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Bachelor thesis Pieter van der Wijk

Authors:
The Kerr-Metric: describing Rotating
Black Holes and Geodesics
Bachelor Thesis
by
P.C. van der Wijk
1469037
Rijksuniversiteit Groningen
September 2007
2
The Kerr-Metric: describing
Rotating Black Holes and
Geodesics
P.C. van der Wijk
(s1469037)
Supervisor: Prof. Dr. E.A. Bergshoeff
Rijksuniversiteit Groningen
Faculty of Mathematics and Natural Sciences
24 September, 2007
2
Contents
1 Introduction 5
2 Black holes in general 7
2.1 How are black holes formed? . . . . . . . . . . . . . . . . . . 7
2.1.1 Stellar black holes . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Primordial black holes . . . . . . . . . . . . . . . . . . 8
2.1.3 Supermassive black holes . . . . . . . . . . . . . . . . 8
2.2 How can black holes be observed? . . . . . . . . . . . . . . . . 8
2.2.1 X-ray ........................... 8
2.2.2 Spectral shift . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Gravitational lensing . . . . . . . . . . . . . . . . . . . 10
2.2.4 Flares ........................... 11
2.2.5 Gravitational waves . . . . . . . . . . . . . . . . . . . 11
2.2.6 Possible candidates . . . . . . . . . . . . . . . . . . . . 11
3 Static black holes 13
3.1 Introduction............................ 13
3.2 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Curvature and time-dilation . . . . . . . . . . . . . . . 14
3.3 Singularities............................ 15
3.4 Event horizon and stationary surface limit . . . . . . . . . . . 16
3.4.1 Event horizon . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.2 Stationary surface limit . . . . . . . . . . . . . . . . . 18
3.5 Approaching a static black hole . . . . . . . . . . . . . . . . . 19
3.6 Geodesics ............................. 19
3.6.1 Derivations ........................ 20
3.6.2 Timelike geodesics . . . . . . . . . . . . . . . . . . . . 21
3.6.3 Null geodesics . . . . . . . . . . . . . . . . . . . . . . . 23
4 Rotating black holes 25
4.1 Introduction............................ 25
4.2 Kerrmetric ............................ 25
4.2.1 Boyes-Lindquist coordinates . . . . . . . . . . . . . . . 26
3
4CONTENTS
4.2.2 Kerr coordinates . . . . . . . . . . . . . . . . . . . . . 27
4.3 Singularities............................ 28
4.4 Symmetries ............................ 28
4.5 Frame-dragging.......................... 29
4.5.1 Stationary observer . . . . . . . . . . . . . . . . . . . . 32
4.6 Stationary limit surface . . . . . . . . . . . . . . . . . . . . . 33
4.6.1 Static Observers . . . . . . . . . . . . . . . . . . . . . 33
4.6.2 Penrose process . . . . . . . . . . . . . . . . . . . . . . 35
4.6.3 Gravitational redshift . . . . . . . . . . . . . . . . . . 36
4.6.4 Different values for aand M.............. 36
4.7 Eventhorizon........................... 37
4.7.1 Choice of coordinates . . . . . . . . . . . . . . . . . . 37
4.7.2 Time-like vs. space-like . . . . . . . . . . . . . . . . . 39
4.7.3 Different values for aand M.............. 39
5 Geodesics around a Kerr-black hole 41
5.1 Four constants of motion . . . . . . . . . . . . . . . . . . . . . 41
5.2 θ-motion.............................. 46
5.2.1 Low energy particles . . . . . . . . . . . . . . . . . . . 46
5.2.2 High-energy particles . . . . . . . . . . . . . . . . . . . 47
5.3 r-motion.............................. 48
5.3.1 Case 1: Q > 0, Γ >0................... 48
5.3.2 Case 2: Q > 0, Γ <0................... 49
5.3.3 Case 3 and 4: Q < 0, Γ >0 or Γ <0.......... 50
5.4 Equatorial motion . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Conclusion 55
7 References 57
Chapter 1
Introduction
In 1795 Laplace proposed, using Newton’s theory of gravity, it was possible
for very dense and massive objects to have an escape velocity larger than
the speed of light. Not even light could escape from such an object: it would
appear black. In 1915 Einstein published his famous theory of general rela-
tivity. This new theory predicted the possibility of such dark objects (called
singularities: objects with infinite curvature due to an infinite density) from
which light would not be able to escape: black holes.
Black holes are caused by singularities, points of infinite density in the
spacetime. The spacetime around such a point is so strongly curved, that it
exerts a very strong ‘gravitational pull’ such that everything nearby is drawn
into the black hole. This ‘pull’ is so strong, not even light is able to escape
from it: it will always ‘fall’ into the singularity. The area of spacetime for
which light will always go to the singularity, is called the black hole. The
singularities are real physical objects, like a book or a drain in a bath. But
the black hole, the blackness around the singularity, is as tangentable as a
whirlpool of the water going into the drain: the black hole tells something
about the curvature of the spacetime, as the whirlpool says something about
the curvature of the water.
During the first halve of the twentieth century, black holes were mere
‘thought experiments’ of the theoretical physicists. In 1916 K. Schwarzschild
gave the first solution for the Einstein equation of general relativity. His
solution described the spacetime around a static massive object and was
called the Schwarzschild-metric. Later, in 1963, R. Kerr discovered another
solution: the Kerr-metric. The Kerr-metric gives the spacetime outside a
massive rotating object. These two solutions describe the static and rotating
black hole respectively.
In the second halve of the twentieth century, astronomers observed some
strange phenomenan in the universe: very small objects that emitted jets of
particles with very high energy. They proposed black holes to be the objects
that were the sources of these jets. A black hole was no longer a theoretical
5
6CHAPTER 1. INTRODUCTION
construct, but a real physical object.
Stars that have exhausted their thermonuclear fuel are no longer able to
maintain their equilibrium with their inward gravitational force. The star
will undergo a gravitational collapse. If the star is massive enough, its fi-
nal state will be as a black hole. And because the star rotates before the
collapse, the collapse will result in a spinning or rotating black hole. The
spacetime around the final state of a very massive star is described by the
Kerr-metric. [Begelman, 1995]
There are several motivations for studying black holes and the metrics that
describe the spacetime around it. First of all, stellar mass black holes tell
us something about the last stage in star evolution. They should have in-
formation about the final moments of the star his life.
Secondly there are (predictions of) very massive black holes in the center
of the galaxies like our own galaxy. These black holes are important to
the theory of cosmology as they are perhaps ‘seeds’ of galaxy formation.
However, how they are formed is unclear, as well is the answer to the question
whether they are properly described by the Kerr-metric. Supermassive black
holes may have been created in the very early universe and tell us things of
that era. They might even play a role in the debate of the dark matter, since
black holes are very hard to observe and it is therefore hard to measure how
many mass they contain in total and how large the portion of dark matter
it has.
Further more, black holes are the extreme for gravitational theory since
singularities are objects of infinite density and curvature. Therefore they
form objects which probably have to be explained in terms of quantum
gravity: large mass at small spacetime. Black holes may form testcases for
this quantum gravity theory. [Begelman, 1995] [Rees, 2007]
This bachelor thesis is about the spacetime around a rotating black hole,
as described by the Kerr-metric. But first it will treat three general types
of black holes and several methods that are used for the detection of black
holes in the universe. In chapter three, the case of a static black hole is
explained: the Schwarzschild-metric. Here are the first important concepts
of black holes discussed, for example the event horizon and time dilation.
This third chapter is used for building a reference frame for the discussion of
rotating black holes in chapter four and five. In the fourth and fifth chapter,
the rotating black hole is discussed. Firstly the features of the black hole
itself. And secondly the geodesics of test particles around the black hole.
The last chapter provides a conclusion about rotating black holes and the
geodesics around it.
Chapter 2
Black holes in general
2.1 How are black holes formed?
There are three general types of black holes: stellar black holes, primordial
black holes and supermassive black holes. Each type of black hole has it’s
own way of formation. These three types are discerned by their mass (and
size). The more massive a particular source of a black hole is, the larger the
curvature of spacetime, the larger the black hole.
2.1.1 Stellar black holes
A stellar black hole is a black hole that is formed at the end of the lifetime of
a star. A star consist of nuclear fuel, mainly hydrogen. During it’s life, there
is nuclear fusion which produces the energy a star radiates. This radiation
causes an outward radiation pressure that is in equilibrium with the inward
gravitational force of the star. After some time, the star is exhausted due
to the fusion and the star is no longer hot enough for further nuclear fusion.
The star can become a red giant to increase the temperature of the core.
But the temperature will eventually not be high enough for further fusion.
The nuclear fusion process is halted and there is no radiation pressure
any more. The star will undergo a gravitational collapse. This collapse may
result in a supernova: the star is ‘detonated’ into a very large explosion in
which large portions of its mass are blown away. After that, the core will
recollapse.
Depending on the mass of the star at the moment of the final recollapse,
the core can become one of the three following products: a white dwarf
(mass less than 1,4 solar mass); a neutron star (mass between 1,4 and 3
solar mass); or a singularity, causing a black hole (mass greater than 3 solar
mass). A black hole formed this way, is a steller black hole. [Zeilik, 1998]
7
8CHAPTER 2. BLACK HOLES IN GENERAL
2.1.2 Primordial black holes
In the early universe, just after the Big Bang, the universe was very hot and
dense (according to the Standard Model). Small quantum fluctuations in
the density at that time are indicated by the galaxies at our present time:
they are the products of inhomogeneities in the density during the early
universe. Some regions might have been so dense, they would be sufficiently
compressed by gravitation to overcome the velocities of the expansion and
possible pressure forces from the inside. These very dense regions could
further collapse to create a black hole: primordial black hole.
These primordial black holes may be less massive than stellar black holes:
105g and upwards. As with all black holes, primordial black holes may
have increased in mass since the early universe by the accretion of matter.
But they may have lossed mass by Hawking radiation as well. [Misner, 1973]
[Carr, 1973]
2.1.3 Supermassive black holes
Supermassive black holes have masses 106solar mass and upwards. They
are found in the center of galaxies.
The formation of the supermassive black holes is less well understood
than that of the steller black hole. One idea on the formation is that they
are formed by the collapse of the first generation of stars after the Big
Bang, and have accreted matter over the time of millions of years. These
supermassive black holes may have been the seeds for the next generation
of galaxies. Interaction between galaxies, like mergers, could have resulted
in the merging of both black holes at the centers. Another idea is that
supermassive black holes are formed by the merging of clusters consisting of
stellar black holes. [Rees, 2007] [web3]
2.2 How can black holes be observed?
There are several ways in which black holes can be observed in a indirect
way: X-ray, spectral shift, gravitational lensing and flares. The only direct
way to observe black holes is via gravitational waves. But this last method
is not fully operational yet. Thus if a black hole is not near any matter, it
will not be observable because all indirect methods use surrounding matter
as an indicator for the presence of a black hole. Up till now, no black hole
as been directly observed.
2.2.1 X-ray
Black holes can absorb material from the interstellar medium or a companion
star. This process of absorbing material is called accretion. Because of the
2.2. HOW CAN BLACK HOLES BE OBSERVED? 9
possible angular momentum of the material, an accretion disk is formed
around the black hole. The material in the accretion disk rotates around
the black hole, the inner parts rotating faster than the outer parts. This
causes friction. The friction has two effects: lowering the angular momentum
of the material such that it can spiral into the black hole; and the material
is heated up to higher temperature as it spirals into the black hole.
The material that falls in, into the gravitational field, is strongly com-
pressed and heated up: it will radiate X-rays. The gravitational potential
energy of the matter is converted into kinetic energy as it falls in, and be-
cause of the friction, kinetic energy is converted into heat and radiation.
Because the infalling particles have very high velocities, they are relativis-
tic and thus is the radiation they emit beamed. This beaming causes the
jet-shape of the X-ray radiation.
The X-ray jets are caused by the gas that falls into the black hole, the
black hole itself does not emit the radiation.
A supermassive black hole with an accretion disk which emits jets of X-
ray are called quasars and they are found in active galactic nucleus (AGN’s).
A stellar black hole with X-ray jets is often located in a binary system in
which the accretion disk is formed from the material of its companion star.
These stellar black holes are called pulsars. [Begelman, 1995]
Figure 2.1: A stellar black hole with an accretion disk from its compan-
ion star. At the axis of rotation of the accretion disk there are two jets
visible.[fig1]
2.2.2 Spectral shift
Black holes have a gravitational influence on their surrounding. In the case
of a black hole in a binary star system or in the center of a galaxy, this
gravitational influence can be measured.
10 CHAPTER 2. BLACK HOLES IN GENERAL
In a binary system, with one visible star and an invisible object (like a
black hole), it is possible to measure the mass of the invisible companion.
Via the shift in the spectral lines of the star it is possible to determine the
speed with which it orbits around its invisible partner. And then applying
Newton’s Law one can estimate the mass of the invisible partner. If the
mass is larger then three solar mass, it is likely to be a black hole. If the
mass is less, it could also be a neutron star. The black hole could also have
X-ray jets as explained above.
At a galactic nucleus, the stars move in random directions, only respond-
ing to the total gravitational force from all the matter. All the stars have
a certain average speed, depending on the total mass and the radius of the
orbit of the star. This average speed varies differently with radius if there
is a supermassive black hole at the center of the galaxy than when there
is none: the stars move faster because of this large (invisible) gravitational
pull. Furthermore, the shape of the orbits of the stars close to the black hole
are more cigar-like shaped than for the case of no black hole. [Begelman,
1995]
2.2.3 Gravitational lensing
Gravitational lensing is the bending of a lightpath by a compact object. For
example, if there is a galaxy far far away and between that galaxy and the
Earth is a very compact and massive object, for example a cluster of galaxies
or a black hole, then the path of the light from that galaxy is bent by the
gravitational field of the compact object. The light is deflected and instead
of the one galaxy, one sees several images of that galaxy positioned around
the original galaxy. The number of images, and the possible distortions of
the images, depend on the shape of the compact object. With gravitational
lensing it is possible to determine the mass of the compact object. [web5]
Figure 2.2: Overview of gravitational lensing: the red lines indicate the true
light paths, the dotted lines are the light paths the observer would think the
light has traveled. The lens can be a cluster of galaxies or (several) black
holes. [web5]
2.2. HOW CAN BLACK HOLES BE OBSERVED? 11
2.2.4 Flares
Stars that are very close to supermassive black holes experience large tidal
forces: the closer part of the star experiences a larger ‘gravitational pull’
than the outer part. If the tidal force is larger than the gravitational force
that holds the star together, the tidal force tears the star apart. A part of
the material from the star will fall into the black hole and give of radiation
(using the same process as for the X-ray jets): the black hole will ‘flare
up’.[Begelman, 1995]
2.2.5 Gravitational waves
If the gravitational field changes by the change in size or shape of a massive
object, or by the acceleration of a massive object (provided the motion is
not perfectly spherically like a spinning disk), the changes of the spacetime
geometry are propagated by gravitational waves: ripples of spacetime. These
waves can be regarded as radiation. The gravitational radiation is very
weak compared to the electromagnetic radiation, and instead of dipole, it is
quadrupole radiation. [web6]
An example of a changing gravitational field is a binary system of a
neutron star and a black hole. Because they orbit around each other and
they are both very massive, there are large changes in the gravitational field;
they radiate strong gravitational waves. The system emits gravitational
waves, therefore the neutron star and the black hole will lose energy, and
eventually they will coalesce. Such a merging of a neutron star with a black
hole will cause large changes in the spacetime as well, when the neutron star
is absorbed by the black hole the gravitational field changes much.
Gravitational radiation, because it is very weak, even for the case of a
binary system of a neutron star with a black hole, has not yet been directly
observed. It has been indirectly proven in 1974 by R. Hulse and J. Taylor.
There are large detectors being build with which one hopes to measure these
gravitational waves (for example LISA). The detectors are large interferom-
eters which should measure the ripples of the spacetime of the gravitational
radiation. [Begelman, 1995]
2.2.6 Possible candidates
Three examples of black hole candidates are Cygnus X-1, A0620-00 and
LMC X-3. These are all found in X-ray binaries. Lower mass limits of the
first two are around 3,2 solar mass, and LMC X-3 has of mass of at least 7
solar masses. [web4]
12 CHAPTER 2. BLACK HOLES IN GENERAL
Chapter 3
Static black holes
3.1 Introduction
Black holes can be fully described in terms of three paramaters: mass, angu-
lar momentum and charge. Because of global conservation laws, these three
properties are conserved during the collapse of the star. All other properties
of the star that has collapsed to the black hole are lost during the collapse.
This follows from the A Black Hole Has no Hair theorem. In addition to
these parameters, there are four laws, derived from standard laws of physics,
which describes the dynamics of a black hole in general. [Hawking, 1973]
The Schwarzschild black hole only has mass, it does not have an angular
momentum or charge. It is a static black hole.
3.2 Schwarzschild metric
The Schwarzschild metric describes the spacetime curvature around static
massive objects. Examples of such an object is a non-rotating star or a static
black hole. In the derivation of the Schwarschild metric, four assumptions
were made: spacetime would be static, thus independent of coordinate time
t; spacetime is spherically symmetric; spacetime is empty, with the exception
of the static massive object there are no other sources of curvature; and
spacetime is asymptotically flat, the gtt-component goes to c2, and the grr -
component goes to 1 as rgoes to infinity. [Foster, 2006]
The Schwarzschild metric is given by:
c22=c212m
rdt212m
r1
dr2r22r2sin2θ2(3.1)
with mMG
c2.Mis the mass of the massive object. For the remainder of
this text, the following units for the speed of light c= 1 and the gravitational
constant G= 1 will be used. As can be seen from the metric, the curvature
is only radial dependent. In the figure below is an illustration of how a
13
14 CHAPTER 3. STATIC BLACK HOLES
threedimensional spacetime would look like around a Schwarzschild black
hole.
Figure 3.1: Three-dimensional spacetime around schwarzschild black hole:
two spatial dimensions and one time dimension (indicated by the arrow
pointing upwards) [fig2]
3.2.1 Curvature and time-dilation
As a consequence of the curvature, the physical distance between two (radial)
coordinates is different from the distance between those two coordinates.
The length of a radial line-element dR (choosing φand θconstant) is given
by:
dR (1 2m/r)1/2dr (3.2)
The measured radial distance dR is larger than the radial coordinate distance
dr. To measure a whole line, one needs to integrate the above expression.
Coordinates are like street numbers: the distance between the 36th St. and
37th St. is not necessarily equal to the distance between 38th St. and 39t
St. [Foster, 2006]
Furthermore, there is a time-dilation as well: for observers close to a
schwarzschild black hole, time flows slower than for observers far away. For
a static observer, which measures propertime , a time-interval is given by:
= (1 2m/r)1/2dt (3.3)
The time-dilation is an important cause of the gravitational redshift.
Say you have a static emitter and a static receiver, both at different radial
distances from the static black hole. The emitter emits signals to the ob-
server, always using the same time-interval Ebetween two signals. The
receiver measures a time-interval Rbetween two signals. The frequency
of the signals measured by the emitter is νEn/τE, where nthe number
of signals is, emitted by the emitter, during a time-interval ∆τE. But the
receiver measures a frequency of νRn/τR: he measures the same num-
ber of signals, but a different time-interval because he is further away from
the black hole, and thus time flows faster for him. The redshift is given by:
νR
νE
=12m/rE
12m/rR1/2
(3.4)
3.3. SINGULARITIES 15
If the receiver is closer to the black hole then the emitter, light is blueshifted.
And if the emitter is closest to the black hole, the light (or any other signal)
is redshifted. Notice that if the rE= 2m(at the event horizon), light is
infinite redshifted.
3.3 Singularities
A singularity is a point in spacetime for which the curvature of the manifold
goes to infinity. This is represented by a term in the metric going to infinity:
it ‘blows up’. In other words, the curvature at that point is not well described
by the metric. But a term in the metric can also ‘blow up’ due to (bad)
choice of coordinates at that point, hench the difference between coordinate
singularities and curvature singularities. Coordinates singularities can be
removed by choosing a more fortunate coordinate system and curvature
singularities can not be removed: they are real properties of the manifold.
[Townsend, 1997]
The Schwarzschild metric has a coordinate singularity at r= 2m, then
the grr term ‘blows up’. By changing to the Eddington-Finkelstein coor-
dinates one can remove this singularity from the metric. The Eddington-
Finkelstein coordinates are based on free falling photons (null geodesics),
ingoing and outgoing. [Foster, 2006][Misner, 1976]
For the ingoing Eddington-Finkelstein coordinates, one needs to replace
the coordinate time tby v:
vt+r+ 2mln (r/2m1) (3.5)
This leads to a Schwarzschild metric of:
2=12m
rdv22dvdr r22r2sin2θdφ2(3.6)
There is no term that blows up at r= 2m. However, ingoing Eddington-
Finkelstein coordinates are only able to describe ingoing photons into the
static black hole. To give the geodesics of outgoing photons, one needs to
use the outgoing Eddington-Finkelstein coordinates. To achieve this, one
needs to replace the plus-sign of the dvdr-term by a minus-sign. [Misner,
1973]
There is also a curvature singularity in the Schwarzschild metric, it is
located at r= 0. The first term in the metric ‘blows up’ for that coordinate.
The shape of the singularity is a point. This point has an infinite density.
It is possible for particles or photons to reach a curvature singularity.
But once one has reached the singularity, it is impossible for it to move
away from that point by extending their path in a continious way. [Misner,
1973]
16 CHAPTER 3. STATIC BLACK HOLES
3.4 Event horizon and stationary surface limit
The first two terms in the Schwarzschild metric changes sign at r= 2m. The
change of sign in the first term is the indication for the stationary surface
limit, and the change in the second term indicates the event horizon. These
surfaces are not a real physical surfaces in the sense that one is able to
touch it, they are mathematical constructs, like the boundary between two
countries.
3.4.1 Event horizon
An event horizon is a surface that can be considered as a one-way-membrane:
it lets signals from the outside in, but it prevents signals from the inside to
go to the outside. The event horizon is the boundary between where the
curvature is strong enough and where it is not strong enough to prevent
photons from within to escape to infinity. Photons emitted within the area
of the event horizon are trapped forever within the black hole, photons
emitted just outside the event horizon (emitted within the right direction)
are able to reach infinity. The event horizon is a sphere-shaped surface
around the black hole singularity (in three spatial dimensions).
Because signals from the inside of the event horizon can not pass the
event horizon, observers at the outside are not able to see inside the event
horizon. Thus they are unable to see the singularity of the rotating black
hole. This is accordance with the theorem of ‘Cosmic Censorship’ [Hawking,
1974] which states that ‘naked singularities’ are forbidden.
The sphere-shaped surface of the rotating black hole is ’generated’, given
form, by photons that are forever trapped at the event horizon: they have
no radial velocity. These photons (or horizon generators) were emitted pre-
cisely at the event horizon in radial outward direction and are not able to
go beyond the event horizon (on both sides: inside and outside the event
horizon) because of their tangential direction. These photons do not have
end-points: they will be there for always. [Misner, 1973]
Say there is a photon that moves in a radial direction of a static black
hole: == 0. The metric is given by:
0 = 12m
rdt212m
r1
dr2(3.7)
Rewriting gives:
dr
dt 2
=12m
r(3.8)
In the limit of rgoing to infinity, the speed of the photon is the speed of
light (the units are such that c= 1). However, at r= 2m, the speed of
3.4. EVENT HORIZON AND STATIONARY SURFACE LIMIT 17
the photon is zero: the event horizon. However, because of the coordinate
singularity, other coordinates are needed for better analyses: ingoing or
outgoing Eddington-Finkelstein coordinates.
For the case of the ingoing Eddington-Finkelstein coordinates, the null-
geodesic of the radial photon is given by:
0 = 12m
rdv22dvdr (3.9)
Rearranging gives:
dv
dr 12m
rdv
dr 2= 0 (3.10)
Solving this equation to dv/dr gives two solutions:
dv
dr = 0 (3.11)
dv
dr =2
(1 2m/r)(3.12)
Differentiating the expression for v(3.5) gives dv/dr =dt/dr +1
12m/r .
Using this in combination with the first solution for dv/dr gives:
dt
dr =1
12m/r (3.13)
This gives the ingoing null geodesic, as it is negative in the region of r > 2m.
Integrating the corresponding dv/dr gives v=A, where Ais a constant.
The other solution gives the following function for dt
dr and is integrated
as follow:
dt
dr =1
12m/r (3.14)
v= 2r+ 4mln |r2m|+B(3.15)
(3.16)
(Bis a constant) This gives the outgoing null geodesic: it is positive for
r > 2m. Notice the behaviour at r= 2m: the outgoing photon is not
able to cross the r= 2mboundary, where as, the ingoing photon can cross
that boundary: ingoing photons can cross the event horizon, but outgoing
photons can not cross it as they are trapped inside.
These geodesics can be pictured in a (two-dimensional) spacetime-dia-
gram, see figure below (3.2). The axes are oblique to let it appear as in flat
spacetime. [Foster, 2006]
18 CHAPTER 3. STATIC BLACK HOLES
Figure 3.2: Eddington-Finkelstein spacetime-diagram. The small arrows
indicate the direction of the outgoing photons. At r=r0an observer is
located: he only observes photons emitted outside the event horizon. The
ingoing photons are the straight lines. [Foster, 2006]
3.4.2 Stationary surface limit
A static observer is an observer that only moves in time: it spatial coordi-
nates are constant (with respect to an inertial frame): dt 6= 0, dr ==
= 0. This gives a line-element for a timelike observer of:
ds2=12m
rdt2>0 (3.17)
For the case of a particle with mass, ds2needs to be time-like: larger than
zero. However, at r= 2m,gtt is equal to zero. Thus for the case of an
observer that follows time-like geodesics, he can not be static at r= 2m,
because dr,or can not be zero at that same point: the observers
needs to move in spatial coordinates as well. And for r < 2m, the gtt-term
is negative: the time-like observer has to move in space within the event
3.5. APPROACHING A STATIC BLACK HOLE 19
horizon.
When an observer is not able to be static for r < 2m,r= 2mis called a
stationary limit surface: beyond that surface, one is not able to be stationary
(or static).
3.5 Approaching a static black hole
Say there is an astronaut who is send from a spaceship far away, into a
schwarzschild black hole. The astronaut carries a flash light which he uses
to create a light flash every second (using his own watch).
From the viewpoint of the spaceship, as the astronaut nearers the black
hole, the time-interval between two flashes increases and the astronaut slows
down as he approaches the event horizon. This is caused by the time-
dilation. And as the astronaut approaches the horizon, he appears to look
more red: his light is red-shifted. Just before he looks to reach the event
horizon, his speed appears to become zero and his light becomes infinite
red-shifted: he will fade from view. The spaceship will never observe the
astronaut crossing the event horizon.
From the viewpoint of the astronaut, he accelerates as he falls to the
black hole. If he would look into space, every thing would seem to be
normal. And he would flash his flashlight every second, as agreed with the
persons on board the spaceship. As he crosses the event horizon, nothing
special happens to him at that moment. What he would see inside the event
horizon is not known to us.
The above story is not complete: it ignored the gravitational effects
of the black hole on the astronaut. As the astronaut comes closer to the
black hole, the gravitational force on his legs becomes substantially larger
than the gravitational force on the upper part of his body (this is because
the distance dependence of the gravitational force), providing his legs are
closest to the black hole: the astronaut becomes stretched, the lower part
more than the upper part of the body. As he nearers, his legs become even
more stretched as the difference in force increases. In the end his body will
be stretched that far, that it is fatal to him. This point is reached outside
the event horizon.[Hawking, 1988]
3.6 Geodesics
To discuss the geodesics of free test particles in the vicinity of a static black
hole, one needs to derive expressions for their paths. The derivations for
the geodesics form the first part of this section. After that, some types of
motion are discussed for timelike and null-geodesics.
20 CHAPTER 3. STATIC BLACK HOLES
3.6.1 Derivations
The Lagrangian for a geodesic is given by
L=1
2gµν ˙xµ˙xν(3.18)
where the dot is an indication for the derivative to some affine parameter λ.
For the case of timelike-geodesics, λcan be set equal to the propertime τ.
The langrangian for the Schwarzschild-metric is given by:
L=1
2"12m
r˙
t212m
r1
˙r2r2˙
θ2+ sin2θ˙
φ2#(3.19)
The corresponding canonical momenta pµ=∂L
˙xµare given by:
pt=12m
r˙
t(3.20)
pr=12m
r1
˙r(3.21)
pθ=r2˙
θ(3.22)
pφ=r2sin2θ˙
φ(3.23)
From the Euler-Lagrange equation
∂L
∂xµ
∂λ
∂L
˙xµ= 0 (3.24)
it follows that ptand pφare constants of motion, because L
∂t =L
∂φ = 0:
pt=E=12m
r˙
t(3.25)
pφ=L=r2˙
φ(3.26)
with Eas the energy of the particle at infinity, and Lit’s angular momentum.
The canonical momenta give rise to the following Hamiltonian:
H=pt˙
t+pr˙r+pθ˙
θ+pφ˙
φL=L=cst (3.27)
The equality between the Hamiltonian and Lagrange means there is no phys-
ical potential involved. And the constant of the equality can be choosen such
that it is equal to 1/2 for timelike geodesics, and zero for null geodesics.
The Hamiltonian gives the following equation (in the equatorial plane
θ=π/2):
2L=E2
12m/r ˙r2
12m/r L2
r2=δ(3.28)
3.6. GEODESICS 21
where δis equal to 1 or 0 for respectively timelike or null geodesics. Every
geodesic can be choosen such that it lies in one plane, and because of the
spherical symmetry of the Schwarzschild-metric, every plane can be choosen
as equatorial plane. Thus there is no loss of generality be choosing to be in
the equatorial plane.
Furthermore, the Euler-Lagrange give an equation for the case of µ= 1
(the radial coordinate):
12m
r1
¨r+m
r2˙
t212m
r2m
r2˙r2r˙
φ2= 0 (3.29)
The equations (3.25), (3.26), (3.28) and (3.29) will be used throughout
the next sections. [Chandrasekhar, 1983], [Foster, 2006]
3.6.2 Timelike geodesics
For the case of a timelike geodesic, equation (3.28) can be rewritten to
dr
2
+12m
rδ+L2
r2=E2(3.30)
Using the above equation with the expression (3.26) for the angular momen-
tum with u1/r one gets
du
2
= 2mu3u2+2m
L2u1E2
L2(3.31)
The above equation determines the shape of the geodesic in one plane. And
by using the expression (3.25) and (3.26) the solution can be completed.
[Chandrasekhar, 1986]
Furthermore, it is possible to rewrite (3.30), using (3.25), to obtain an
integral for the coordinate time [Misner, 1976]:
t=ZrEdr
(1 2m/r) [E2(1 2m/r)(1 + L2/r2)]1/2(3.32)
Vertical free-fall
A special case of a timelike geodesic is the vertical free-fall. In a free-fall, φ
is constant ( ˙
φ=L= 0). Using equation (3.30) one gets
dr
2
+ (1 2m/r)E2= 0 (3.33)
Further differentiation and then divide by ˙rgives:
2 ˙r¨r+2m
r2˙r= 0 (3.34)
¨r+m
r2= 0 (3.35)
22 CHAPTER 3. STATIC BLACK HOLES
which can be rewritten by using m=GM/c2to
¨r+GM
r2= 0 (3.36)
Which is the Newtonian equation for gravity. [Foster, 2006]
Effective potential
The second term in equation (3.30) can be interpreted as an effective poten-
tial
dr
2
+V2=E2(3.37)
V2=12m
r1 + L2
r2(3.38)
Drawing a graph of the potential as a function of rgives:
Figure 3.3: The effective potential. The dot indicates a minimum. [Misner,
1973]
The potential has an attractive part corresponding to the ‘gravitational
force’ on the particle, but also a repulsive part caused by the conservation
of angular momentum (similar to the centrifugal force). Different values of
the angular momentum of the particle gives different effective potentials (see
figure 3.4).
A particle with a effective potential as in figure 3.3 has for different
values of its energy Edifferent orbits. If his energy is equal to the effective
3.6. GEODESICS 23
Figure 3.4: The effective potential for different values of the angular mo-
mentum Lof the particle. [Misner, 1973]
potential energy at the minimum, then the particle follows a stable circular
orbit. If the particle has a bit more energy, it will follow elliptical orbits,
where the radii at which the energy equals the potential energy are the
turning points: the radial velocity changes direction. For the case when the
energy equals the potential energy at maximum, the particle moves along a
so-called ‘knife-edge’ orbit: it is a unstable orbit and a small perturbation
results in going to infinity or fall into the black hole. The last case, is when
the energy Eis larger than the maximum potential energy: the particle will
directly go into the black hole. [Misner, 1973] [Taylor, 2000]
3.6.3 Null geodesics
For photons, δ= 0, and equation (3.30 becomes:
dr
2
+12m
rL2
r2=E2(3.39)
24 CHAPTER 3. STATIC BLACK HOLES
Again, one can rewrite this to an equation which gives the shape in a plane,
using u= 1/r and equation (3.26):
du
2
= 2mu3u2+E2
L2(3.40)
There are two special cases for photon geodesics that will be treated
below: radial and circular geodesics.
Radial geodesic
A radial geodesic (L= 0) for a photon is described by
dr
=±E(3.41)
using equation (3.25) this becomes
dr
=±12m
r(3.42)
Integrating it gives:
t=±hr+ 2mlog r
2m1i+C(3.43)
where Cis a constant. Integrating the expression (3.41) with respect to
gives
r=±+C(3.44)
Meaning, a photon is able to cross the event horizon in it’s own affine pa-
rameter (3.44), but it would take an infinite amount of coordinate time tto
reach the event horizon (3.43): an observer outside the event horizon will
never observe the photon crossing the horizon. [Foster, 2006]
Circular orbit
For a circular orbit, ˙r= ¨r= 0, one gets the following from equation (3.29):
˙
φ
˙
t!2
=m
r3(3.45)
In combination with equations (3.39), (3.26) and (3.25) (giving ˙
φ
˙
t2
=
12m/r
r2) this equality becomes:
12m/r
r2=m
r3
r2m=m
r= 3m(3.46)
Giving a circular orbit for a photon at r= 3m. [Foster, 2006]
Chapter 4
Rotating black holes
4.1 Introduction
A rotating black hole has rotation in addition the static black hole. The
description of a rotating black hole uses two of the three parameters: mass
and rotation. As such, a rotating black hole is not described by the Schwarz-
schild-metric but by an other metric: the Kerr-metric. The Kerr-metric
describes the spacetime around a rotating singularity.
Stellar black holes are caused by the collapse of stars. A star is a very
massive, rotating but chargeless object. Because charges of opposite sign
cancels, stars are neutral. Hench, the spacetime around a stellar black hole
is described by the Kerr-metric.
Although the Kerr-metric gives the spacetime around rotating massive
objects and the Schwarzschild-metric that of static massive objects, there are
similarities between them. Both metrics are able to describe black holes that
are caused by curvature singularities; they share a coordinate singularity at
their event horizon; in both metric two observer experience a time-dilation
and curvature. Furthermore, the spacetime is both in empty (with the
exception of the one massive object) and asymptotically flat.
4.2 Kerr metric
The solution to the Einstein equation for a spinning, rotating massive object
without charge, is given by the Kerr-metric. The object rotates around its
z-axis (or its θ-axis). A rotating or spinning black hole without charge, is
such an object.
The Kerr-metric can be described in different coordinate systems. The
most common coordinate system used is the Boyes-Lindquist coordinate
system. The advantage of this coordinate system is that it is written in
spherical coordinates, which are easy to work with and some features are
easily noticed. But the drawback of this coordinate system is that it has a
25
26 CHAPTER 4. ROTATING BLACK HOLES
coordinate singularity at the event horizon (see next section). To ‘remove’
this coordinate singularity, one changes from coordinate system. An often
used alternative is the Kerr coordinate system. Both coordinate systems
are given below because both will be used in the discussion of the spacetime
and the geodesics of particles around black holes.
4.2.1 Boyes-Lindquist coordinates
The Kerr metric in Boyes-Lindquist coordinates is given by [Foster, 2006]:
c22=12mr
ρ2c2dt2+4mcra sin2θ
ρ2dtdφ ρ2
dr2
ρ22r2+a2sin2θ+2mra2sin2θ
ρ22(4.1)
In which:
ρ2r2+a2cos2θ(4.2)
r2+a22mr (4.3)
Furthermore: mGM/c2in which M is the mass of the rotating black hole
and G the gravitational constant. The parameter ais the rotation parameter
and is connected to the angular momentum Jof the rotating black hole as
follows:
a=J/Mc (4.4)
The parameter agives the direction and speed of the rotation. A positive
value indicates a clockwise rotation of the object, a negative value indicate
a counterclockwise rotation. The larger angular momentum, the larger the
value for a, the faster the rotation. For the remainder of this, and the next
chapters, special units are chosen such that c= 1 and G= 1. [Adler, 1967]
[Foster, 2006]
In the special case of J= 0, abecomes zero and the Kerr-metric reduces
to the Schwarzschild metric: the solution to the Einstein equations of a
nonrotating mass. If a2=m2then the Kerr-metric describes the special
case of the ’Extreme Kerr geometry’. A rotation parameter a2> m2gives
a ‘naked singularity’: a singularity that has no event horizon to hide it
from sight. In this case the singularity would be observable. But that is
in violation with physics because it would then be possible to observe a
point of infinite density (‘Cosmic censorship theorem). As for now, naked
singularities have never been observed. [Hawking, 1974]
The different terms in the metric are ‘responsible’ for different features
of a rotating black hole. The first term is the gtt-term and creates the
4.2. KERR METRIC 27
time dilation. If an observer would be completely (spatially) static, then
dr === 0, then
2=12mr
ρ2dt2(4.5)
giving =dtp12mr/ρ. Thus the closer the observer is to the black
hole, the slower time would flow compared to an observer at infinity (he
would measure =dt). Thus close to the rotating black hole, time flows
less fast. Furthermore, the surface for which this metric term is zero, is the
stationary limit surface.
The second term in the metric, the g-term, is the only off-diagonal
component of the metric. This term creates the frame-dragging. Frame-
dragging is the effect that spacetime appears to twist around the black hole.
The geodesics of objects in the neighborhood of the rotating black hole are
twisted around the black hole because of the frame-dragging. Beyond the
stationary limit surface, the gtt-term is negative, and to have timelike or null
paths, this second-term needs to be positive: particles and photon need to
rotate around the black hole. And for the case of a= 0 (no rotation), this
term vanishes.
The third term, the grr -term, is the indicator of the event horizon. When
this component is zero at the event horizon, particles and photons do not
change their radial coordinate: they are not able to move further away from
the black hole. They can not escape from the black hole.
4.2.2 Kerr coordinates
The Boyes-Lindquist being the spherical coordinate system of a rotating
black hole, the Kerr coordinates being the coordinates that follow the path
of a ‘radial’ infalling photon [Misner, 1976]:
d˜
V=dt +r2+a2
dr, d ˜
φ=+a
dr
By using these coordinates for the Kerr-metric one obtains the following
metric [Novikov, 1989]:
ds2=12mr
ρ2d˜
V22drd ˜
Vρ22
ρ2hr2+a22a2sin2θisin2θd ˜
φ2+ 2asin2θd ˜
φdr
+4amr sin2θ
ρ2d˜
φd ˜
V(4.6)
with
ρ2r2+a2cos2θ(4.7)
r2+a22mr (4.8)
28 CHAPTER 4. ROTATING BLACK HOLES
(ρ2and ∆ are the same as for the Boyes-Lindquist coordinates.) This form is
similar to the ingoing Eddington-Finkelstein coordinates of the Schwarzschild-
metric, both describe the ingoing, freely falling photon. The advantage of
the Kerr-coordinates over the Boyes-Lindquist coordinates, is that it does
not have a coordinate singularity at the event horizon. [Misner, 1973]
4.3 Singularities
The Kerr-metric in Boyes-Lindquist coordinates has a coordinate singularity
at r=m+m2a2(the event horizon). For this value of r, ∆ = 0 and
the coefficient of dr2goes to infinity. For a particle approaching the event
horizon, the grr -term goes to infinity and the coordinate time needs to go to
infinity for the particle to reach the event horizon. This creates an infinite
twisting of the path of the particle around the black hole, because dθ > 0.
[Misner, 1973]
This coordinate singularity at the event horizon can be removed or
avoided by choosing an other coordinate system, for example the Kerr co-
ordinates.
A rotating black hole has a curvature singularity as well. This singularity
has the shape of a ring. One can transform the Kerr-metric to the Kerr-
Schild coordinates, and from these coordinates it follows that for r= 0,
there is a ring given by x2+y2=a2(in which the xand yare Euclidian
coordinates and ais the rotation parameter) which is the singularity. If
there is no rotation, a= 0, the ring is just a point, as is the case for the
Schwarzschild-metric. [Hawking, 1974]
4.4 Symmetries
As can be clearly seen from the Kerr-metric in Boyes-Lindquist coordinates:
the metric terms are independent of coordinate time tand axial coordinate
φ. Thus the solution is stationary and axial-symmetric: an observer on a
worldline of constant θand r, and with a uniform angular velocity sees a
spacetime geometry that does not change while traveling along this world-
line.
Because of these symmetries there are three coordinate transformations
for which the Kerr-metric is invariant:
t0=t+cst; r,θ,φunchanged;
φ0=φ+cst; t,r,θ, unchanged;
t0=t,φ0=φ; r and θunchanged.;
These symmetry properties are similar to the symmetry of an ordinary rotat-
ing spinner: if you take two photo‘s at two different moments, both pictures
4.5. FRAME-DRAGGING 29
will look the same. The same is true for the angle-symmetry. These three
symmetry properties are general properties of homogenous spinning objects.
Bear in mind that the Kerr-metric describes a black hole in an empty space-
time. As soon as one include a (non-test) particle in the spacetime (or
metric), it loses its symmetry because one would be able to see the particle
travel along its worldline and be able to discern the two pictures with the
use of the position of the particle.
A tool for describing symmetries are Killing vectors. If you have a metric
gµν on some coordinate system dxaand that metric is independent of a
coordinate xk(for example xk=φ) such that
∂gµν
∂xk= 0 (4.9)
for k=a, then the vector
ξ
∂xk(4.10)
is called the ‘Killing vector’. Killing vectors are coordinate independent as
they are a properties of the spacetime itself. Killing vector ξis an infinites-
imal displacement which is length-conserved: a curve can be displaced in
the direction of xkby a shift of ∆xk, then the new curve has the same
length as the original curve. If a geometry has a Killing vector, then the
scalar product of the tangent vector of any geodesic with the Killing vector
is constant:
pk=~p ·~
ξ=constant (4.11)
Since the Kerr-metric has two symmetries, it has two Killing vectors
associated with coordinate time tand axial coordinate φ:
ξt=
∂t r,θ,φ
and ξφ=
∂φ t,r,θ
(4.12)
The three scalar products of the Killing vectors gives three terms of the
metric: ~
ξt·~
ξt=gtt ~
ξt·~
ξφ=g~
ξφ·~
ξφ=gφφ (4.13)
These equalities follow from the definition of the metric-terms. [Foster, 2006]
[Misner, 1973]
4.5 Frame-dragging
A rotating black hole causes frame-dragging of the spacetime geometry.
Frame-dragging is the twisting of spacetime around the black hole. It is
like water in a bath that goes down the drain: before going through the
drain it circles around the drain (because of the ‘frame-dragging’ of the wa-
ter). Below is a figure in which the frame-dragging around a rotating black
30 CHAPTER 4. ROTATING BLACK HOLES
hole is illustrated. It is as if you look on the xy-plane of a black hole that
rotates clockwise around its z-axis. The radial lines spiral clockwise around
the center of the black hole.
Figure 4.1: Frame-dragging of a rotating black hole, as viewed from above.
[Web2]
Suppose somewhere is a rotating black hole. Then one could build a
large steel frame around the rotating black hole, such that the frame is
fixed in infinity at the distant stars. Thus in infinity, this steel frame is
considered a lorentz frame. Near the black hole, gyroscopes are fixed onto
the frame. From real experiments it follows that these gyroscopes rotate
around the same axis as the rotation axis of the spinning black hole: the
g0φ-term causes a rotation of the gyroscopes with respect to the basis-vector
∂φ , and since these basis-vectors are fixed to the frame, and thus fixed to
the stars at ‘infinity’, these gyroscopes rotate with respect to these distant
stars. Gyroscopes have the ability not to change direction due to external
forces: thus they rotate due to the frame-dragging of the spacetime geome-
try. Thus the rotation of a black hole causes the twisting or frame-dragging
of spacetime around the black hole itself. [Misner, 1973] [Sexl, 1979] [Foster,
2006]
The Lagrangian for a free test-particle following a geodesic is given by
L=1
2gµν ˙xµ˙xν, with the time-deriviatives to proper-time τ. This gives:
L=1
212mr
ρ2˙
t2+2mra sin2θ
ρ2˙
t˙
φρ2
2∆ ˙r2
ρ2
2˙
θ21
2r2+a2sin2θ+2mra2sin2θ
ρ2˙
φ2(4.14)
Because Lis not explicitly dependent on φ(φis cyclic), the Euler-Lagrange
equation with respect to the φcoordinate is:
∂L
˙
φ= constant (4.15)
4.5. FRAME-DRAGGING 31
2mra sin2θ
ρ2˙
tr2+a2sin2θ+2mra2sin2θ
ρ2˙
φ= constant (4.16)
Since the Kerr-metric describes a totally isolated rotating black hole, it
is possible to choose the constant such that for rgoing to infinity, where
dφ/dτ = 0 (no frame-dragging at infinity), the constant is equal to zero.
Because of the properties of the Euler-Lagrange equations is the above equa-
tion valid for every point in spacetime and thus is the constant zero at the
black hole as well. Bringing ˙
φto the other side and dividing by ˙
tgives:
˙
φ
˙
t=
dt
=2mra
(r2+a2)ρ2+ 2mra2sin2θ(4.17)
This is the observed angular velocity from far away, where t
=τ. Thus the
observer which observes the free particle at the rotating black hole, would see
it rotate around the black hole due to frame-dragging. Even if the particle
has only radial velocity when is it far away from the black hole, when it
comes closer tot the rotating black hole, it begins to rotate around the black
hole. This is illustrated in the figure 4.5 below [Foster, 2006]
Figure 4.2: Orbits of two particles near a rotating black hole. The rotation
of the black hole is counterclockwise. Because of the frame-dragging, both
particles will eventually rotate with the direction of the rotation of the black
hole, even though one particle first rotates in opposite direction.[Begelman,
1995]
The effect of frame-dragging increases with increasing a(a larger angular
momentum) (in the domain of 0 < a2< m2) and decreases with increasing
r.
32 CHAPTER 4. ROTATING BLACK HOLES
4.5.1 Stationary observer
A stationary observer is an observer that moves along the Killing vectors: it
moves, but it sees an unchanged spacetime in its neighborhood. There are
no changes in the other direction components dr and . Following Misner
(1973) it is possible to determine the possible values of the angular velocity
of stationary observers. The angular velocity Ω relative to the rest frame of
the distant stars is defined as:
dt =dφ/dτ
dt/dτ =uφ
ut(4.18)
where utand uφare two components of the 4-velocity. The 4-velocity of a
stationary observer (dr == 0) is, in terms of the Killing vectors, equal
to:
~u =ut
∂t +uφ
∂φ =ut
∂t + Ω
∂φ =
=ξt+ Ωξφ
|ξt+ Ωξφ|=ξt+ Ωξφ
(gtt + 2Ωg+ Ω2gφφ)
1
2
(4.19)
An observer can not have every value of Ω: the 4-velocity must be within
the future light cone since he follows timelike paths:
gtt + 2Ωg+ Ω2gφφ <0 (4.20)
Solving this quadratic equation for Ω gives:
Ω = g
gφφ ±sg
gφφ 2
gtt
gφφ
(4.21)
As can be seen from the above equation for Ω, there is an upperbound and
a lowerbound on Ω. Defining a variable ω
ω≡ −g
gφφ
=2mcra
(r2+a2)ρ2+ 2mra2sin2θ=1
2(Ωmin + Ωmax) (4.22)
with
min =ωqω2gtt/gφφ (4.23)
max =ω+qω2gtt/gφφ (4.24)
min <<max (4.25)
Notice that ωis the frame-dragging of the rotating black hole (see the ex-
pression for the angular velocity of the frame-dragging (4.17)).
4.6. STATIONARY LIMIT SURFACE 33
min and Ωmax take the values of c/r and c/r respectively for far
away from the black hole. Far away, ω(and ω2) go to zero because of the r4
dependence in the denominator. The fraction gtt /gφφ goes to c2/r2since the
denominator has a r4-term, but the nominator has a r2term in it. These
limits are in agreement with special relativity: rmin =cand rmax =c.
For a decreasing radius, the minimum angular velocity Ωmin increases:
the gtt/gφφ-term increases faster than the ω2. When gtt reaches zero at
the outer stationary limit surface, Ωmin becomes 0. As the radial distance
increases even further, the range between Ωmin and Ωmax decreases and
ωincreases. At the event horizon r=m+m2a2the minimum and
maximum angular velocities take the same value (gtt/gφφ =ω2). Therefore
there is no possibility there for an observer to be stationary since his angular
velocity ought to be larger than Ωmin and smaller than Ωmax for the 4-
velocity to lie within the future light-cone (see equation 4.25): thus timelike
worldlines point inward to the event horizon.
4.6 Stationary limit surface
The stationary limit surface (also known as the ‘static limit’) is the outer-
most surface of a rotating black hole. It is the boundary between the two
areas where observers can be static (outside the stationary limit surface) and
where observers can no longer be static due to the strong frame-dragging
(inside the stationary limit surface). In the region between the stationary
surface limit and the event horizon, observers can be stationary.
The stationary limit surface is located at r=m+m2a2cos2θ. Inside
the stationary limit surface, every observer, particle or photon rotates with
the same direction as the rotation of the black hole.
The region between the stationary limit surface and the event horizon is
called the ergosphere (the stationary limit surface itself is called the ergo-
surface). This name originates from the possibility of the Penrose process
inside the ergosphere. Particles in the ergosphere are still able to escape
from the rotating black hole to infinity.
Because of the gravitational curvature of the spacetime geometry, the
black hole causes gravitational redshift. Photons emitted by an emitter
close to the static limit, send to an observer farther away, are redshifted.
4.6.1 Static Observers
A static observer is an observer that only moves in time: it spatial coordi-
nates are constant (with respect to an inertial frame): dt 6= 0, dr ==
= 0. This gives a line-element for a timelike observer of:
ds2=gttdt2=12mr
ρ2dt2>0 (4.26)
34 CHAPTER 4. ROTATING BLACK HOLES
Figure 4.3: Overview of a rotating black hole. (Adapted from [fig3])
Thus in the case of a static observer, gtt needs to be larger than zero (dt2is
always positive). gtt is zero if:
12mr
ρ2= 0
ρ22mr = 0
r2+a2cos2θ2mr = 0 (4.27)
Solving this quadratic eguation to rgives:
r=m±pm2a2cos2θ(4.28)
This gives two surfaces which depend on the mass mof the black hole
and the rotational parameter a. The outer surface (r=m+m2a2cos2θ)
is the static limit.
If a static observer goes through the ergosurface, then gtt changes sign:
the term becomes smaller than zero and timelike paths are only possible if
6= 0. Thus an observer inside the static limit needs to rotate to follow
timelike paths. If the observer follows a geodesic, he would rotate with the
same direction as the frame-dragging of the black hole.
Thus every particle and photon needs to rotate with the frame-dragging
inside the ergosphere. Below is a figure which gives an illustration of the
influence of the frame-dragging on the light cone. As you can see, the closer
a light cones comes to the static limit, the more it tilts to the direction
of the rotation of the black hole due to the frame-dragging. And inside
in the egrosphere, light can no longer go in the opposite direction of the
frame-dragging. [Misner, 1973]
Although dφ/dt needs to be larger than zero for timelike (or null) curves
inside the ergosphere, dr/dt can have any sign in that region: particles can
4.6. STATIONARY LIMIT SURFACE 35
Figure 4.4: Lightcones near a rotating black hole. The view is from above:
the small circles are the light cones as seen from above; the small dots within
these circles are the tips (origins) of the cones. Inside the event horizon, light
cones are so heavily tilted towards the singularity, light can not escape from
the singularity; light cones in the ergosphere are tilted in the direction of
the rotation and slightly to the singularity, but light can still go to infinity.
[D’Eath, 1973]
go into the ergosphere from infinity, but they can also leave the ergosphere
and go to infinity. [D’Eath, 1996]
4.6.2 Penrose process
The region between the static limit and the event horizon is called the er-
gosphere because the rotating black hole can do work on particles in this
region (‘ergo’ is Greek for ‘work’). This process of work done by the black
hole on a particle is called the Penrose process.
A particle following a geodesic that enters the ergosphere under some
specific circumstances can decay into two particles A and B inside the ergo-
sphere. The ingoing particle has an energy E:
E=p·ξt(4.29)
which is equal to p0at infinity. This particle decays into two particles A and
B, with energies EAand EB:E=EA+EB. The decay can be done in such
a way that particle B goes through the event horizon into the black hole,
and particle A escapes from the black hole to infinity. Because of (global)
energy conservation
Eblackhole,initial +E=Eblackhole,final +EA(4.30)
Particle B, crossing the event horizon, has a negative energy because
within the ergosphere, the sign of the killing vector ξtchanges. The black
36 CHAPTER 4. ROTATING BLACK HOLES
hole absorbs a negative energy. Paricle A, that goes to infinity will gain that
amount of energy because of energy conservation: EA> E. [D’Eath, 1996]
[Townsend, 1997]
4.6.3 Gravitational redshift
Outside the stationary limit surface it is possible for an observer to remain
static with respect to the distant stars. Analogue to the derivation made
in Foster (2006) for the redshift in the Schwarzschild-metric, it is possible
to do the same derivation for the Kerr-metric, outside the stationary limit
surface. The redshift in the Kerr-metric is given by:
λR
λE
=1(2mr/ρ2)R
1(2mr/ρ2)E
(4.31)
This redshift formulae is only valid if both observer and emitter are outside
the stationary limit surface since both need to be static for the derivation.
Inside the stationary surface it is impossible to remain static due to the
frame-dragging.
Because of different proper-times of the emitter and the observer (assum-
ing they are at different distances to the rotating black hole), both measure
a different frequency of the light one sends to the other. If the receiver
is closest to the black hole, he observes the light to be blueshifted. If the
emitter is closest to the black hole, the receiver observes the light to be
redshifted. In the limit of the emitter going to the stationary limit surface,
the light is redshifted to infinity.
The derivation of the redshift formula above makes explicit use of static
emitters (and receivers). But particles at the stationary limit surface or
inside the ergosphere can not remain static with respect to the distant stars
(lorentz frame at infinity) because of the frame-dragging. And thus the
equation for the redshift does not hold in this case. [Adler, 1976].
4.6.4 Different values for aand M
How does the stationary limit surface depend on the angular momentum
parameter a, mass Mand angle θ? For a= 0 the ergosphere vanishes
because there is no frame-dragging: the stationary limit surface will coalesce
with the event horizon at r= 2m(the inner stationary surface goes to
r= 0). As aincreases, the ergosphere becomes more flattened on top.
If mass Mincreases, the angular momentum parameter adecreases: the
ergosphere becomes more spherical by increment in the z-direction, and
the inner stationary limit surface decreases. The ergosphere is an ellipsoid:
because of the cos2θdependence it flattens at the z-axis as it coalesce with
the event horizon at the z-axis. At the equatorial plane, the ergosphere has
its maximum radius of r= 2m.
4.7. EVENT HORIZON 37
4.7 Event horizon
As stated in Section (3.4.1), an event horizon is a surface that can be con-
sidered as a one-way-membrane: it lets signals from the outside in, but it
prevents signals from the inside to go to the outside. The curvature within
the event horizon is that strong, that particles or photons can not escape
from there to infinity. As well for the Kerr-metric as for the Schwarzschild-
metric is the event horizon a sphere-shaped surface around the black hole
singularity.
The horizon generators are the photons that have no-endpoints and will
for always stay on the horizon. Whereas they follow straight lines for the
Schwarzschild black hole, the null-geodesics are twisted for the Kerr black
hole: they twist around the horizon, as the twists on a barber-pole (see
figure 4.5). This twisting is caused by the frame-dragging: the photons are
within the ergosphere and thus they can not be static. [Misner, 1973]
The Boyes-Lindquist coordinates have a coordinate singularity at the
event horizon, therefore it is not an adequate coordinate system to describe
the rotating black hole at that location. By choosing Kerr-coordinates, the
properties of the event horizon can be made more clear.
4.7.1 Choice of coordinates
The line-element ds2of a photon in Boyes-Lindquist coordinates can be
written as:
0 = gttdt2+ 2g dtdφ +grr dr2+gθθ2+gφφ2(4.32)
Bringing the dr2-term to the other side and divide by dt2and grr gives:
dr
dt 2
=1
grr "gtt + 2g
dt +gθθ
dt 2
+gφφ
dt 2#(4.33)
with 1
grr
=
ρ2(4.34)
If ∆ = 0 then the radial coordinate velocity becomes zero and thus is the
photon unable to move radially any further:
∆=0
r2+a22mr = 0
r=2m±2m2a2
2
r=m±pm2a2(4.35)
r=m1±p1a2/m2(4.36)
38 CHAPTER 4. ROTATING BLACK HOLES
Thus there are two surfaces of null geodesics that have no future endpoints.
The inner surface is called the Cauchy horizon and the outer surface is the
event horizon.
A closer inspection of the expression for the radial coordinate velocity
at the event horizon (eqn. 4.33) gives that if a photon approaches the event
horizon from both sides, it is halted at the horizon: a photon that comes
from infinity and enters the event horizon, will never go further into the black
hole! It will spiral an infinite time around the horizon, as the coordinate time
tgoes to infinity. But that is not wat physical happens, this is caused by
the coordinate singularity of the Boyes-Lindquist coordinates: the grr -term
goes to infinity.
By changing the coordinates of the Kerr-metric to Kerr-coordinates, one
is able to remove this coordinate singularity. It is possible to re-express the
Kerr-metric (using Kerr coordinates) in a different way then in section 4.2.2:
ds2=c2d˜
t2r2
(a2+r2)2a24drd˜
tmr3
(a2+r2)2a2
dr2r2(2mr +r2)
(a2+r2)2a2
"d˜
φ2adtmr
(a2+r2)2a2adr 2mr +r2
(a2+r2)2a2#2
×2amr
(a2+r2)2a22(4.37)
(For checking, it is easiest to rewrite the above expression into equation
(4.6).)
As one can see from this equation, if ∆ = 0 (thus at the event horizon), dr
needs to become smaller than zero for timelike curves (which have ds2>0):
the first term is zero, and all other terms are positive because they are
quadratic. [Thorne, 2005]
The above situation is illustrated in figure 4.5 below. Picture (a) and (b)
correspond to the light cones one gets in Boyes-Lindquist coordinates. In the
figure one is able to see the pinch-off of the light cones when they come nearer
to the event horizon. At the horizon they permit only rotational movement:
all photons become horizon generators. Figures (c) and (d) correspond to
the case of the adapted Kerr-coordinates. As the light cones come closer to
the horizon they do not pinch-off but they tilt over in the direction of the
black hole. And at the event horizon, photons can only move further into
the black hole or be a horizon generator.
4.7. EVENT HORIZON 39
Figure 4.5: Lightcones near the event horizon: (a) and (b) are for Boyes-
Lindquist coordinates, (c) and (d) for Kerr-coordinates. (b) and (d) are
spacetime diagrams, and (a) and (c) are respectively their views from above.
The light-cones in (a) and (b) are pinched-off, in (c) and (d) they tilt over
towards the singularity. [Thorne, 2005]
4.7.2 Time-like vs. space-like
When one passes the static limit, the gtt term changes sign. At the event
horizon, the grr term changes sign as well. One can speculate what this
means for the paths inside the event horizon, for example that the radial
coordinate rcorresponds to a time-paramater and the time-coordinate t
corresponds to a spatial coordinate inside the event horizon.
But the inside of a real black hole is not properly described by the Kerr-
metric. Because after the collapse, the inside does not tend to go to the
Kerr-metric due to gravitational radiation [Novikov 1976].
4.7.3 Different values for aand M
A larger value for the rotation paramater agives a decrease in the radius
of the event horizon and thus means a stronger curvature: the spacetime
manifold has the same curvature, but a shorter distance over which this
curvature is spread. If a= 0 then the event horizon has r= 2mand
the Cauchy horizon has r= 0. A larger value for the mass-parameter m
increases the radius of the outer horizon, but decreases the Cauchy horizon.
40 CHAPTER 4. ROTATING BLACK HOLES
Chapter 5
Geodesics around a
Kerr-black hole
This chapter is about the paths of free test particles outside a Kerr black
hole. As these are freely falling particles, they are described by geodesics. To
give these geodesics, one needs to find the expressions for the four-velocity
uµ. These expressions are derived in the first section of this chapter. After
obtaininig uµ, two types of motion are discussed for different values of the
parameters: radial r-motion and axial θ-motion.
5.1 Four constants of motion
To find expressions for the geodesics, one needs four constants of motion.
Geodesics are often derived with the use of the Lagrangian Land the Euler-
Langrange equations, however, here we will use the Hamilton-Jacobi ap-
proach because this method will give us the fourth constant of motion. The
derivation of the four constants of motion presented below, is analogue to
the derivation given in [Carter, 1968] (and [Misner, 1973]).
The Hamiltonian is given by:
H(xµ, pµ) = pµ˙
xµL(xµ,˙
xµ) (5.1)
with the dot indicating a derivative with respect to the affine parameter λ.
For the case of a free particle, the Lagrangian Lis
L(xµ,˙
xµ;t) = 1
2gµν ˙
xµ˙
xν(5.2)
The conjugate momenta pµis defined as
pµ∂L
˙
xµ=gµν ˙
xν(5.3)
41
42 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
The leads to the expression
H=gµν ˙
xµ˙
xν1
2gµν ˙
xµ˙
xν=1
2gµν ˙
xµ˙
xν=L(5.4)
The equality of the Hamiltonian to the Lagrange indicates there is no po-
tential energy involved, as would be expected since this was the case for the
Schwarzschild black hole as well.
Inverting the expression for the conjugate momentum above gives an
equation for ˙xµin terms of the conjugate momentum:
˙xµ=gµν pν(5.5)
This gives the following expression for the Hamiltonian (using (5.3) and
(5.5)
H=1
2gµν ˙
xµ˙
xν=1
2pµ˙xµ=1
2gµν pµpν(5.6)
To find the explicit equation for the Hamiltonian, one has to invert
the metric gµν to gµν . The expression for the inverse metric (in Kerr-
coordinates) is given by the inverse of the line-element ds =gµνdxµdxν:
∂s 2
=ρ2
∂θ 2
+ 2ρ2r2+a2
∂r
˜
V
+ 2ρ2a
∂r
˜
φ+ 2ρ2a
˜
V
˜
φ
+ρ2asin2θ
˜
V2
+ρ2sin2θ
˜
φ2
+ρ2
∂r 2
(5.7)
Replacing
∂xµwith pµone obtains the following equation for the Hamil-
tonian:
H=1
2ρ2np2
r+ 2 h(r2+a2)p˜
V+ap˜
φipr+p2
θo
+1
2ρ2hasin θp ˜
V+ sin1θp ˜
φi2(5.8)
A constant of motion is defined as the pµfor which H
∂xµ= 0. In the
case of a rotating black hole, the Hamiltonian is not explicitly dependent
on tand φsince the Kerr spacetime is symmetric in coordinate time and
the axial coordinate. Therefore one is able to define these two constants of
motion:
p˜
V=g˜
V ν ˙xν=E(5.9)
p˜
φ=g˜
φν ˙xν=L(5.10)
5.1. FOUR CONSTANTS OF MOTION 43
Where Ethe energy of the test-particle at infinity is, and Lthe angular
momentum around the symmetry axis. Thus the first two constants of
motion are Eand L
A third constant of motion follows from the relation gµν ˙xµ˙xν=δwhere
δ= 1 for the case of a timelike geodesic, δ= 0 for null geodesic and δ=1
for spacelike geodesics (this is the same as in the Schwarzschild case). It can
be considered as a constant related to the rest mass of the particle. This
leads to
H=1
2gµν pµpν=1
2δ(5.11)
So far, there are three constants of motion: energy E, angular momen-
tum Land rest-mass δ. With the Hamilton-Jacobi method is it possible to
obtain a fourth constant of motion, named after its ‘inventor’ Carter. The
solution of the Hamilton-Jacobi method will be formulated in terms of all
the constants of motion. [web1]
By the definition of the Hamilton-Jacobi method, the Hamilton-Jacobi
equation is given by:
∂S
∂λ =H=1
2gµν pµpν=1
2δ(5.12)
Where Sthe Jacobi action is. If S is a solution to the Hamilton-Jacobi
equation, then S
∂xi=pi, where piis a constant of motion. Assuming the
case of a solution for S, consisting of variables that can be seperated, Scan
be expressed in the constants of motion:
S=1
2δλ E˜
V+L˜
φ+Sθ+Sr(5.13)
In which Sθis a function of θand Sra function of r.
Inserting the expression of S into the Hamiltonian by making the partial
derivatives of S(5.8); using pi=S
∂xiand multiplying everything with 2ρ2,
gives:
2ρ2∂S
∂λ = Sr
∂r 2
+ 2 (r2+a2)2(E) + aLSr
∂r
+∂Sθ
∂θ 2
+asin θ(E) + sin1θL2=ρ2δ2(5.14)
Rearranging gives:
∂Sθ
∂θ 2
+asin θE sin1θL2+a2δ2cos2θ=
∂Sr
∂r 2
+ 2 (r2+a2)2EaLSr
∂r r2δ2(5.15)
44 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
Since both sides depend on different variables, for the expression to hold
along the geodesic, both sides must be equal to the same constant. This
constant is called the Carter constant Kand is the fourth constant of motion.
While using again the relation pi=∂S
∂xi, the two equations for Klook like:
p2
θ+aE sin θLsin1θ2+a2δ2cos2θ=K(5.16)
p2
r2r2+a2EaLpr+δ2r2=K(5.17)
Solving these two quadratic equations, gives two expressions for pθand pr:
pθ=Θ (5.18)
pr= ∆1(P+R) (5.19)
with
Θ = K(LaE)2cos2θ[a2(δ2E2) + L2sin2θ] (5.20)
P=E(r2+a2)La (5.21)
R=P2∆(δ2r2+K) (5.22)
This then gives (using ˙xµ=gµν pνand equations (5.8), (5.9), (5.10),
(5.18), (5.19): the equations for the Hamiltonian and the constants of mo-
tion):
ρ2˙
θ=Θ (5.23)
ρ2˙r=R(5.24)
ρ2˙
˜
V=a(aE sin2θL)+(r2+a2)∆1[R+P] (5.25)
ρ2˙
˜
φ=(aE Lsin2θ) + a1[R+P] (5.26)
which gives the four-velocity uµ. The signs of square roots of Θ and Rcan
be chosen indepentdently, but one must be consistent in that choice.
The final solution (integrating equations (5.18) and (5.19) to θand r
respectively to obtain Sθand Sr) for the Jacobi action is then given by
S=1
2δλ E˜
V+L˜
φ+Sθ+Sr
=1
2δ2λE˜
V+L˜
φ
+Zθ
(Θ)+Zr
1P dr +Zr
1(R)dr (5.27)
Differentiating the Jacobi action with respect to the four constants of
5.1. FOUR CONSTANTS OF MOTION 45
motion (K, δ, E and L) gives respectively:
Zθ
Θ=Zrdr
R(5.28)
λ=Zθa2cos2θ
Θ+Zrr2
Rdr (5.29)
˜
V=Zθa(aE sin2θL)
Θ
+Zrr2+a2
1 + P
Rdr (5.30)
˜
φ=Zθ(aE Lsin2θ)
Θ
+Zra
1 + P
Rdr (5.31)
Which are the first-integrals of motion. Again, the signs of the two squares
can be chosen indepently.
As for a check on the results obtained, one can rewrite the first-integral
for the coordinate time and compare it to the Schwarzschild first-integral
for coordinate time (equation (3.32)) given below:
t=ZrEdr
(1 2m/r) [E2(1 2m/r)(δ+L2/r2)]1/2
By using the relation for the coordinate transformation from Boyes-Lindquist
coordinate time to Kerr-coordinates (equation (4.6):
dt =d˜
Vr2+a2
dr
while setting the rotation parameter ato zero, one obtains the following:
t=Zrr2
dr +Zr(r2)P
Rdr Zrr2
dr
=Zr(r2+a2)
Rdr (5.32)
the equation becomes
t=Zrr2P
Rdr
=Zrr4Edr
r2(1 2m/r) [E2r4r4(1 2m/r)(δ2+K/r2)]1/2
=ZrEdr
(1 2m/r) [E2(1 2m/r)(δ2+K/r2)]1/2(5.33)
46 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
This is equal to the expression for a massive test-particle of the Schwarzschild
black hole. Notice that in this case, the Carter’s constant Kequals L2.
However, the geometrical interpretation of the Kis unclear.
Now the equations for the motion along geodesics are given it is possible
to say some things about the geodesics.
5.2 θ-motion
The equation describing the θ-motion of particles is equation (5.23):
ρ4˙
θ2=K(LaE)2cos2θ[a2(δ2E) + L2sin2θ] (5.34)
Note that it is the ˙
θsquared: a positive value indicate a positive or negative
θ-velocity, but a negative ˙
θ2corresponds to an imaginary axial velocity.
It is convenient to write ˙
θ2as a function of u, where ucos θ. Then
u= 1 corresponds to the z-axis (θ= 0), u= 0 corresponds to the equatorial
plane and 0 < u2<1 corresponds to 0 < θ < π without the point θ=π/2.
f(u) = ρ4˙
θ2=QQ+L2a2Γu2a2Γu4(5.35)
In which QK(aE L)2and Γ E2δ2. [Stewart, 1973]
5.2.1 Low energy particles
In this section, the discussed particles have low energies, around the order
of one mass-energy.
Figure 5.1: θ-motion for the case of low energy particles. Figure A corre-
sponds to L= 0, B to L > 0 and C to L2> a2Γ. [Stewart, 1973]
If a particle has a low energy and no angular momentum (L= 0), then
the radial velocity is zero at the z-axis (f(u) = 1): a stationary point. (See
the figure 5.1 A). The second derivative of θis zero at u= 1, thus a particle
can have an orbit at the z-axis. However, it is an unstable orbit for small
θ-perturbations. This is the only case of θ-motion that a particle can have
an orbit at the z-axis.
5.2. θ-MOTION 47
As the angular momentum increases, the stationary point will shift to-
wards z-axis, away from the equatorial plane. But for these cases, the
stationary point is not a point of an orbit. The black hole is pulling the
particles towards the equatorial plane due to the frame-dragging. (See fig-
ure 5.1 B). The particle will follow an oscillatory motion that crosses the
equatorial plane repeatedly, with θlying in the range θ0θπθ0, where
u0= cos θ0, for which f(u0) = 0.
A further increase in the angular momentum, decreases the value for
f(u0) = Q: the difference between Eand Lincreases. The particle will
cross the equatorial plane with a lower θ-velocity. If Q= 0 (and L2> a2Γ),
then f(u0) = 0: the particle has a stable orbit at the equatorial plane, since
u= 0 the only valid value for uis. In this case, there is no θvelocity. (See
figure 5.1 C.)
If L= Γ = 0 in addition to Q= 0, then the particle can have any θ
value as a constant value since ˙
θ= 0 for all u.
A larger value for the angular momentum would mean a further decrease
in Q,Qbecoming more negative. Then f(u) is negative for every θ, which
is not a physical possible situation. It is not possible for a particle to have
such a large angular momentum and at the same time low energy. [Stewart,
1973] [Carter, 1968]
5.2.2 High-energy particles
Figure 5.2: θ-motion for the case of high energy particles. Figure A corre-
sponds to L= 0, B to L2< a2Γ and C to L > 0. [Stewart, 1973]
A high energy particle with a small angular momentum L, has a small Q:
Q < 0. The particle will have an oscillatory motion between two angles θ1
and θ2in case of the following inequality: (Q+L2a2Γ)2+4a2ΓQ0. This
inequality must hold to have such a maximum. If the inequality does not
hold, there is no real solution to the quadratic equation for which u-value
f(u) = 0 and thus no possible physical situation.
The motion will not cross the equatorial plane. (See for reference, figure
5.2 A.) (Again, in the case of L= 0, it is able to cross the z-axis.) This
48 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
motion also has a stable orbit for the case θ1=θ2, if the inequality is an
equality.
As for particles with a larger angular momentum, the next special case
is when Q= 0. For particles with high energies (L2< a2Γ), this situation
is different than for particles with low energies. The point u1has shifted
towards the equatorial plane. And it is possible to have an unstable orbit
at the equatorial plane. But the general motion is still oscillatory between
0uu0. The maximum umis caused by the rotation of the black hole.
Further increase in the angular momentum would lead to a situation
which is equal to the second case of the low energy particles (figure 5.1 B.):
an oscillatory motion between the equatorial plane and θ0. [Stewart, 1973]
[Carter, 1968]
5.3 r-motion
Just as one can do for the θ-motion, ii is possible to rewrite the expression
for ρ2˙r2:
ρ2˙r2R(r)=Γr4+ 22r3+ (a2E2L2a2δ2Q)r2
+ 2m[(aE L)2+Q]ra2Q(5.36)
Recall that Q=K(aE L)2and Γ E2δ2.R(r) is the square of radial
velocity, and can therefore not be negative: a negative value for R(r) gives
a non-physical (imaginary) value for the radial velocity. Because ∆ = 0 at
the event horizon r+,R(r+) needs to be equal or larger than zero at the
event horizon (this follows easily from the original equation for ρ2˙r2(5.26)).
Some specific values for R(r) are: R(0) = a2Q, and R(r)Γr4as
r→ ∞. A negative value for Γ corresponds to E < δ. Such a particle does
not have enough energy to go to infinity as it is bound by the black hole (E
is not large enough compared to the ’effective potential’ of the black hole).
There are four different cases which will be discussed, depending on the
signs of Γ and Q. In this treatment, a distinction is being made between
prograde orbits (Lis positive and same direction as the rotation of the black
hole) and retrograde orbits (Lis negative and opposite to the direction or
black hole rotation), because the frame-dragging forces the particle to rotate
with the rotation of the black hole. This results that retrograde orbits
further away then prograde orbits, will result in the particle being trapped
into the black hole. Frame-dragging slows particles in retrogrades orbits
down. [Chandrasekhar, 1983]
5.3.1 Case 1: Q > 0,Γ>0
If the energy E is large enough, all the coefficients in equation (5.36) are
non-negative, with the exception of the coefficient of r0: the particles are
not able to reach the singularity, due to their low energy. [Carter, 1986]
5.3. R-MOTION 49
Figure 5.3: r-motion for the case of Q > 0, Γ >0 on the left, and Q > 0,
Γ<0 on the right. F.O. means free orbis, T.O. trapped orbit, B.O. bound
orbit. A solid line corresponds to retrograde orbits (large negative L), and
a dotted line corresponds to prograde orbits (large positive L). [Stewart,
1973]
Since R(0) <0 and R(r+)0 there has to be a zero between those
points, at r=r1, in or on the horizon (see figure 5.3). Consider a particle
moving from infinity inwards in the case of the solid line. Because ˙r2>0,
the particle will move inwards ( ˙r < 0), cross the event horizon and move up
to the point r1. Then it will reverse in motion, but it is not able to cross the
event horizon again and it is trapped. (Trapped orbits are orbits which are
partially within the event horizon: the particles with this orbit will become
trapped into the black hole.) If a particle has ˙r > 0 (it moves away from
the black hole), it is able to go to infinity.
Suppose Lis increased to a large positive value (prograde orbits), leaving
Eand Qfixed. It follows that the shape of R(r) will change to the dotted
line in the graph. There are two additional zeros at r2
=2mand at r3>2m.
For the case of a dotted line, no particle is allowed in the range of r2<
r < r3. Any particle with r < r2is trapped inside: because of its prograde
orbit, it is not able to resist the frame-dragging, and because of its close
proximity it will cross the event horizon in the end. However, particles at
r > r3move along a parabolic orbit around the black hole: they stay far
away from the black hole to prevent being trapped by it. In the special case
of r2=r3, a particle at r2is in an unstable, spherical orbit around the black
hole. This orbit is unstable for small r-perturbations.
5.3.2 Case 2: Q > 0,Γ<0
As the energy of the particles of Case 1 decrease, they will eventually have
energies of E < δ. These particles have a lower velocity and are less able
to escape the gravitational ‘pull’ from the black hole. Since R(0) <0,
R(r+)0 and R(r)→ −∞ as r→ ∞, there have to be at least two real
50 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
zeros. In figure 5.3 above the case for two zeros (solid line) is illustrated.
Particles with r1r+r2are trapped, other values for rare not
allowed. If the particle has ˙r > 0, then he would follow an elliptical orbit
largely within the ergosphere, but once inside the event horizon, it will not
escape. In the case of r1=r2there is a special case where the particle orbits
on the event horizon. This orbit is not stable, because small perturbations
will cause the particle to cross the event horizon and disappear within it.
By changing the value for Lto larger values, keeping Eand Qfixed, one
gets the dotted line. As in the previous case, particles withing r1rr2
are trapped. There are two additional zeros r3and r4. Particles in the range
of r3rr4are bound and they will oscillate within that range and follow
elliptical shaped orbits. The case of r3=r4gives a stable spherical orbit.
5.3.3 Case 3 and 4: Q < 0,Γ>0or Γ<0
Figure 5.4: r-motion for the case of Q < 0, Γ >0 on the left, and Q < 0,
Γ<0 on the right. F.O. means free orbis, T.O. trapped orbit, B.O. bound
orbit. A solid line corresponds to retrograde orbits (large negative L), and
a dotted line corresponds to prograde orbits (large positive L). [Stewart,
1973]
These two cases are rather similar to the first cases, but are about par-
ticles with a larger angular momentum (and energy), causing Qto become
negative. In these cases there is a positive crossing with the R(r) axis:
trapped particles are able to reach r= 0. In both cases, there is a zero less
than in the corresponding cases of positive Q.
5.4 Equatorial motion
For orbits in the equatorial plane, particles must have Q= 0 and L2> a2Γ,
because those particles are stable versus θ-perturbations. [Stewart, 1973]
For deriving the expression of the effective potential V(r), similar as the
effective potential for the Schwarzschild black hole (see section 3.6.2), one
5.4. EQUATORIAL MOTION 51
should rewrite equations (5.23) and (5.24) to obtain respectively
K=ρ2˙
θ2
+L2+a2E22aLE +a2δ2cos2θ
a2E2cos2θ+L2cos2/sin θ(5.37)
K=ρ2˙r2+E2(r2+a2) + L2a2
2LEa(r2+a2)δ2r2(5.38)
When these equations are made equal, it is possible to bring all terms to
one side and come to an expression of the form
αE22βE +γr4˙r2= 0 (5.39)
using θ=π/2 for the equatorial plane ( ˙
θ=¨
θ= 0)
α=r2+a22a2sin2θ(5.40)
β= 2mrLa (5.41)
γ=L2a2δ2r2L2(5.42)
(5.43)
Solving equation (5.39) to find an equation for Eleads to:
E=β+pβ2αγ +αr4˙r2
α(5.44)
The allowed regions for a particle are those with EV(r), where V(r) is
the effective potential, the minimum allowed value of Eat a radius r:
V(r) = β+pβ2αγ
α(5.45)
The effective potential gives information about the radial motion. It depends
on the angular momentum Land the radial distance r. If E > V for every
rand some L0, then all particles with energy Eand angular momentum
L0are able to just fall into the black hole. However, if the energy E1of a
particle with L1is lower than some V(r1, L1), then the particle is not able
to come more closely to the black hole then r1, and depending on the shape
of the effective potential, it could follow a circular, elliptical or parabolic
orbit with r1as one turning point. [Misner, 1973]
Null-geodesics
Photons (δ= 0) follow null-geodesics. Two plots are made for two differ-
ent values for the angular momentum of a photon are in figure 5.5 below:
L= 2mE (prograde) and L=2mE (retrograde). As can be seen, these
photons are able to follow circular orbits close to r= 2m(stationary surface
52 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
Figure 5.5: Two effective potentials for photons with different angular mo-
mentum: prograde (L= 2mE) and retrograde (L=2mE).
limit) and r= 4mrespectively. Both are unstable against r-perturbations.
[Misner, 1973] [Stewart, 1973]
The radius of the retrograde photon is larger because of the rotation of
the black hole. Would this photon come nearer to the black hole, then the
rotation of the spacetime geometry would force the photon to change its
direction of rotation towards the direction of rotation of the black hole. As
the photon loses it’s rotational velocity, it will fall into the black hole. (See
figure 5.6.)
Figure 5.6: The orbit of a retrograde photon. [Chandrasekhar, 1983]
5.4. EQUATORIAL MOTION 53
Timelike-geodesics
A particle with a test-mass has δ= 1. A specific case is the innermost
stable circular orbit for timelike particles at r=mand r= 9m(depending
on their angular momentum). [Misner, 1973]
Figure 5.7: Two effective potentials for timelike geodesics in an Extreme
Kerr geometry (a=m). As can be seen in the diagram, the potential of
L= 2m/3 has an stable orbit at r=m, and for L=22m/33 it has a
minimum at r= 9m.
The same reasoning about why the retrograde orbit for a photon has a
larger radius can be applied to the retrograde orbit of a timelike particle.
Schwarzschild case
Setting the rotation paramater ain the expression for the effective potential
to zero, gives the effective potential for the Schwarzschild black hole:
α=r4(5.46)
β= 0 (5.47)
γ=δ2r2+L2=r22mrδ2r2+L2(5.48)
(5.49)
Giving:
V2=αγ
α2=r4r22mrδ2r2+L2
r8
=
r212m
rr2L2
r2+δ2
r4
=12m
rL2
r2+δ2(5.50)
Which is the same expression as equation (3.38) for timelike particles.
54 CHAPTER 5. GEODESICS AROUND A KERR-BLACK HOLE
Chapter 6
Conclusion
A rotating black hole is described by the Kerr-metric. Using different co-
ordinate systems, different properties or features of a rotating black hole
can be described. One feature of such a (theoretical) black hole is it’s ring
singularity. This ring can only be reached by particles following specific
geodesics. Around this singularity is the Cauchy-horizon. With an increas-
ing radial distance one reaches the event horizon and then the stationary
limit surface. Once inside the event horizon, particles are not able to escape
to the outside of the event horizon. Within the ergosphere it is impossible
to be static due to the strong frame-dragging caused by the rotation of the
black hole. Particles are forced to rotate with the direction of the rotation
of the black hole. Outside the stationary limit surface it is possible to be
static, but such a particle would not follow a geodesic (only at infinity is the
frame-dragging zero).
The frame-dragging, or rotation of spacetime, has a very strong influence
on the geodesics of particles (with and without mass), their orbits are for
example twisted around the black hole; depend strongly on the sign of the
angular momentum of the particle (whether it follows a retrograde or pro-
grade orbit); and there are no stable θ-motions outside the equatorial plane.
The possible geodesics put restraints on the energy and angular momentum
of the particles in an accretion disk in the equatorial plane of the black hole:
all particles rotate with the direction of the rotation of the black hole, other-
wise all the retrograde particles would disrupt the disk; the particles should
have energy larger then their rest-energy; and in combination with their
angular momentum should satisfy the conditions for a stable orbit in the
equatorial plane. Via collisions, the disk should be able to accrete particles
which do initially not satisfy those conditions. The rotation of the black hole
also influences the ‘images’ of a galaxy caused by the gravitational lensing:
photons that pass by the black hole from the left are differently bend by
the black hole than photons that pass by on the right. The image will be
distorted because of the difference between retrograde and prograde orbits.
55
56 CHAPTER 6. CONCLUSION
Although a rotating black hole has large influence on the surrounding
matter, it has never been directly observed since it does not emit photons.
But black holes are perhaps directly observed when large interferometers
are build for the detection of gravitational waves. And that could lead to
answers to questions about black holes, for example about supermassive and
primordial black holes.
Chapter 7
References
Adler, 1975 R. Adler, M. Bazin & M. Schiffer, Introduction to general
relativity, second edition, McGraw-Hill, 1975
Begelman, 1995 M. Begelman & M.Rees, Gravity’s Fatal Attraction, Black
Holes in the Universe, New York, Scientific American Library, 1995
Carr, 1974 B.J. Carr & S.W. Hawking, ‘Black holes in the early universe’,
Royal Astronomical Society (1974) p399-415
Carter, 1968 B. Carter, ‘Global Structure of the Kerr Family of Gravita-
tional Fields’, Physical Review (1968) p1559-1571
Chandrasekhar, 1983 S. Chandrasekhar, The Mathematical Theory of
Black holes, New York, Oxford University Press, 1983
D’Eath, 1996 P.D. D’Eath, Black Holes, Gravitational Interactions, New
York, Oxford University Press, 1996
Foster, 2006 J. Foster & J.D. Nightingale, A Short Course in General
Relativity, third edition, New York, Springer, 2006
Misner, 1973 C.W. Misner, K.S. Thorne & J.A. Wheeler, Gravitation,
New York, W.H. Freeman and Company, 1973
Hawking, 1974 S.W. Hawking & G.F.R. Ellis, The Large Scale Structures
of Space-Time Cambridge, Cambridge University Press, 1974
Hawking, 1988 S. Hawking, Het Heelal, Amsterdam, Uitgeverij Bert Bak-
ker, 1988
Rees, 2007 M.J. Rees & M. Volonteri, ‘Massive Black holes: formation
and evolution’, from: arXiv:astro-ph/0701512v1, 2007
Sexl, 1979 R. Sexl & H. Sexl, White Dwarfs-Black Holes, An Introduction
to Relativistic Astrophysics, New York, Academic Press Inc, 1979
57
58 CHAPTER 7. REFERENCES
Stewart, 1973 J.Stewart & M. Walker, ‘Black Holes: the Outside Story’,
Springer Tracts in Modern Physics, 1973, volume 69
Taylor, 2000 E.F. Taylor & J.A. Wheeler, Exploring Black Holes-Intro-
duction to General Relativity, Addison Wesley Longman, 2000
Townsend, 1997 P.K. Townsend, Black Holes, from: http://arXiv:gr-qc/-
9707012v1, 1997
web1 Hamilton-Jacobi Equation, retrieved June 26, 2007 from: http://-
beige.ucs.indiana.edu/B673/node48.html
web2 Kerr’s rotating Black Holes, retrieved May, 13, 2007 from: http://-
www.astro.ku.dk/cramer/RelViz/text/exhib3/exhib3.html
web3 Chandra, Field Guide to X-ray Sources, Supermassive Black Holes
(2006) Retrieved July, 2, 2007 from: http://chandra.harvard.edu/-
xray sources/blackholes sm.html
web4 Beth Brown - Howard University Senior Thesis (1991) Retrieved
June, 21, 2007 from: http://www.physics.howard.edu/students/beth/
bh front.html
web5 http://www.astro.rug.nl/onderwys/sterIIproject98/louise/
web6 http://en.wikipedia.org/wiki/Gravitational waves
Zeilik, 1998 M. Zeilik & S.A. Gregory, ‘Introductory Astronomy & Astro-
physics’,United States, Thomson Learning, 1998.
fig1 http://www.ifa.hawaii.edu/barnes/ast110 06/bhaq/Accretion disk.png
fig2 http://tycho.bgsu.edu/laird/cp images/black holes.html
fig3 http:// www.644net.com/V1-5 files/image074.gif
Cover image: a rotating black hole with an accretion disk and
two jets. From: http://esamultimedia.esa.int/images/spcs/xmm-
newton/xmm20050110b.jpg.
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