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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:

10−5g 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:

c2dτ2=c21−2m

rdt2−1−2m

r−1

dr2−r2dθ2−r2sin2θdφ2(3.1)

with m≡MG

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 dτ, a time-interval is given by:

dτ = (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 dτEbetween two signals. The

receiver measures a time-interval dτRbetween two signals. The frequency

of the signals measured by the emitter is νE≡n/∆τE, where nthe number

of signals is, emitted by the emitter, during a time-interval ∆τE. But the

receiver measures a frequency of νR≡n/∆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

=1−2m/rE

1−2m/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:

v≡t+r+ 2mln (r/2m−1) (3.5)

This leads to a Schwarzschild metric of:

dτ2=1−2m

rdv2−2dvdr −r2dθ2−r2sin2θdφ2(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: dθ =dφ = 0. The metric is given by:

0 = 1−2m

rdt2−1−2m

r−1

dr2(3.7)

Rewriting gives:

dr

dt 2

=1−2m

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 = 1−2m

rdv2−2dvdr (3.9)

Rearranging gives:

dv

dr 1−2m

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

1−2m/r .

Using this in combination with the ﬁrst solution for dv/dr gives:

dt

dr =−1

1−2m/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

1−2m/r (3.14)

v= 2r+ 4mln |r−2m|+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 =dθ =

dφ = 0. This gives a line-element for a timelike observer of:

ds2=1−2m

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,dθ or dφ 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"1−2m

r˙

t2−1−2m

r−1

˙r2−r2˙

θ2+ sin2θ˙

φ2#(3.19)

The corresponding canonical momenta pµ=∂L

∂˙xµare given by:

pt=1−2m

r˙

t(3.20)

pr=−1−2m

r−1

˙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=1−2m

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˙

φ−L=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

1−2m/r −˙r2

1−2m/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):

1−2m

r−1

¨r+m

r2˙

t2−1−2m

r−2m

r2˙r2−r˙

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

dτ 2

+1−2m

rδ+L2

r2=E2(3.30)

Using the above equation with the expression (3.26) for the angular momen-

tum with u≡1/r one gets

du

dφ2

= 2mu3−u2+2m

L2u−1−E2

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

dτ 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

dτ 2

+V2=E2(3.37)

V2=1−2m

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

dλ2

+1−2m

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

dφ2

= 2mu3−u2+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

dλ =±E(3.41)

using equation (3.25) this becomes

dr

dλ =±1−2m

r(3.42)

Integrating it gives:

t=±hr+ 2mlog r

2m−1i+C(3.43)

where Cis a constant. Integrating the expression (3.41) with respect to dλ

gives

r=±Eλ +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

=

1−2m/r

r2) this equality becomes:

1−2m/r

r2=m

r3

r−2m=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]:

c2dτ2=1−2mr

ρ2c2dt2+4mcra sin2θ

ρ2dtdφ −ρ2

∆dr2

−ρ2dθ2−r2+a2sin2θ+2mra2sin2θ

ρ2dφ2(4.1)

In which:

ρ2≡r2+a2cos2θ(4.2)

∆≡r2+a2−2mr (4.3)

Furthermore: m≡GM/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 =dθ =dφ = 0, then

dτ2=1−2mr

ρ2dt2(4.5)

giving dτ =dtp1−2mr/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 dτ =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 gtφ-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=dφ +a

∆dr

By using these coordinates for the Kerr-metric one obtains the following

metric [Novikov, 1989]:

ds2=1−2mr

ρ2d˜

V2−2drd ˜

V−ρ2dθ2

−ρ−2hr2+a22−∆a2sin2θisin2θd ˜

φ2+ 2asin2θd ˜

φdr

+4amr sin2θ

ρ2d˜

φd ˜

V(4.6)

with

ρ2≡r2+a2cos2θ(4.7)

∆≡r2+a2−2mr (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+√m2−a2(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=gtφ ~

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

21−2mr

ρ2˙

t2+2mra sin2θ

ρ2˙

t˙

φ−ρ2

2∆ ˙r2

−ρ2

2˙

θ2−1

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˙

t−r2+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=dφ

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 dθ. 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:

Ω≡dφ

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 =dφ = 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 + 2Ωgtφ + 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 + 2Ωgtφ + Ω2gφφ <0 (4.20)

Solving this quadratic equation for Ω gives:

Ω = −gtφ

gφφ ±sgtφ

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≡ −gtφ

gφφ

=2mcra

(r2+a2)ρ2+ 2mra2sin2θ=1

2(Ωmin + Ωmax) (4.22)

with

Ωmin =ω−qω2−gtt/gφφ (4.23)

Ωmax =ω+qω2−gtt/gφφ (4.24)

Ωmin <Ω<Ωmax (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

Ωmin 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: rΩmin =−cand rΩmax =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+√m2−a2the 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+√m2−a2cos2θ. 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 =dθ =

dφ = 0. This gives a line-element for a timelike observer of:

ds2=gttdt2=1−2mr

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:

1−2mr

ρ2= 0

ρ2−2mr = 0

r2+a2cos2θ−2mr = 0 (4.27)

Solving this quadratic eguation to rgives:

r=m±pm2−a2cos2θ(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+√m2−a2cos2θ)

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

dφ 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+ 2gtφ dtdφ +grr dr2+gθθdθ2+gφφdφ2(4.32)

Bringing the dr2-term to the other side and divide by dt2and grr gives:

dr

dt 2

=1

grr "gtt + 2gtφ

dφ

dt +gθθ dθ

dt 2

+gφφ dφ

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+a2−2mr = 0

r=2m±2√m2−a2

2

r=m±pm2−a2(4.35)

r=m1±p1−a2/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)2−a2∆−4drd˜

tmr3

(a2+r2)2−a2∆

−dr2r2(2mr +r2)

(a2+r2)2−a2∆

−"d˜

φ−2adtmr

(a2+r2)2−a2∆−adr 2mr +r2

(a2+r2)2−a2∆#2

×2amr

(a2+r2)2−a2∆2(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˙

xµ−L(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˙

xν−1

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

=ρ−2∂

∂θ 2

+ 2ρ−2r2+a2∂

∂r ∂

∂˜

V

+ 2ρ−2a∂

∂r ∂

∂˜

φ+ 2ρ−2a∂

∂˜

V ∂

∂˜

φ

+ρ−2asin2θ∂

∂˜

V2

+ρ−2sin2θ∂

∂˜

φ2

+ρ−2∆∂

∂r 2

(5.7)

Replacing ∂

∂xµwith pµone obtains the following equation for the Hamil-

tonian:

H=1

2ρ−2n∆p2

r+ 2 h(r2+a2)p˜

V+ap˜

φipr+p2

θo

+1

2ρ−2hasin θp ˜

V+ sin−1θp ˜

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) + aL∂Sr

∂r

+∂Sθ

∂θ 2

+asin θ(−E) + sin−1θL2=ρ2δ2(5.14)

Rearranging gives:

∂Sθ

∂θ 2

+asin θE −sin−1θL2+a2δ2cos2θ=

−∆∂Sr

∂r 2

+ 2 (r2+a2)2E−aL∂Sr

∂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 θ−Lsin−1θ2+a2δ2cos2θ=K(5.16)

∆p2

r−2r2+a2E−aLpr+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−(L−aE)2−cos2θ[a2(δ2−E2) + L2sin−2θ] (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 sin2θ−L)+(r2+a2)∆−1[√R+P] (5.25)

ρ2˙

˜

φ=−(aE −Lsin−2θ) + a∆−1[√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

2δ2λ−E˜

V+L˜

φ

+Zθ

(√Θ)dθ +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:

Zθdθ

√Θ=Zrdr

√R(5.28)

λ=Zθa2cos2θ

√Θdθ +Zrr2

√Rdr (5.29)

˜

V=Zθ−a(aE sin2θ−L)

√Θdθ

+Zrr2+a2

∆1 + P

√Rdr (5.30)

˜

φ=Zθ−(aE −Lsin−2θ)

√Θdθ

+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˜

V−r2+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) [E2r4−r4(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−(L−aE)2−cos2θ[a2(δ2−E) + L2sin−2θ] (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 u≡cos 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=Q−Q+L2−a2Γu2−a2Γu4(5.35)

In which Q≡K−(aE −L)2and Γ ≡E2−δ2. [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 θ0≤θ≤π−θ0, 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+L2−a2Γ)2+4a2ΓQ≤0. 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

0≤u≤u0. 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˙r2≡R(r)=Γr4+ 2mδ2r3+ (a2E2−L2−a2δ2−Q)r2

+ 2m[(aE −L)2+Q]r−a2Q(5.36)

Recall that Q=K−(aE −L)2and Γ ≡E2−δ2.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 r1≤r+≤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 r1≤r≤r2

are trapped. There are two additional zeros r3and r4. Particles in the range

of r3≤r≤r4are 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+a2E2−2aLE +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

αE2−2βE +γ−r4˙r2= 0 (5.39)

using θ=π/2 for the equatorial plane ( ˙

θ=¨

θ= 0)

α=r2+a22−∆a2sin2θ(5.40)

β= 2mrLa (5.41)

γ=L2a2−∆δ2r2−∆L2(5.42)

(5.43)

Solving equation (5.39) to ﬁnd an equation for Eleads to:

E=β+pβ2−αγ +αr4˙r2

α(5.44)

The allowed regions for a particle are those with E≥V(r), where V(r) is

the eﬀective potential, the minimum allowed value of Eat a radius r:

V(r) = β+pβ2−αγ

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/3√3 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=−r2−2mrδ2r2+L2(5.48)

(5.49)

Giving:

V2=−αγ

α2=r4r2−2mrδ2r2+L2

r8

=

r21−2m

rr2L2

r2+δ2

r4

=1−2m

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

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Royal Astronomical Society (1974) p399-415

Carter, 1968 B. Carter, ‘Global Structure of the Kerr Family of Gravita-

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Chandrasekhar, 1983 S. Chandrasekhar, The Mathematical Theory of

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58 CHAPTER 7. REFERENCES

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