentum parameter
a = J/M nearly equal to M; J nearly equal to M2).”
The metric for the equatorial plane of the extremespin black hole results if
we set a = M in equation [1], which then becomes
[3. extreme Kerr]
Note how the denominator of the dr2 term in the Kerr metric differs in two
ways from the dr2 term in the Schwarzschild metric: here the denominator
is squared and also contains M/r instead of 2M/r.
Equation [3] has been simpliﬁed by deﬁning
[4. extreme Kerr]
The form R2dφ2 of the last term on the right side of equation [3] tells us
that R is the reduced circumference for extreme Kerr spacetime. That is,
the value of R is determined by measuring the circumference of a station
ary ring in the equatorial plane concentric to the black hole and dividing
this circumference by 2π. This means that r is not the reduced circumfer
ence but has a value derived from equation [4]. Finding an explicit
expression for r in terms of R requires us to solve an equation in the third
power of r, which leads to an algebraic mess. Rather than solving such an
equation, we carry along expressions containing both R and r. Note from
equation [4] that R is not equal to r even for large values of r, although the
percentage difference between R and r does decrease as r increases.
QUERY 5 Maximum angular momentum of Sun? A recent estimate of the angular
momentum of Sun is 1.91 × 1041 kilogram meters2 per second (see the ref
erences). What is the value of the angular momentum parameter a = J/M
for Sun, in meters? (Hint: Divide the numerical value above by Mkg, the
mass of Sun in kilograms, to obtain an intermediate result in units of
meter2/second. What conversion factor do you then use to obtain the
result in meters?) What fraction a/M is this of the maximum possible value
permitted by the Kerr metric?
dτ212M
r
–
dt24M2
r
dtdφdr2
1M
r
–
2
–R2dφ2
–+=
R2r2M22M3
r
++≡
QUERY 6 Limiting values of R. What is rH, the value of r at the horizon of an
extreme spinning black hole? What is RH, the value of R at the horizon?
Find the approximate range of rvalues for which the value of R differs
from the value of r by less than one part in a million.
Section 4 The Kerr Metric for Extreme Angular Momentum F7
Now move beyond the new rvalue for the horizon—the ﬁrst new feature
of the Kerr metric—to the second new feature of the Kerr metric, which is
the presence of the product dtdφ of two different spacetime coordinates,
called a cross product. The cross product implies that coordinates φ and t
are intimately related. In the following section we show that the Kerr met
ric predicts frame dragging. What does “frame dragging” mean? Near
any center of attraction, radial rocket thrust is required to keep a station
ary observer at a ﬁxed radius. Near a spinning black hole an additional
tangential rocket thrust is required during initial placement of an object in a
stationary position, a position from which the ﬁxed stars do not appear to
move overhead. (See box page F20.) One might say that spacetime is
swept around by the rotating black hole: spacetime itself on the move!
Unless otherwise noted, everything that follows applies to the equatorial
plane around an extreme Kerr black hole.
5 The Static Limit
The third new feature of the Kerr metric is the presence of a socalled
static limit. The horizon of a rotating black hole lies at an rvalue less than
2M (equation [2] with the plus sign). The horizon is where the metric coef
ﬁcient of dr2 blows up. In contrast, for the equatorial plane, the coefﬁcient
of dt2, namely, (1 – 2M/r), goes to zero at r = 2M, just as it does in the
Schwarzschild metric for a nonrotating black hole. The rvalue r = 2M in
the equatorial plane at which the coefﬁcient of the dt2 term goes to zero is
called the static limit. An examination of equations [3] and [1] shows that
the expression for the static limit in the equatorial plane is the same what
ever the value of the angular momentum parameter a, namely
[5]
The static limit gets its name from the prediction that for radii smaller than
rS (but greater than that of the horizon rH) an observer cannot remain at
rest, cannot stay static. The space between the static limit and the horizon
is called the ergosphere. Inside the ergosphere you are inexorably
dragged along in the direction of rotation of the black hole. Not even a
tangential rocket allows you to stand at one ﬁxed angle φ. For you the
ﬁxed stars cannot remain at rest overhead. In principle, a small amount of
frame dragging is detectable near any spinning astronomical object. An
experimental Earth satellite (Gravity Probe B), now under construction at
Stanford University, will measure the extremely small framedragging
effects predicted near the spinning Earth. Inside the static limit of a rotat
QUERY 7 More general Ra. Consider the more general case of arbitrary angular
momentum parameter a given in equation [1]. What is the expression for
R2 (call it Ra2) in this case? What is the value of Ra in the limiting case of
the nonspinning black hole?
rS2M=
F8 PROJECT F The Spinning Black Hole
ing black hole, in contrast, the frame dragging is irresistible, as will be
described on the following page.
The Kerr metric for three space dimensions—not discussed in this book—
reveals that the horizon has a constant rvalue in all directions (is a sphere)
while the static limit has cusps at the poles. Figure 1 shows this result. This
ﬁgure is drawn using the Kerr bookkeeper (BoyerLindquist) rcoordinate,
which shows only one possible way to view these structures. When
Figure 1 is plotted in terms of the reduced circumference R/M instead of
r/M, then the radius of the horizon is greater in the equatorial plane than
along the axis of rotation, giving the horizon the approximate shape of a
hamburger bun.
Figure 1 Computer plot of the crosssection of an extreme black hole showing the static limit
and horizon using the Kerr bookkeeper (BoyerLindquist) coordinate r (not R). From inside the
horizon no object can escape, even one traveling at the speed of light. Between the horizon
and the static limit lies the ergosphere, shaded in the ﬁgure. Within this ergosphere
everything—even light—is swept along by the rotation of the black hole. Inside the
ergosphere, too, a stone can have a negative total energy (Section 10).
QUERY 8 Reduced circumference of the static limit. For the extreme black hole,
ﬁnd an expression for RS, the reduced circumference of the static limit, in
the equatorial plane.
QUERY 9 Displaying the spinning black hole from above. Draw a crosssection of
the extreme black hole in the equatorial plane. That is, display the static
limit and horizon in bookkeeper coordinates on a plane cut through the
horizontal axis of Figure 1, as if viewing that ﬁgure downward along the
vertical axis from above. Label the static limit, horizon, and ergosphere
and put in expressions for their radii.
Section 4 The Kerr Metric for Extreme Angular Momentum F9
Now look more closely at the nature of the static limit in the equatorial
plane. Examine the Kerr metric for the case of a light ﬂash moving initially
in the φ direction (dr = 0). (Only the initial motion in the equatorial plane
will be tangential; later the ﬂash may be deﬂected radially away from the
tangential direction.) Because this is light, the proper time is zero between
adjacent events on its path: dτ = 0. Make these substitutions in the metric
[3], divide through by dt2, and rearrange to obtain
[6. light, dr = 0]
Equation [6] is quadratic in the angular velocity dφ/dt.
Look closely at this expression at the static limit, namely, where r = 2M
and R2 = 6M2. The two solutions are
[8. light, dr = 0]
To paraphrase Schutz (see references), the second solution in [8] represents
light sent off in the same direction as the hole is rotating. The ﬁrst solution
says that the other light ﬂash—the one sent “backward”—does not move
at all as recorded by the faraway bookkeeper. The dragging of orbits has
become so strong that this light cannot move in the direction opposite to
the rotation! Clearly, any material particle, which must move slower than
light, will therefore have to rotate with the hole, even if it has an angular
momentum arbitrarily large in the sense opposite to that of hole rotation.
The static limit creates a difﬁculty of principle in measuring the reduced
circumference R, deﬁned by equation [4] on page F6. According to that
deﬁnition, one measures R by laying off the total distance—the circumfer
ence—around a stationary ring in the equatorial plane concentric to the
black hole, then dividing that circumference by 2π to ﬁnd the value of R.
But inside the static limit no such ring can remain stationary; it is inevita
R2dφ
dt

24M2
r
dφ
dt

–1
2M
r
–
–0=
QUERY 10 Tangential motion of light. Solve equation [6] for dφ/dt. Show that the
result has two possible values (simpliﬁed in equation [11], page F10):
[7. light, dr = 0]
dφ
dt
2M2
rR2
2M2
rR2
1r2R2
4M4
 12M
r
–
+
12⁄
±=
dφ
dt
0=and
dφ
dt
4M2
rR2
1
3M
==
QUERY 11 Light dragging in the ergosphere. Show that inside the ergosphere (r such
that rH < r < rS), light launched in either tangential direction in the equa
torial plane moves in the direction of rotation of the black hole as
recorded by the faraway bookkeeper. That is, show that the initial
tangential angular velocity dφ/dt is always positive.
F10 PROJECT F The Spinning Black Hole
bly swept along in a tangential direction, even if we ﬁre powerful rockets
tangentially trying to keep it stationary. Thus, for the present, we have no
practical deﬁnition for R inside the static limit. We will overcome this difﬁ
culty in principle in Section 9.
To anticipate a later result, we mention here the fourth new feature of the
Kerr metric, which is analyzed further in Sections 10 and 11.
The fourth new feature of the Kerr metric is available energy. No net
energy can be extracted from a nonspinning black hole (except for the
quantum “Hawking radiation,” page 24, which is entirely negligible for
starmass black holes). For this reason, the nonspinning black hole carries
the name dead. In contrast, energy of rotation is available from a spinning
black hole, which therefore deserves its name live. See Section 12.
6 Radial and Tangential Motion of Light
For light (dτ = 0) moving in the tangential direction (dr = 0), we call the
tangential velocity Rdφ/dt as recorded by the Kerr bookkeeper. From
equation [7], this tangential velocity is given by
[10. light, dr = 0]
The second term on the right side of [10] can be simpliﬁed by substituting
for R2 in the numerator from equation [4]. (Trust us or work it out for
yourself!) Equation [10] becomes
[11. light, dr = 0]
QUERY 12 Radial motion of light. For light (dτ = 0) moving in the radial direction
(dφ = 0), show from the metric that
[9. light, dφ = 0]
Show that this radial speed goes to zero at the static limit and is imaginary
(therefore unreal) inside the ergosphere. Meaning: No purely radial
motion is possible inside the ergosphere. See Figure 2.
dr
dt
 1 M
r
–
12M
r
–
12⁄
±=
Rdφ
dt
2M2
rR
2M2
rR
1r2R2
4M4
 12M
r
–
+
12⁄
±=
Rdφ
dt
2M2
rR
rM–
R

±=
Section 7 Wholesale Results, Extreme Kerr Black Hole F11
The radial and tangential velocities of light in equations [9] and [11] are
bookkeeper velocities, reckoned by the Kerr bookkeeper using the coordi
nates r and φ and the faraway time t. Nobody measures the Kerr
bookkeeper velocities directly, just as nobody measured directly book
keeper velocities near a nonspinning black hole (Chapters 3 through 5).
Figure 2 shows the radial and tangential bookkeeper velocities of light for
the extreme Kerr metric. Note again that these plots show the initial veloc
ity of a light ﬂash launched in the various directions. After launch, a
radially moving light ﬂash may be dragged sideways, or a tangentially
moving ﬂash may be deﬂected inward.
7 Wholesale Results, Extreme Kerr Black Hole
Now suppose that you have never heard of the Kerr metric and someone
presents you with the “anonymous” metric [3] (which we know to be the
metric for the extreme Kerr black hole) plus the deﬁnition of R:
[3]
[4]
You say to yourself, “This equation is just a crazy kind of mixedup
Schwarzschildlike metric, with a nutty denominator for the dr2 term, a
crossterm in dtdφ, and R2 instead of r2 as a coefﬁcient for dφ2. Still, it’s a
metric. So let’s try deriving expressions for angular momentum, energy,
and so forth for a particle moving in a region described by this metric in
analogy to similar derivations for the Schwarzschild metric.” So saying,
QUERY 13 Light dragging at the horizon. What happens to the light dragging at the
horizon (rH given by equation [2] with the plus sign and a = M, and RH
derived in Query 6)? Show that at the horizon the initial tangential rota
tion dφ/dt for light has a single value whichever way the pulse is launched.
Show that the bookkeeper initial tangential velocity Rdφ/dt for this light
at the horizon has the value shown in Figure 2.
QUERY 14 Lockedin motion? (Optional) Kip Thorne says, “I guarantee that, if you
send a robot probe down near the horizon of a spinning hole, blast as it
may it will never be able to move forward or backward [in either tangen
tial direction] at any speed other than the hole’s own spin speed. . . .”
What evidence do equation [11] and Figure 2 give for this conclusion?
What is “the hole’s own spin speed”? (See Kip S. Thorne, Black Holes and
Time Warps, W. W. Norton & Co., New York, 1994, page 57.)
dτ212M
r
–
dt24M2
r
dtdφdr2
1M
r
–
2
–R2dφ2
–+=
R2r2M22M3
r
++≡
F12 PROJECT F The Spinning Black Hole
Figure 2 Computer plot of bookkeeper radial and tangential velocities of light near an extreme Kerr
black hole (a = J/M = M). Note that as r/M becomes large, the different bookkeeper velocities all
approach plus or minus unity. Note also that purely radial motion of light is not possible inside the static
limit. Important: These are initial velocities of light just after launch in the given direction. After launch,
the light will generally change direction. For the case of a nonrotating black hole, see Figures 6 and 7,
pages B18–19.
Section 8 Plunging: The “StraightIn Spiral” F13
you use the Principle of Extremal Aging and other methods of Chapters 2
through 5 to derive expressions similar to results in those chapters and
enter them in the right hand column of Table 1.
Notes: (1) We limit ourselves to the equatorial plane. (2) Outside the static
limit we can still set up stationary spherical shells (which we have limited
to stationary rings in the equatorial plane). However, equation [21] with
dφ/dτ = 0 tells us that a stationary ring has negative angular momentum. So
during construction we need to provide an initial tangential rocket blast to
give negative angular momentum to the ring structure in order to make it
stationary. (See box page F20.)
8 Plunging: The “StraightIn Spiral”
Near the nonrotating black hole, the simplest motion was radial plunge
(Chapter 3). What is the simplest motion near a spinning black hole? By
analogy, examine the motion of a stone dropped from rest at a great dis
tance which thereafter falls inward, maintaining zero angular momentum.
Table 1 Comparison of results of nonspinning and extremespin black holes
Quantity Nonspinning Schwarzschild
black hole Extremespin Kerr black hole
(“shell” = stationary ring outside static limit)
Deﬁne r and RReduced circumference =
[12]
Reduced circumference R given by:
[13]
Shell time vs.
faraway time:
(gravitational
red shift) [14] [15. stationary]
drshell vs. dr [16] [17. stationary]
Energy
(constant of
the motion) [18] [19]
Angular
momentum
(constant of
the motion) [20] [21]
rcircumference of shell()
2π

≡R2r2M22M3
r
++≡
dtshell 12M
r
–
12⁄dt=dtshell 12M
r
–
12⁄dt=
drshell 12M
r
–
12⁄–dr=drshell 1M
r
–
1– dr=
E
m
1
2M
r
–
dt
dτ

=E
m
1
2M
r
–
dt
dτ
 2M2
r
dφ
dτ

+=
L
m
r2dφ
dτ

=L
m
R2dφ
dτ
2M2
r
dt
dτ

–=
QUERY 15 Energy and angular momentum as constants of the motion. Derive
entries [19] and [21] in Table 1 for energy and angular momentum of a
free object moving in the equatorial plane of an extreme Kerr black hole.
F14 PROJECT F The Spinning Black Hole
Equation [22] gives the remarkable result that a particle with zero angular
momentum nevertheless circulates around the black hole! This result is
evidence for our interpretation that the black hole drags nearby spacetime
around with it. Figure 3 shows the trajectory of an inward plunger with
zero angular momentum, as calculated in what follows.
Let’s see if we can set up the equations to describe a stone that starts at rest
far from a rotating black hole and moves inward with zero angular
momentum. At remote distance, in ﬂat spacetime, the stone has energy
E/m = 1. It keeps the same energy as it falls inward. From equation [19] in
Table 1,
[23]
Equations [22] and [23] are two equations in the four unknowns dr, dt, dτ,
and dφ. A third equation is the metric [3] for the extremespin black hole.
With these three independent equations, we can eliminate three of the four
unknowns to ﬁnd a relation between any two remaining differentials. We
QUERY 16 No angular momentum. But angular motion! Set angular momentum [21]
equal to zero and verify the following equation:
[22. L = 0]
dφ
dt
2M2
rR2
=
Figure 3 Computer plot: Kerr map (Kerr bookkeeper plot) of the trajectory in space of a stone
dropped from rest far from a black hole (therefore with zero angular momentum). According to
the faraway bookkeeper, the stone spirals in to the horizon at r = M and circulates there forever.
E
m
1 1
2M
r
–
dt
dτ
 2M2
r
dφ
dτ

+==
Section 8 Plunging: The “StraightIn Spiral” F15
choose to solve for the quantities dr and dφ, because we want to draw the
trajectory, the Kerr map. Don’t bother doing the algebra—it is a mess.
After substituting equation [4] for R2 into the result, one obtains the rela
tion between dr and dφ:
[24. L = 0]
The computer has no difﬁculty integrating and plotting this equation, as
shown in Figure 3. Since we used the Kerr bookkeeper angular velocity
[22], the resulting picture is that of the Kerr bookkeeper. For her, the zero
angularmomentum stone spirals around the black hole and settles down
in a tight circular path at r = M, there to circle forever.
Remember that for the nonspinning black hole an object plunging inward
slows down as it approaches the horizon, according to the records of the
Schwarzschild bookkeeper. For both spinning and nonspinning black
holes, the infalling stone with L = 0 never crosses the horizon when
clocked in faraway time.
The observer who has fallen from rest at inﬁnity has quite a different per
ception of the trip inward! For her there is no pause at the horizon; she has
a quick, smooth trip to the center (assuming that the Kerr metric holds all
the way to the center!). An algebra orgy similar to the previous one gives a
relation between dr and dτ, where dτ is the wristwatch time increment of
the infaller:
[25. L = 0]
dr rM–
r
 r5
2M3
r4
M2
– r3
M
r2
–Mr
2
++
12⁄dφ=
rM–()
2
r
r3
2M3
r
2M
+12⁄dφ=
QUERY 17 Final circle according to the bookkeeper. Verify that dr goes to zero (that is,
r does not change) once this stone reaches the horizon.
QUERY 18 Bookkeeper speed in the “ﬁnal circle.” Guess: At the horizon, what is the
value of the tangential speed Rdφ/dt of the stone dropped from rest at
inﬁnity, as measured by the Kerr bookkeeper? Now derive a formula that
gives you this numerical value. Was your guess correct?
dr
dτ

22Mr34M2r2
–4M3r4M4
–2M5
r
++
r2rM–()
2
=
2M
r
 1M2
r2
+
=
F16 PROJECT F The Spinning Black Hole
Figure 4 compares the magnitude of the square root of this expression
with the magnitude of the velocity of the stone dropped from rest at a
great distance in the Schwarzschild case (equation [32], page 322):
[26. L = 0 Schwarzschild]
Both equations [25] and [26] show bookkeeper radial components of speed
greater than unity in the region of small radius. The resulting speed is
even more impressive when one adds the tangential component of motion
forced on the diver descending into the spinning black hole (Figure 3).
Does such motion violate the “cosmic speed limit” of unity for light? A
similar question is debated for the Schwarzschild black hole in Section 3 of
Project B, Inside the Black Hole, pages B6–12.
Research note: When applied inside the horizon, equation [25] assumes that
the Kerr metric correctly describes spacetime all the way to the center of
the extreme Kerr black hole. This may not be the case. See the box Egg
beater Spacetime? on page B5.
9 Ring Riders
Equation [22] on page F14 describes the angular rotation rate ω of an in
falling stone that has zero angular momentum:
[27. L = 0]
In some way, ω in this equation describes the angular rate at which space
is “swept along” by the nearby spinning black hole. What happens if we
“go with the ﬂow,” moving tangentially at angular rate ω given by this
equation? How do we guarantee that our rotation is at the correct rate to
yield zero angular momentum? What happens to us at the static limit?
To pursue these ideas, we envision a set of nested rings in the equatorial
plane and concentric to the black hole (Figure 5). Each of these rings
revolves at an angular rate given by equation [27] as reckoned by the Kerr
bookkeeper. Rings at different values of r rotate at different angular rates.
The result of this construction is a set of observers in the equatorial plane
whom we call ring riders. A ring rider is an observer who stands at rest on
one of the rotating rings with zero angular momentum. In times past, ring
riders were known as locally nonrotating observers, but now the custom
ary name is zero angular momentum observers or ZAMOs. Each ring
rider, like each shell observer in Schwarzschild geometry, is subject to a
gravitational acceleration directed toward the center of the black hole. In
both cases the radially inward gravitational acceleration becomes inﬁnite
at the horizon, destroying any possible circumferential ring structures at
or inside the horizon. According to ring rider measurements, light has
speed unity, the same speed in both tangential directions, as we shall see.
dr
dτ
 2M
r

12⁄
–=
dφ
dt
ω≡ 2M2
rR2
=
Section 9 Ring Riders F17
Figure 4 Computer plot: comparison of radial components of plunge velocities experienced by
different infallers who drop from rest (so with L = 0) at a great distance from Schwarzschild and
extreme Kerr black holes.
Figure 5 Kerr map (perspective plot) of rings surrounding a spinning black
hole. The rings rotate in the same direction as the black hole but at angular
rates that differ from ring to ring, as given by equation [27], page F16.
QUERY 19 Ring slippage. Will the inner rings rotate with larger or smaller angular
velocity than the rings farther out? Justify your choice.
QUERY 20 Ring speed according to the bookkeeper. What are the units of ω in equa
tion [27]? What is the numerical value of the bookkeeper speed Rω for
each of the rings r = 100M, r = 10M, r = 2M, and r = M? Express each
answer as a fraction of the speed of light.
QUERY 21 Does rain fall vertically? Present an argument that a stone dropped from
rest starting at a great radial distance falls vertically past the rider on
every zero angular momentum ring. Guess: Is the same true if the stone is
ﬂung radially inward from a great distance? Guess: What about light?
F18 PROJECT F The Spinning Black Hole
Can we write a simpliﬁed metric for the rider on the zero angular momen
tum ring? Probably not for events separated radially because of shearing,
the slippage between adjacent rings. So limit attention to events separated
tangentially along the ring. According to the remote observer, each ring
revolves with an angular velocity ω given by equation [27]. Deﬁne an
azimuthal angle increment dφring measured along the ring with respect to
some zero mark on the ring. Let an object move uniformly along the ring.
Then, as recorded by the Kerr bookkeeper, the object’s total angular
velocity dφ/dt is the angular velocity dφring/dt with respect to the ring
added to the bookkeeper angular velocity ω of the ring, or
[28]
The positive direction of both dφ and dφring is in the direction of rotation of
the black hole.
Now think of two events separated by the angle dφring along the ring and
at faraway time separation dt. Then the angular separation dφ between
these two events for the faraway observer is, from [27] and [28],
[29. dr = 0]
The metric [3] with the same limitation to motion along the ring (dr = 0) is
[30. dr = 0]
dφ
dt
dφring
dt
 ω+=
dφdφring 2M2
rR2
dt+=
dτ212M
r
–
dt24M2
r
dtdφR2dφ2
–+=
QUERY 22 New metric for the ring. Substitute equation [29] into [30]. Show ﬁrst that
the coefﬁcient of the crossterm in dtdφring is equal to zero. Second, collect
terms in dt2 and dφring2 to show that the resulting metric is given by equa
tion [31] for motion along the ring. Hint: Group over a common
denominator r2R2, then substitute in the numerator for R2 (equation [4]):
[31. dr = 0]
QUERY 23 Time on the ring rider clock. A ring rider is at rest on a (zero angular
momentum) ring. Show that the time dtring between ticks on his clock and
the time dt between ticks on the faraway clock are related by the equation
[32. dr = dφring = 0]
Show that, with this substitution, the metric for dr = 0 becomes
[33. dr = 0]
dτ2r2
R2
1M
r
–
2dt2R2dφring
2
–=
dtring rM–
R
 dt=
dτ2dtring
2R2dφring
2
–=
Section 9 Ring Riders F19
In brief, for nearby events along the ring the metric [33] looks like that of
ﬂat spacetime. But spacetime is not ﬂat near a spinning black hole. Equa
tion [33] describes a local frame useful only in analyzing events that are
limited in space and time and for which the “local gravitational force” in
the radial direction can be neglected. However, this equation is useful for
analyzing events that occur near to one another along the same ring.
Now (ﬁnally!) we can deﬁne the reduced circumference R everywhere
external to the horizon, even inside the static limit. A ring rider measures
the circumference of his ring and then divides this circumference by 2π.
[34]
The result is a formal deﬁnition of the reduced circumference R for this
zero angular momentum ring. The value of R, along with the value of r
from equation [4], is then stamped on each rotating ring for all to see and
everyone to use. The same values of R and r can also be stamped on each
stationary ring that coincides with an already measured rotating ring. (Of
course, nonrotating rings can exist only outside the static limit.)
This set of zero angular momentum rotating rings can extend from the
horizon to inﬁnite radius. For a pair of events near one another along a
given ring, the proper distance dσ between them is given by the equation
[35. dr = dt = 0]
circumference of
freely rotating ring
2πR≡
dσRdφring
=
Figure 6 Silvered inner surface of rotating zero
angular momentum ring allows signaling along
ring with light ﬂashes. Lightpath segments
shown as straight will be curved. We assume
that each segment is arbitrarily short so that
light skims along close to the ring. Equal time
for light transmission in opposite directions
around the ring veriﬁes that the ring has zero
angular momentum (equation [27] and Query
24). Then light signals at locallymeasured speed
v ring = 1 allow synchronization of clocks around
the ring.
Ring
Light
paths
QUERY 24 Speed of light along the ring is unity for ring riders. From the metric [33],
show that the ring rider measures the speed of light along the ring to have
the magnitude unity. Is this value the same for motion of the light in both
directions along the ring (Figure 6)?
QUERY 25 Is motion along ring free or locked? Hard thought question; optional.
Equation [33] says that the ring rider on every ring can use special relativity
in analyzing motion along the ring. So he must be able to move freely back
and forth along the ring, even on a ring near the horizon. In contrast,
Query 14 asserts that the tangential motion near the horizon is rigidly
locked to the rotation of the black hole. Locked or free? What’s going on?
F20 PROJECT F The Spinning Black Hole
Tornado Without a Wind?
In what sense does spacetime near a spinning black hole
“circulate like a tornado”? In the vicinity of a spinning black
hole can we feel this rotation of spacetime?
For comparison, think about trying to stand still on the surface
of Earth as the circulating wind of a tornado passes over you.
You must lean into the wind or hang onto something ﬁxed
and solid in order to keep from being swept along in the
tangential direction in which the wind moves around the
center of the tornado. While standing still you feel a force in
the direction in which the wind blows.
Now suppose you stand still, at rest on a stationary ring
concentric to a spinning black hole. “Stationary” and “at
rest” mean that for you the remote stars do not move
overhead. (Such a stationary ring can be constructed only
outside the static limit.) You experience the same kind of
radiallyinward “gravitational force” you felt while resisting
the tornado on Earth. But is there an additional tangential
“tornado force” due to swirling spacetime, a “force” pushing
you in the direction of rotation of the black hole? We said so
in early printings of this book, but we made an error. Standing
at rest on a stationary ring concentric to the center of a black
hole, you experience NO sideways tangential force.
We were misled by the analogy to a tornado. To begin to
understand the difference between an Earthtornado and
spacetime near a spinning black hole, look at the graph
below. This graph plots the angular momentum of an object
AT REST outside an extreme spinning black hole as a function
of radius (equation at the end of this box). Note that for an
object at rest this angular momentum is negative. Key idea:
An object at rest already has the angular momentum
appropriate for that radius and does not need a tangential
force to maintain this angular momentum. The situation is
similar to that of a stone in a circular orbit around Earth; the
stone feels no force in the tangential direction and does not
need such a force in order to continue in its orbit with
constant angular momentum. The essential difference
between the Earthorbiting stone and the person standing on
a stationary ring outside a spinning black hole is that the
person has (negative) angular momentum while standing still.
So no tangential force is needed for you to stand still on a
stationary ring concentric to a spinning black hole.
The graph below also shows that the stationary observer has a
different value of angular momentum at each different radius.
(In our analysis we have neglected the difference in angular
momentum between the head and feet of a stationary
observer.) In order to change radius without moving sideways,
you must change your angular momentum. Changing angular
momentum does require a tangential force, but only
temporarily, while you are changing radius. For example,
suppose that you descend from a great distance along a radial
line ﬁxed with respect to the remote stars. As you move
radially inward along this ﬁxed line, the magnitude of your
(negative) angular momentum must increase. So to stay on
the ﬁxed radial path of your descent without being swept
sideways, you must ﬁre a rocket tangentially in order to
change your angular momentum, but only while you continue
to move inward. Once you stop descending, for example by
stepping onto a stationary ring, there is no sideways force and
no need for a tangential rocket to maintain your position.
By how much will you have to increase your negative angular
momentum as you descend along a ﬁxed radial line? The
answer comes from equation [21], page F13 with dφ/dτ = 0.
For an object at rest we have dτ = dtshell. Use equation [15] on
the same page to eliminate dt/dτ = dt/dtshell from equation
[21]. Divide through by M. The result is the equation
The ﬁgure below plots the quantity L/(mM) for a stationary
object as a function of r/M. Notice the result that the
magnitude of the (negative) angular momentum increases
without limit as you descend to the static limit at r/M = 2.
L
mM

2M
r
 dt
dτ

–
2M
r
 dt
dtshell

–
2M
r
 12M
r
–
12⁄–
–== =
Section 10 Negative Energy: The Penrose Process F21
10 Negative Energy: The Penrose Process
Roger Penrose devised a scheme for milking energy from a spinning black
hole. This scheme is called the Penrose process (see references). The Pen
rose process depends on the prediction that in some orbits inside the
ergosphere a particle can have negative total energy. Before we detail the
Penrose process, we need to describe negative total energy.
Negative Total Energy
What can negative total energy possibly mean? Negative energy is
nothing new. In Newtonian mechanics the potential energy of a particle at
rest far from Sun is usually taken to be zero by convention. Then a particle
at rest near Sun has zero kinetic energy and negative potential energy,
yielding a total energy less than zero. But in Newtonian mechanics the
zero point of potential energy is arbitrary, and all reasonable choices of
this zero point lead to the same description of motion. In contrast, special
relativity determines the rest energy of a free material particle in ﬂat
spacetime, setting its rest energy equal to its mass. So the arbitrary choice
of a zero point for energy is lost, and a particle far from a center of gravita
tional attraction always has an energy that is positive.
For Schwarzschild geometry the physical system differs from Newtonian.
A particle at rest near the horizon of a nonspinning black hole has zero
total energy (from equation [18] in Sample Problem 1, page 312). The
meaning? That it takes an energy equal to its rest energy (= m) to remove
this particle to rest at a large distance from the black hole (where it has the
energy m). As a consequence, if the particle drops into the black hole from
its stationary position next to the horizon, then the mass of the combined
blackholeparticle system (measured by a faraway observer, Figure 4,
page 311) does not change.
For Kerr geometry the physical system differs from that in Schwarzschild
geometry. A particle can have a negative energy near a spinning black
hole. The meaning? An energy greater than its rest energy (greater than m)
is required to remove such a particle to rest at a great distance from the
black hole. If the particle with negative energy is captured by the spinning
black hole, the black hole’s mass and angular momentum decrease. (See
Section 11.) This process can be repeated until the black hole has zero
angular momentum. Then it becomes a “dead” Schwarzschild black hole,
from which only Hawking radiation can extract energy (box page 24).
F22 PROJECT F The Spinning Black Hole
Strategy of the Penrose Process
The strategy of the Penrose process is similar to the following unethical
series of ﬁnancial transactions:
1. You and I decide to share our money. Our combined net worth
is positive.
2. I give you all my money, then borrow money from a bank and
give that to you as well. My bank debt is a negative entry on
my accounting balance sheet, so now my net worth is negative.
3. I declare bankruptcy and the bank is stuck with my debt.
The net result is the transfer of money from me and from the bank to you.
The bank provides the mechanism by which I can enter a state of negative
net worth.
The Penrose process is similar:
1. Starting at a distant radius, you and I together descend to a
position inside the ergosphere.
2. We are moving together tangentially inside the ergosphere in
the rotation direction. You push me away violently in a direction
opposite to the direction of rotation. This push puts you into a
new trajectory and puts me into a state of negative energy.
3. I drop into the black hole, which is stuck with my negative
energy. You continue in your new trajectory, arriving at a
distant radius with augmented energy.
The net result is the transfer of energy from me and from the black hole to
you. The spinning black hole provides the mechanism by which I can
enter a state of negative energy.
This entire strategy rests on the assumption that an object can achieve a
state of negative energy in the space surrounding a spinning black hole. Is
this assumption correct? Look again at expression [19] for the energy of a
stone near an extreme Kerr black hole:
[19]
Can this energy be negative? Start to answer this question by ﬁnding the
“critical” condition under which the energy is zero.
E
m
1
2M
r
–
dt
dτ
 2M2
r
dφ
dτ

+=
Section 10 Negative Energy: The Penrose Process F23
Figure 7 shows a plot of equation [37] along with plots of the positive and
negative tangential velocities of light from Figure 2. The tangential motion
of any particle must be bounded by the curves of tangential light motion.
(Inside the ergosphere even light moving “in the negative tangential direc
tion” moves forward, in the direction of rotation, according to the remote
bookkeeper.) In addition, equation [38] tells us that a particle with nega
tive energy must have a tangential velocity that lies below the heavy line
in the Figure 7. The shaded area in that ﬁgure conforms to these conditions
and shows the range of bookkeeper tangential velocities of a stone for
which the stone has negative energy. Next we turn our attention away
from the bookkeeper to what the ring rider measures (Query 29).
QUERY 26 Conditions for zero energy. Set E/m = 0 in equation [19] and show that the
resulting expression for the bookkeeper rate of change of angle is
[36]
Under what conditions is this angular velocity negative? positive?
QUERY 27 Bookkeeper tangential velocity for zero energy. Now assume that the
direction of motion is tangential and show that the bookkeeper velocity is
given by the expression
[37. dr = 0]
QUERY 28 Bookkeeper tangential velocities for negative energy. Now redo the analy
sis for the circumstance that the particle energy is negative. Show that the
condition is
[38. dr = 0]
dφ
dt

E=0
2Mr–
2M2
=
vbkkpr E=0 Rdφ
dt

E=0
R2Mr–()
2M2
==
vbkkpr E=neg R2Mr–()
2M2

<
The essence of newer physics
Of all the entities I have encountered in my life in physics, none
approaches the black hole in fascination. And none, I think, is a more
important constituent of this universe we call home. The black hole
epitomizes the revolution wrought by general relativity. It pushes to an
extreme—and therefore tests to the limit—the features of general
relativity (the dynamics of curved spacetime) that set it apart from special
relativity (the physics of static, “ﬂat” spacetime) and the earlier
mechanics of Newton. Spacetime curvature. Geometry as part of physics.
Gravitational radiation. All of these things become, with black holes, not
tiny corrections to older physics, but the essence of newer physics.
—John Archibald Wheeler
F24 PROJECT F The Spinning Black Hole
Figure 7 Computer plot showing bookkeeper tangential velocities of light (thin curves) and
tangential velocity of a stone with zero energy (thick curve), calculated using equation [37].
For r greater than 2M, the static limit, the particle cannot have zero energy (or negative
energy), because it would have to be moving in a negative tangential direction with a speed
greater than that of light in that direction. Only inside the ergosphere is this critical tangential
velocity possible. The shaded area shows the range of bookkeeper velocities for which the
stone has negative energy.
QUERY 29 Ring rider velocity for zero energy. Optional—messy algebra! A stone
moves tangentially along a rotating ring. For what values of the ring
velocity vring will the energy measured at inﬁnity be negative? Set E/m = 0
in equation [19]. Then make substitutions from equations [29] and [32] to
convert variables to dφring and dtring. Simplify using equation [4]. Show
that the result is
[39]vring, E =0 Rdφring
dtring
 1
2
r
M
1–
r
M

–==
Section 11 Quasar Power F25
Figure 8 plots equation [39] for ring velocity. Energy measured at inﬁnity
E/m will be negative for values of the ring velocity in the shaded region of
the plot. The range of ring velocities for which energy is negative depends
on the radius of the ring. Limiting cases are interesting: For a ring at the
static limit, motion backward along the ring with the speed of light leads
to zero energy. In contrast, for a ring near the horizon, any nonzero back
ward ring velocity, no matter how small, leads to negative energy.
11 Quasar Power
How much total energy can be extracted from a rotating black hole? In
general relativity, energy is a seamless whole; we cannot separate the
kinetic from the rest energy of a rotating object. Milking energy from a
rotating black hole changes its mass M along with its angular momentum
J. Analysis has identiﬁed a socalled irreducible mass Mirr that is the
smallest residual mass that results when all the angular momentum is
milked out of a rotating black hole. This irreducible mass Mirr of an
uncharged rotating black hole with angular momentum parameter
a = J/M is given by the equation
Figure 8 Computer plot showing the range of ring velocities (shaded region) for which the energy
measured at inﬁnity is negative. Negative ring velocity means motion along the ring in a direction
opposite to the direction of rotation of the black hole.
F26 PROJECT F The Spinning Black Hole
[40]
or equivalently
[41]
(Wald, page 326. Misner, Thorne, and Wheeler, page 913) This result was
discovered in Princeton by a 19yearold Athenian, Demetrios
Christodoulou, who never ﬁnished high school.
The ﬁnal state is a nonrotating Schwarzschild black hole of mass Mirr. The
net result is that a total energy M – Mirr has been extracted from an
uncharged rotating black hole.
From where do quasars get their power (box page F4)? Probably not
directly from the Penrose process (Section 10). One set of theories has the
quasar radiation coming from the gravitational energy of matter descend
ing toward the black hole as it orbits in an accretion disk. This matter
interacts with other matter in the disk in a complicated manner not well
understood. As debris in the disk moves toward the center, it is com
pressed along with its magnetic ﬁelds, is heated, and emits radiation
copiously. The net result is to convert its gravitational energy into radia
tion with high efﬁciency (high compared with nuclear reactions on Earth).
Note that the angular momentum of the black hole may actually be
increased during this process, depending on the initial angular momen
tum of the gas and clouds that swirl into the black hole. Another theory
derives the quasar output from the rotation energy of the black hole itself,
employing magnetic ﬁeld lines to couple black hole rotation energy to the
matter swirling around exterior to the horizon of the black hole. Such a
model leads to reduction in the rotation rate of the black hole.
Mirr
21
2
M2MM
2a2
–()
12⁄
+[]=
M2Mirr
2J2
4Mirr
2
+=
QUERY 30 Irreducible mass of extreme Kerr black hole. What is the irreducible mass
of an uncharged extreme Kerr black hole of mass M? What fraction of the
mass M of an extreme Kerr black hole can be extracted in the form of
energy by an advanced civilization (deﬁned as a civilization that can
accomplish any engineering feat not forbidden by the laws of Nature)?
QUERY 31 How much energy is available from the monster in our galaxy? Imagine
that the black hole of mass M = 2.6 x 106 MSun thought to exist at the cen
ter of our galaxy is an extreme Kerr black hole. How much total energy
can be milked from it? Express your answer as a multiple of the mass MSun
of our Sun.
Section 11 Quasar Power F27
The details of the emission of radiation by quasars may be complicated,
but the analysis in the present project provides the basis for an estimate of
the energy available for such processes.
Suppose that each element of the accretion disk circles the black hole at the
same rate of rotation as the local ring (an unrealistic assumption, since
rotating with the ring does not place the particle in a stable circular orbit).
As a given bit of debris moves inward, let it radiate energy sufﬁcient to
keep it at rest with respect to the local ring. For a bit of debris riding on the
ring, the time dτ between ticks on its wristwatch is the same as time dtring
between ticks of the ring clocks, since they are relatively at rest. Equation
[19] for the energy of this bit of debris then becomes
[42]
Now, the relation between ring time increments and bookkeeper time
increments is given by equation [32]:
[32]
QUERY 32 Quasar output. How much energy does a quasar put out each second?
Suppose that the quasar emits energy at a rate 100 times the emission
rate of our entire galaxy, which contains approximately 1011 stars similar
to our Sun. How much light energy does Sun put out per second? Lumi
nous energy from Sun pours down on the outer atmosphere of Earth at
a rate of 1370 watts per square meter (called the solar constant). From
the solar constant, estimate the energy production rate of our Sun in
watts, then of our galaxy, and then of a quasar that emits energy at 100
times the rate of our galaxy. This rate corresponds to the total conver
sion to energy of how many Sun masses per Earthyear?
E
m
1
2M
r
–
dt
dtring
2M2
r
dφ
dtring

+=
dtring rM–
R
 dt=
QUERY 33 Energy of stone riding on the ring. Substitute equation [32] into equa
tion [42], use equation [27] for the resulting dφ/dt, and collect terms
over a common denominator R(r – M) to obtain
[43. riding on ring]
For the expression for R2 in the numerator (only) substitute from equa
tion [4] and simplify to show that, for a stone riding on the ring,
[44. riding on ring]
E
m

12M
r
–
R24M4
r2
+
Rr M–()
=
E
m
rM–
R
=
F28 PROJECT F The Spinning Black Hole
Equation [44] is a simple expression but awkward to calculate because R is
a function of r (equations [4] and [13]). However, the computer has no dif
ﬁculty with these complications and plots the result in Figure 9.
Figure 9 does not lead to a correct estimate of the emission rate of a quasar.
In practice the rings do not rotate at the same rate as the accretion disk,
and the accretion disk itself is not a perfectly efﬁcient emitter of radiation.
A few percent of the rest energy of swirling particles may be emitted in the
form of radiation before they plunge across the horizon. Still, a few per
cent is far greater than the efﬁciency of nuclear reactors on Earth.
Figure 9 Computer plot of energy measured at inﬁnity for an object riding at rest with
respect to an L = 0 ring rotating at various radii around an extreme Kerr black hole. Example
shown by dot on the diagram: Stone riding at rest on a ring at r = 4M has total energy mea
sured at inﬁnity of E = 0.72m.
QUERY 34 Brilliant garbage. A blob of matter starts at rest at a great distance from
a black hole and gradually descends, riding at rest on each local ring and
emitting any change of energy as radiation. Now this matter rides on the
ring at r = 2M, the static limit. From Figure 9, determine what fraction of
its original rest energy it has radiated thus far. In principle, what is the
maximum fraction of its original rest energy that can be radiated before
it disappears inward across the horizon of the black hole?
Section 12 A “Practical” Penrose Process F29
12 A “Practical” Penrose Process
Using results of Sections 10 and 11, we can devise a “practical” Penrose
process by which energy can be milked from an extreme spinning black
hole. Actually, this process is “practical” only for an advanced civilization,
one that can accomplish any engineering feat not forbidden by the laws of
Nature. Outline of the strategy: Equal quantities of matter and antimatter
(say positrons and electrons united in positronium molecules, in bulk as
liquid positronium) are carried down to a rotating ring just outside the
horizon of an extreme Kerr black hole. There the matter and antimatter are
combined (annihilated) to create two oppositely moving pulses of electro
magnetic radiation. One pulse has negative energy and drops into the
black hole, robbing the black hole of some of its massenergy of rotation.
The other pulse has positive energy and escapes to a distant observer who
uses this energy for practical purposes. Now for the details.
The generalization of equation [44] for a particle moving along a rotating
ring is given by the equation
[45]
where vring = Rdφring/dtring. Equation [45] comes from applying a boatload
of algebra to equations [19], [29], and [32] and simplifying using equation
[4]. In addition, the derivation of [45] employs the following results of spe
cial relativity:
[46. special relativity]
where
[47. special relativity]
A ﬁnal transformation (time stretching) from special relativity tells us that
[48. special relativity]
where dτ is the wristwatch time of the stone moving along the ring.
Note that in equation [45], vring can be positive or negative, corresponding
to motion in either direction along the ring. Under some circumstances
this results in negative energy for the particle.
Now apply some simplifying circumstances. First keep constant the value
of Ering = mγring in equation [46] while letting m go to zero and vring go to
plus or minus one. The result signiﬁes a pulse of electromagnetic
radiation.
E
Ering
 rM–
R
 2M2
rR
vring
+=
Ering mγring
=
γring 1
1vring
2
–()
12⁄

≡
dtring γringdτ=
F30 PROJECT F The Spinning Black Hole
Second, apply equation [45] to a rotating ring very close to the horizon, as
a limiting case. In other words r —> M and R —> 2M. Equation [45]
becomes
[49. light ﬂash moving along ring as r –> M]
With these equations we can analyze the following idealized method for
milking energy from the black hole. Start with a mass m of matter and an
equal mass m of antimatter. Total mass: 2m.
Phase 1. Take the total load of mass 2m down to a position at rest on a ring
of zero angular momentum near to the horizon of an extreme Kerr black
hole, milking off the energy as it makes successive moves from rest on one
ring to rest on the next lower ring.
Phase 2. Combine the matter and antimatter at rest with respect to the
nearhorizon ring and direct the resulting light pulses in opposite direc
tions along the ring at the horizon.
Now the light ﬂash with negative energy drops across the horizon into the
black hole, thereby reducing the angular momentum (and mass) of the
spinning hole. In contrast, the light ﬂash with positive energy ﬂies out to a
great distance and its energy is employed for useful purposes.
13 Challenges
Nothing but algebra stands in the way of completing a full analysis of
orbits of stones and light in the equatorial plane of the extreme Kerr black
hole. The strategies required are analogous to those that led to similar
results for the nonrotating Schwarzschild black hole (Chapters 4 and 5).
E
Ering
 1±=
QUERY 35 Energy extracted in Phase 1. When Phase 1 is completed, how much
energy will have been milked off for use at a distant location?
QUERY 36 Energies of tangential light ﬂashes. Just after Phase 2 is completed, what
is the ring energy Ering of each of the two light pulses moving along the
ring as measured locally by a rider on that ring? What is the energy mea
sured at inﬁnity E of each of these ﬂashes?
QUERY 37 Total energy extracted. In summary, what is the total useful energy made
available to distant engineers as a result of this entire procedure? How
much mass/energy was the input for this process?
QUERY 38 Phase 1 reduction of angular momentum? Thought question, optional.
Does the energy extracted in Phase 1 by itself reduce the rotation rate of
the black hole? In answering, recall the analogous extraction of energy
from a nonrotating black hole (Exercise 6, Chapter 3).
Section 13 Challenges F31
• Computing the orbits of a stone from equations that relate dr and dφ to
the passage of wristwatch time dτ (similar to equations [21] and [22],
page 49).
• Carrying out qualitative descriptions of different classes of orbits
using an effective potential (similar to equation [32], page 418).
• Finding stable circular orbits, similar to those analyzed in the exercises
of Chapter 4. (Stable orbits allow a more realistic analysis of the behav
ior and energy of a particle orbiting with the accretion disk.)
• Predicting orbits of light as done in Chapter 5.
• Predicting details of life inside the horizon, comparable to the analysis
carried out for the Schwarzschild black hole in Project B, Inside the
Black Hole. Such an analysis is probably fantasy, since inside the
Cauchy horizon (choosing the minus sign in equation [2]) spacetime
appears to be unstable, hence not described by the Kerr metric, and
possibly lethal to incautious divers. (See Research Note, page F5.)
• Verifying that an extreme spinning black hole cannot accept additional
angular momentum. Can an object moving in the direction of rotation
of an extreme black hole cross the horizon and thus increase the angu
lar momentum of this structure which already has maximum angular
momentum?
Much of the complicated algebra that lies on the way to these outcomes
springs from the relation between the radius r and the reduced circumfer
ence R given by equation [4]. Once the algebra is mastered, results can be
plotted using a simple computer graphing program.
• For readers with unfettered ambition or for those skilled in the use of
computer algebra manipulation programs, the outcomes of this project
can be rederived for a black hole that spins with angular momentum
parameter a = J/M less than its maximum value. Start with the metric
[1] and use the more general reduced circumference Ra, deﬁned by the
equation (valid in the equatorial plane)
[50]
The resulting equations are easy to check at the extremes: They go to
the Schwarzschild limit when a —> 0 and to the expressions derived in
this project when a —> M.
• We have studied two important metrics: the Schwarzschild metric for
a nonspinning black hole and the Kerr metric for a spinning black
hole. You can apply the skills you have now mastered to analyze the
consequences of a third metric, the socalled ReissnerNordstrøm
metric for an electrically charged nonspinning black hole. For a pair of
events that occur near one another on a plane through the center of
Ra
2r2a22Ma2
r
++=
F32 PROJECT F The Spinning Black Hole
such a charged black hole, the ReissnerNordstrøm metric has the
form
[51]
Here Q is the electric charge of the black hole in units of length.
Good luck!
14 Basic References to the Spinning Black Hole
Introductory references to the spinning black hole
For the human and scientiﬁc story of the spinning black hole, read Kip S.
Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W.
Norton, New York, 1994, pages 46–54 and pages 286–299.
Bernard F. Schutz has an excellent analytic treatment in A First Course in
General Relativity, Cambridge University Press, New York, 1985, pages
294–305.
Chapter 33 of Misner, Thorne, and Wheeler’s Gravitation, W. H. Freeman
and Company, San Francisco (now New York), 1973, is very thorough,
with wonderful summary boxes, though beset with the mathematics of
tensors and differential forms. It is also approximately 30 years old.
Chapter 12 of Robert M. Wald’s General Relativity (University of Chicago
Press, Chicago, 1984) is authoritative and straightforward. The mathemat
ics is deep; you have to “read around the mathematics” to ﬁnd the
physical conclusions, which are clearly stated.
Section 12.7 of Black Holes, White Dwarfs, and Neutron Stars by Stuart L.
Shapiro and Saul A. Teukolsky (John Wiley, New York, 1983) pages 357–
364, covers the spinning black hole, mostly with algebra rather than ten
sors, and discusses orbits in some detail.
Steven Detweiler, editor, Black Holes: Selected Reprints, American Associa
tion of Physics Teachers, New York, 1982. This collection may be out of
print but is available in some physics libraries.
Original references to the spinning black hole
The ﬁrst paper: R. P. Kerr, “Gravitational Field of a Spinning Mass as an
Example of Algebraically Special Metrics,” Physical Review Letters, Volume
11, pages 237–238 (1963).
dτ212M
r
–Q2
r2
+
dt2dr2
12M
r
–Q2
r2
+
– r2dφ2
–=
Section 15 Further References and Acknowledgments F33
Choice of coordinate system can make thinking about the physics conve
nient or awkward. Boyer and Lindquist devised the coordinates that
illuminate our analysis in this project. Robert H. Boyer and Richard W.
Lindquist, “Maximum Analytic Extension of the Kerr Metric,” Journal of
Mathematical Physics, Volume 8, Number 2, pages 265–281 (February 1967).
See also Brandon Carter, “Global Structure of the Kerr Family of Gravita
tional Fields,” Physical Review, Volume 174, Number 5, pages 1559–1571
(1968).
For completeness, the Newman electrically charged black hole: E. T.
Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R.
Torrence, “Metric of a Rotating, Charged Mass,” Journal of Mathematical
Physics; Volume 6, Number 6, pages 918–919 (1965); also E. T. Newman
and A. I. Janis, “Note on the Kerr SpinningParticle Metric,” Journal of
Mathematical Physics, Volume 6, Number 6, pages 915–917 (1965).
The Penrose process, to help you milk the energy of rotation from the
spinning black hole: R. Penrose, “Gravitational Collapse: The Role of Gen
eral Relativity,” Revista del Nuovo Cimento, Volume 1, pages 252–276 (1969).
15 Further References and Acknowledgments
Initial quote: S. Chandrasekhar, Truth and Beauty: Aesthetics and Motivations
in Science, University of Chicago Press, 1987, pages 153–154.
Quote in reader objection, page F3: Stephen Hawking, Black Holes and
Baby Universes, Bantam Books, New York, 1993, pages 91–92.
Quote about Cauchy horizon, page F5: E. Poisson and W. Israel, “Inner
Horizon Instability and Mass Inﬂation in Black Holes,” Physical Review
Letters, Volume 63, Number 16, pages 1663–1666 (16 October 1989).
The value of the angular momentum of Sun (page F6) was provided by
Douglas Gough, private communication.
Quote on page F23 from John Archibald Wheeler with Kenneth Ford,
Geons, Black Holes, and Quantum Foam, A Life in Physics, W. W. Norton &
Company, New York, 1998, page 312.
Stephan Jay Olson suggested using light ﬂashes as part of the practical
Penrose process in Section 12.
For more on the ReissnerNordstrøm metric for a charged black hole
(equation [51], page F32), see entries in the Subject Index and the Bibliog
raphy and Index of Names in Misner, Thorne, and Wheeler’s Gravitation,
W. H. Freeman and Company, San Francisco (now New York), 1973.