Trajectory of a Lightlike Particle Around a Black Hole

Seyyed Mohammad Reza Taheri ([email protected])

Mohammad Saber Naderi ([email protected])

Supervisor: Dr. Hossein Farajollahi

September-2007

Abstract

This text is about motion of lightlike particles around massive objects in area’s of General

Relativity, in this way a computer software (Maple 10) is used to solve equations and plot

trajectories.

Motion of nulllike particles are analyzed in three cases (spacetime):

1. Schwarzschild spacetime.

2. Kerr spacetime.

3. Kerr–Newman spacetime.

and equations of motion of them are obtained by three methods:

1. Lagrangian Method.

2. Hamiltonian Method.

3. Geodesic Equation.

At last results of diﬀerent methods and diﬀerent spacetimes are compared.

1

Acknowledgements

At ﬁrst we would like to thank our supervisor Dr. H. Farojollahi for his advices and guidance

in our research.

We’re also thankful for funding provided by the Guilan University. And ﬁnally, we’d like

to acknowledge encouragement and support of our parents, prior to and during our study.

In addition we’d like to say thank to all of our friends who helped us preparing this

report more or less, specially my (S. M. R. Taheri) brother: Mr. S. M. Taheri for his help.

2

Contents

Abstract 1

Acknowledgements 2

1 Introduction 8

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 A Note on Unites and Notations . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 The Law of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . 15

2 General Relativity, Foundations 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Gravity Is Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 The Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 The Spacetime Metric, and Gravity as a Curvature of Spacetime . . . . . . 29

2.5 Free-fall Motion and Geodesics of Spacetime . . . . . . . . . . . . . . . . . . 31

2.6 The Einstein Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 An Another Way to Learning General Relativity . . . . . . . . . . . . . . . 38

2.7.1 The Experimental Evidence . . . . . . . . . . . . . . . . . . . . . . . 42

3

4

3 Special Cases of Spacetime 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Coordinates and Metric for a Static, Spherical System . . . . . . . . 46

3.2.2 Birkhoﬀ’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Other Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Kerr Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Kerr-Newman Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 The Many-Fingered Nature of Time . . . . . . . . . . . . . . . . . . . . . . 58

4 Lagrangian and Hamiltonian Formalism 62

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Lagrangian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Computer Algebra in General Relativity 67

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Lagrangeian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.1 Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.2 Kerr Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.3 Kerr-Newman Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Hamiltonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.1 Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 Kerr Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Kerr-Newman Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4.1 Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4.2 Kerr and Kerr-Newman Geometry . . . . . . . . . . . . . . . . . . . 77

5.5 Compare Diﬀerent Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6 Compare Diﬀerent Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A Maple Codes for Lagrangian Method 83

5

B Maple Codes for Hamiltonian method 90

C Maple Codes For Geodesic Equation 99

Bibliography 104

List of Figures

1.1 Schematic of Physics Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 An Example of Curved Spacetime . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Time and Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1 Trajectory of a null particle in Schwarzschild sapcetime (Lagrange’s method) 69

5.2 Trajectory of a null particle in Kerr sapcetime (Lagrange’s method) . . . . 71

5.3 Trajectory of a null particle in Kerr-Newman sapcetime (Lagrange’s method) 72

5.4 Trajectory of a null particle in Schwarzschild sapcetime (Hamiltonian method) 74

5.5 Trajectory of a null particle in Kerr sapcetime (Hamiltonian method) . . . 74

5.6 Trajectory of a null particle in Kerr-Newman sapcetime (Hamiltonian method) 75

5.7 Trajectory of a null particle in Schwarzschild sapcetime (Geodesis equation) 76

5.8 Trajectory of a null particle in Kerr sapcetime (Geodesis equation) . . . . . 80

5.9 Trajectory of a null particle in Kerr-Newman sapcetime (Geodesis equation) 80

5.10 Comparison of result trajectories of diﬀerent metrics . . . . . . . . . . . . . 82

6

List of Tables

1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Some useful quantities in conventional and geometrized units . . . . . . . . 39

7

Chapter 1

Introduction

1.1 Overview

At the end of ninetieth century, physics considered essentially of classical mechanics, the

theory of electromagnetism, and thermodynamics. Classical mechanics was used to pre-

dict the dynamics of material bodies, and Maxwell’s electromagnetism provided the proper

framework to study radiation; matter and radiation is described in terms of particles and

waves, respectively. As for the interaction between matter and radiation, they were well

explained by the Lorentz force or by thermodynamics. The overwhelming success of classi-

cal physics made people believe that the ultimate description of nature had been achieved.

It seemed that all known physical phenomena could be explained within the framework of

the general theories of matter and radiation. At the turn of the twentieth century, however,

classical physics, which had been quite unassailable, was seriously challenged on two major

fronts:

1. Relativistic domain: Einstein’s theory of relativity showed that the validity of New-

tonian mechanics ceases at some cases (e.g. at very high speeds comparable to that

of the light).

2. As soon as new experimental techniques were developed to the point of probing atomic

and subatomic structures, it turn out that classical physics fails miserably in provid-

ing the proper explanation for several newly discovered phenomena. It thus became

evident that that the validity of classical physics ceases at the microscopic level.

8

9

Figure 1.1: The frameworks and arenas for the laws of physics, and their relationship to

each other.

When these challenges were found, new theories -Relativity and Quantum mechanics- were

designed to solved problems, and after it Mechanics is divided to three parts: Newtonian,

Relativistic, and Quantum Mechanics, each of them having its own fundamental concepts

and math, and one of them (or two, e.g. in quantum-gravity) is applied according to the

corresponding situation and properties of event, to describe it. Relativity is divided to tow

subdirectory, Special Relativity (SR) and General Relativity (GR); SR is a special case of

GR. If relativity and quantum mechanics are used simultaneously, most accurate description

is obtained. Newtonian mechanics is a special case of them (ﬁg 1.1). Generally, quantum

is known as a theory to study of physical phenomena on the small scale (atom, electron,

etc), and Relativity is known as a theory for large scale physical phenomena. Although

these descriptions about these theories are not completely correct, but are used often (it is

possible to use both of them in both of large and small scale). In this text, motion of light (or

any other mass less particles) around massive objects (e.g. black hole, neutron star, etc) is

studied numerically (by computer algebra). In chapter 1 Newtonian and quantum mechanics

10

and their fundamental concepts are presented brieﬂy, and in next chapters we focus on the

GR. In chapter 2 we conﬁne ourselves to fundamental concepts and language of GR. In

chapter 3, Einstein equation of motion and some special cases of it is studied. Chapter 4 is

about methods that are used to obtain equations of motion, i.e. Euler-Lagrange equation,

Hamilton equations and Geodesic equation (geodesic equation is analyzed in chapter 3).

Lagrange method is selected, because it is a common method in Newtonian mechanics

and most of readers have learned it, and its application is similar (except deﬁnition of

Lagrangian) to Newtonian mechanics. Hamilton method is chosen for its capability to be

quantized, i.e. by quantization of Hamiltonian we can start Quantum-Gravity. Further more

when Lagrange method was presented, it is easy to use it and explain Hamilton equations.

About Geodesic equation, although there are not fundamental diﬀerences between it and

other methods are mentioned above, but in this method Relativistic concepts (e.g. metric)

are used directly. In last chapter, results that obtain by computer algebra are compared

and it is shown that answers from diﬀerent methods are unique. In numerical analysis,

Maple 10 computer software-one of the most power full software for handling Tensors-is

used. Maple codes are located in appendix and it is referred wherever it is necessary. We

use some boxes to noted historical notes, special and attractive cases, mathematical notes,

and concluding remarks.

1.2 A Note on Unites and Notations

In this text we will be dealing with complicated equations that appear in General Rela-

tivity, that can simpliﬁed by uniﬁed constants, such as c (speed of light) and G (Newton’s

gravitation constant), G = c = 1 (geometrical unites). If it is Instructive or necessary

we use them, but when it is not necessary, uniﬁed them. Einstein summation convention

(summation over repeated indices) is understood throughout in this text. In table 1.1 we

list notations that used.

1.3 Newtonian Mechanics

The arena for the Newtonian laws is a spacetime composed of the familiar 3-dimensional

Euclidean space of everyday experience (which we shall call 3-space), and a universal time

11

Table 1.1: Notations

Object Example or Definition

Scalar c, G, k

Vector p, u

Second rank contravariant

tensor

G

αβ

, G

αβ

=

∂x

α

∂x

µ

∂x

β

∂x

ν

G

µν

Second rank covariant ten-

sor

G

αβ

, G

αβ

=

∂x

µ

∂x

α

∂x

ν

∂x

β

G

µν

Tensor without indices G

αβ

→G

Partial derivative of a ten-

sor

∂X

a

∂x

b

= ∂

b

X

a

= X

a

,b

Einstein summation con-

vention

Any index that is repeated in a product

is automatically summed on

∂

α

X

α

=

3

i=0

∂

∂x

i

X

i

Covariant derivative ∇

γ

T

α

β

= T

α

β

;β

= ∂

γ

T

α

β

+Γ

α

δγ

T

δ

β

−Γ

δ

βγ

T

α

δ

∇.A = A

a

;a

Connection coeﬃcients

(Christoﬀel symbols)

Γ

α

βγ

=

1

2

g

αδ

(∂

γ

g

βδ

+ ∂

β

g

γδ

− ∂δg

βγ

)

when g

ab

is metric

12

t. “Absolute, true, and mathematical time . . . ﬂows at a constant rate without relation

to anything external . . . Absolute space . . . without relation to anything external, remains

always similar and immovable”. Isaac Newton (tr. Andrew Motte) [1].

There is 2 laws in Newtonian mechanics that should be presented: Law of inertia and

Newton’s laws of motion (however Newton’s laws of motion include law of inertia).

1.3.1 The Law of Inertia

Galileo (1544-1642) was the ﬁrst to develop a quantitative approach to the study of motion.

He addressed the question —what property of motion is related to force? Is it the position

of the moving object? Is it the velocity of the moving object? Is it the rate of change of

its velocity? . . . The answer to the question can be obtained only from observations; this is

a basic feature of Physics that sets it apart from Philosophy proper. Galileo observed that

force inﬂuences the changes in velocity (accelerations) of an object and that, in the absence

of external forces (e.g. friction), no force is needed to keep an object in motion that is

traveling in a straight line with constant speed. This observationally based law is called the

Law of Inertia. It is, perhaps, diﬃcult for us to appreciate the impact of Galileo’s new ideas

concerning motion. The fact that an object resting on a horizontal surface remains at rest

unless something we call force is applied to change its state of rest was, of course, well-known

before Galileo’s time. However, the fact that the object continues to move after the force

ceases to be applied caused considerable conceptual diﬃculties for the early Philosophers

(see Feynman The Character of Physical Law). The observation that, in practice, an object

comes to rest due to frictional forces and air resistance was recognized by Galileo to be a

side eﬀect, and not germane to the fundamental question of motion. Aristotle, for example,

believed that the true or natural state of motion is one of rest [2].

1.3.2 Newton’s Laws of Motion

During his early twenties, Newton postulated three Laws of Motion that form the basis

of Classical Dynamics. He used them to solve a wide variety of problems including the

dynamics of the planets. The Laws of Motion, ﬁrst published in the Principia in 1687, play

a fundamental role in Newton’s Theory of Gravitation; they are:

1. In the absence of an applied force, an object will remain at rest or in its present state

13

of constant speed in a straight line (Galileo’s Law of Inertia)

2. In the presence of an applied force, an object will be accelerated in the direction of

the applied force and the product of its mass multiplied by its acceleration is equal to

the force.

F = ma (1.1)

3. If a body A exerts a force of magnitude |

F

AB

| on a body B, then B exerts a force of

equal magnitude |

F

BA

| on A. The forces act in opposite directions so that

F

AB

=

F

BA

(1.2)

In law number 2, the acceleration lasts only while the applied force lasts. The applied

force need not, however, be constant in time —the law is true at all times during the

motion. Law number 3 applies to “contact” interactions. If the bodies are separated,

and the interaction takes a ﬁnite time to propagate between the bodies, the law must be

modiﬁed to include the properties of the “ﬁeld” between the bodies. It is a fundamental

(though often ignored) principle of physics that in the Newtonian physics laws must all do

not depend upon any coordinate system or orientation of axes or the time [3].

14

Box 1.1

Standard formulation of Newtonian gravitational force

1. There exist a universal time t, a set of Cartesian space coor-

dinates x

j

(called “Galilean coordinates”), and a Newtonian

gravitational potential Φ .

2. The density of mass ρ generates the Newtonian potential by

Poisson’s equation,

∇

2

Φ ≡

∂

2

Φ

∂t

2

= 4πρ

3. The equation of motion for a freely falling particles is

d

2

x

j

dt

2

+

∂Φ

∂x

j

= 0

4. “Ideal rods” measure the Galilean coordinate lengths; “ideal

clocks” measure universal time.

1.4 Quantum Mechanics

The failure of classical physics to explain several microscopic phenomena -such as blackbody

radiation- had cleared the way for seeking new ideas outside its purview.

The ﬁrst real breakthrough came in 1900 when Max Plank introduced the concept of

the quantum of energy. In his eﬀorts to explain the phenomenon of blackbody radiation,

the he succeeded in reproducing the experimental results only after postulating that the

energy exchange between radiation and its surrounding take place in discrete, or quantized,

amounts. He argued that the energy exchange between an electromagnetic wave of fre-

quency v and matter occurs only in integer multiples of hν, which he called the energy of

quantum, where h is a fundamental constant called Plank’s constant. The quantization of

electromagnetic radiation turned out to be an idea with far-reaching consequences. This is

15

the concept that waves exhibit particle behavior at the microscopic scale. Other experiment

and interpretation showed radiation-like behavior of particles -de Borglie postulate (you can

read complete story in quantum mechanics text books, for example [4]), and it was birth of

quantum theory.

Historically there were two independence formulations of quantum mechanics. The

ﬁrst formulation, called matrix mechanics, was developed by Heisenberg (1925) to describe

atomic structure starting from the observed spectral lines. The second formulation, called

wave mechanics, was due to Schr¨odinger (1926); it is a generalization of the de Borglie

postulate. This method, more intuitive than matrix mechanics, describes the dynamics of

microscopic matter by means of a wave function, called the Schr¨ odinger equation; instead of

the matrix eigenvalue problem of Heisenberg, Schr¨ odinger obtained a diﬀerential equation.

Dirac then suggested a more general formulation of quantum mechanics which deals with

abstract objects such as kets (state vectors), bars, and operators. Quantum mechanics work

in 3-dimentional space with universal time, and have some postulates that are presented

below.

1.4.1 Postulates of Quantum Mechanics

Postulate 1: To each state of a physical system there corresponds a wave function ψ(x; t) .

That’s simple enough. In classical mechanics each state of a physical system is specialized

by two variables, namely position x(t) and momentum p(t) which are both functions of

the one variable time t. (And we all know what position and momentum mean, so we

don’t need fancy postulates to say what they are.) In quantum mechanics each state of a

physical system is specialized by only one variable, namely the wave function ψ(x; t) which

is a function of the two variables position x and time t. At this stage we don’t know what

ψ(x; t) means but we will specify its meaning in a later postulate.

Postulate 2: To every physically measurable quantity A, to be called an observable or

dynamical variable, there corresponds a linear Hermitian operator A whose eigenvectors

form a complete basis.

Postulate 3: The time development of the wave function is determined by the Schr¨ odinger

equation

(−

2

2m

∂

2

∂x

2

+ U)ψ(x; t) = iψ(x; t) (1.3)

16

where U = U(x) (potential). Again this is simple enough. The equation governing the

behavior of the wave function is the Schr¨ odinger equation. (Here we have written it for a

single particle of mass m in 1-dimension).

Postulate 4: (Born hypothesis): |ψ|

2

is the probability density. This postulate states

that the wave function is actually related to a probability density. The basic postulate in

quantum mechanics is that the wave function ψ(x; t) is related to the probability for ﬁnding

a particle at position x. The actual probability for this is, in 1-dimension,

P =

_

A

−A

|ψ|

2

, dx (1.4)

P is the probability for ﬁnding the particle somewhere between A and −A. This means

that |ψ|

2

dx = probability of ﬁnding a particle between position x and x + dx at time t (of

course when ψ(x; t) is normalized). The probabilistic interpretation of the wave function

is what sets quantum mechanics apart from all other classical theories. It is totally unlike

anything you will have studied in your other physics courses. The acceleration or position

of a particle, represented by the symbols a and x, are well deﬁned quantities in classical

mechanics. However with the interpretation of the wave function as a probability density

we shall see that the concept of the deﬁnite position of a particle no longer applies. Thus

particles will be represented by wave functions and we already know that a wave is not

localized in space but spread out. So too is a particle’s wave property spread out over some

distance and so we cannot say exactly where the particle is, but only the probability of

ﬁnding it somewhere.

Box 1.2

Fundamental equations and quantities in physics

Most physical theories are based on just a couple of fundamental

equations. For instance, Newtonian mechanics is based on

F = ma,

classical electrodynamics is based on Maxwell’s equations and general

relativity is based on the Einstein equations G

µν

= 8πGT

µν

.

17

Box 1.2 (continued)

When you take a course on Newtonian mechanics, all you ever do is

solve

F = ma. In a course on electromagnetism you spend all your

time just solving Maxwell’s equations. Thus these fundamental equa-

tions are the theory. All the rest is just learning how to solve these

fundamental equations in a wide variety of circumstances. The funda-

mental equation of quantum mechanics is the Schro¨ odinger equation

(−

2

2m

∂

2

∂x

2

+ U)ψ(x; t) = iψ(x; t)

which is written for a single particle (of mass m) moving in a poten-

tial U in one dimension x. It’s important to understand that these

fundamental equations cannot be derived from anywhere else. They

are physicists’ guesses (or to be fancy, postulates) as to how nature

works. We check that the guesses (postulates) are correct by com-

paring their predictions to experiment. Nevertheless, you will often

ﬁnd derivations of the fundamental equations scattered throughout

physics books. This is OK. The authors are simply trying to pro-

vide deeper understanding, but it is good to remember that these are

not fundamental derivations. Our good old equations like

F = ma,

Maxwell’s equations and the Schr¨ odinger equation are postulates and

that’s that. Nothing more. They are sort of like the deﬁnitions that

mathematicians state at the beginning of the proof of a theorem.

They cannot be derived from anything else.

The wave function ψ(x; t) is the fundamental quantity that we al-

ways wish to calculate in quantum mechanics. Actually all of the

fundamental equations of physical theories usually have a fundamen-

tal quantity that we wish to calculate given a fundamental input.

In Newtonian physics,

F = ma is the fundamental equation and the

acceleration a is the fundamental quantity that we always want to

know given an input force

F.

18

Box 1.2 (continued)

The acceleration a is diﬀerent for diﬀerent forces

F. Once we have

obtained the acceleration we can calculate lots of other interesting

goodies such as the velocity and the displacement as a function of

time.

In classical electromagnetism the Maxwell equations are the funda-

mental equations and the fundamental quantities that we always want

are the electric (

E) and magnetic (

B) ﬁelds. These always depend

on the fundamental input which is the charge (q) and current (

j)

distribution. Diﬀerent q and

j produce diﬀerent

E and

B. In gen-

eral relativity, the fundamental equations are the Einstein equations

(G

µν

= 8πGT

µν

) and the fundamental quantity that we always want

is the metric tensor g

µν

, which tells us how spacetime is curved.

(g

µν

is buried inside G

µν

). The fundamental input is the energy-

momentum tensor T

µν

which describes the distribution of matter.

Diﬀerent T

µν

produces diﬀerent g

µν

.

Similarly the fundamental equation of quantum mechanics is the

Schr¨ odinger equation and the fundamental input is the potential U.

(This is related to force via

F = −

∇U. Diﬀerent input potentials U

give diﬀerent values of the fundamental quantity which is the wave

function ψ(x; t) . Once we have the wave function we can calculate

all sorts of other interesting goodies such as energies, lifetimes, tun-

nelling probabilities, cross sections, etc. The meaning of the wave

function has occupied some of the greatest minds in physics (Heisen-

berg, Einstein, Dirac, Feynman, Born and others).

Chapter 2

General Relativity, Foundations

2.1 Introduction

General relativity (GR) is one of the most beautiful physical theories ever invented. Never-

theless, it has a reputation of being extremely diﬃcult, primarily for two reasons: tensors

are everywhere, and spacetime is curved. These two facts force GR people to use a diﬀerent

language than everyone else, which makes the theory somewhat inaccessible. It is clear that

some deﬁnitions should be presented before beginning theory.

An inertial reference frame (or Lorentz reference frame) is a (conceptual) three-dimensional

latticework of measuring rods and clocks with the following properties:

1. The latticework moves freely through spacetime (i.e., no forces act on it), and is

attached to gyroscopes so it does not rotate with respect to distant, celestial objects.

2. The measuring rods form an orthogonal lattice and the length intervals marked on

them are uniform when compared to, e.g., the wavelength of light emitted by some

standard type of atom or molecule; and therefore the rods form an orthonormal,

Cartesian coordinate system with the coordinate x measured along one axis, y along

another, and z along the third.

3. The clocks are densely packed throughout the latticework so that, ideally, there is a

separate clock at every lattice point.

4. The clocks tick uniformly when compared, e.g., to the period of the light emitted by

19

20

some standard type of atom or molecule; i.e., they are ideal clocks

5. The clocks are synchronized by the Einstein synchronization process: If a pulse of

light, emitted by one of the clocks, bounces oﬀ a mirror attached to another and then

returns, the time of bounce t

b

as measured by the clock that does the bouncing is the

average of the times of emission t

e

and reception t

r

as measured by the emitting and

receiving clock: t

b

=

1

2

(t

e

+ t

r

).

Our second fundamental relativistic concept is the event. An event is a precise location

in space at a precise moment of time; i.e., a precise location (or “point”) in 4-dimensional

spacetime (in GR space and time are similar -spacetime- and separation of them is for our

ordinary experience in life) [5].

It is assumed that readers know tensor and tensor algebra. GR can be summed up in

two statements [6]:

1. Spacetime is a curved pseudo-Riemannian manifold

2. The relationship between matter and the curvature of spacetime is contained in the

equation (Einstein’s equation)

R

µν

−

1

2

Rg

µν

= 8πGT

µν

(2.1)

This equation will be analyzed later. GR is not merely a theory of gravity. Like special

relativity before it, the general theory is a framework within which to formulate all the

laws of physics, classical and quantum -but now with gravity included. However, there is

one remaining, crucial, gaping hole in this framework: It is incapable of functioning, indeed

it fails completely, when conditions become so extreme that space and time themselves

must be quantized. In those extreme conditions GR must be married in some deep, as-

yet-ill-understood way, with quantum theory, to produce an all-inclusive quantum theory

of gravity -a theory which, one may hope, will be a “theory of everything.” [7].

In Newton’s 300-year-old theory of gravity, a mass attracts other masses with a force of

gravity that decreases as the inverse of the square of the distance between them. Masses

move in response to the forces acting on them, including gravitational forces, according to

Newton’s laws of motion.

21

In Einstein’s 1915 general theory of relativity, a mass curves the one time dimension

and three space dimensions of spacetime according to Einstein’s equation. The spacetime

curvature is greatest near the mass and vanishes at a distance. Other masses move along

the straightest possible paths in this curved spacetime. Einstein’s theory thus expresses

both the gravitational eﬀect of mass and the response of mass to that eﬀect in terms of the

geometry of spacetime. The Newtonian idea of a gravitational force acting at a distance

between bodies was replaced by the idea of a body moving in response to the curvature of

spacetime, in other words, Mass Produces Spacetime Curvature, and Spacetime Curvature

Determines the Motion of Mass [8].

Now third relativistic concept is expressed: a metric; a spacetime metric; a curved

spacetime metric; a locally Lorentz, curved spacetime metric. This is the foundation of

spacetime geometry in the real, physical world. Metric described in three languages. In

the language of elementary geometry, “metric” is a table giving the interval between every

event and every other event. In the language of coordinates, “metric” is a set of functions

of position, g

µν

(x

α

), such that the expression

∆s

2

= −∆τ

2

= g

µν

(x

α

)∆x

µ

∆x

ν

(2.2)

or in modern style

ds

2

= g

µν

dx

µ

dx

ν

(2.3)

gives the intervals between any event x

α

and any near by event x

α

+∆x

α

(or x

α

+dx

α

). In

the language of abstract diﬀerential geometry, metric is a bilinear machine, g ≡ (. . . . . . .),

to produce a number [“scalar product g(u,v) ≡ (u.v)”] out of two tangent vectors, u and

v [9](Box 2.1).

In relativity, mass and energy are the same thing according to Einstein’s famous E = mc

2

relation. Not only mass but also any form of energy will curve spacetime. Gravity itself

carries energy, and even small propagating ripples in spacetime cause further curvature.

The equations of Einstein’s theory keep track of this complex feedback interrelationship

between energy and curvature.

Newton’s theory of gravity is not wrong. It is a correct approximation to Einstein’s

theory when spacetime curvature is small and the velocities of masses are much smaller

than the velocity of light. The ﬁrst general relativistic corrections beyond Newtonian the-

ory (called “post-Newtonian”) are responsible for small deviations to the motion of light

22

and to the orbits of the planets from those predicted by Newton. Measurements of these

deviations are among the most precise tests of GR [10].

Box 2.1

Metric distilled from distance [11]

Raw data on distances

Imagine the earth in your mind, by giving distances between some

of the principle identiﬁable points: buoys, ships, icebergs, lighthouse,

peaks, and ﬂags: points to a total of n = 2 ×107. The total number

of distances to be given is n(n−1)/2 = 2×1014. With 200 distances

per page of printout, this means 1012 pages weighing 6 gram each,

or 6 × 106 metric ton of data. With 6 tons per truck this means

106 truckloads of data; or with on truck passing by every 5 seconds,

5 ×106 seconds or 2 months of night and day traﬃc to get in data.

Figure Box2-1-1

First distillation: distances to nearby points only

Get distances between faraway points by adding distances covered on

the elementary short legs of the trip. Boil down the table of distances

to give only the distance between each point and the hundred nearest

points. Now have 100n = 2 × 109 distances, or 2 × 109/200 = 107

pages of data, or 60 tons of records, or 10 truckloads.

Second distillation: distances between nearby points in

terms of metric

Idealize the surface of the earth as smooth. Then in any suﬃciently

23

Box 2.1 (continued)

limited region the geometry is Euclidean. This circumstance has a

happy consequence. It is enough to know a few distances between

the nearby points to be able to determine all the distances between

the nearby points. Locate point 2 so that (102) is a right triangle;

thus, (12)

2

= (10)

2

+ (20)

2

. Consider a point 3 close to 0. Deﬁne

x(3) = (13) −(10)

and

y(3) = (23) −(20)

Then the distance (03) dose not has to be supplied independently; it

can be calculated from the formula

(03)

2

= [x(3)]

2

+ [y(3)]

2

Figure Box2-1-2

Similarly for a point 4 and its distance (04) from the local origin 0.

Similarly for the distance (mn) between any tow points m and n that

are close to 0:

(mn)

2

= [x(m) −x(n)]

2

+ [y(m) −y(n)]

2

Thus it is only needful to have the distance (1m) (from point 1) and

(2m) (from point 2) for each point m close to 0 (m = 3, 4, . . . , N +2)

to be able to work out its distance from every point n close to 0. The

principle to determine the N(N −1)/2 distances between these N

24

Box 2.1 (continued)

nearby points can be reexpressed to advantage in these words: (1)

each point has two coordinates, x and y; and (2) the distance is given

in terms of these coordinates by standard Euclidean metric; thus

(∆s)

2

= (∆x)

2

+ (∆y)

2

Having gone this far on the basis of “distance geometry”, one can

generalize from a small region (Euclidean) to a large region (not

Euclidean). Introduce any arbitrary smooth pair of everywhere-

independent curvilinear coordinates x

k

, and express distance, not

only in the immediate neighborhood of the point 0, but also in the

immediate neighborhood of every point of the surface (except places

where one has to go to another coordinate patch; at least two patches

needed for 2-sphere) in terms of the formula

ds

2

= g

jk

dx

j

dx

k

Thus out of the table of distances between nearby points one has

distilled now ﬁve numbers per point (two coordinates, x

1

, x

2

, and

three metric coeﬃcients, g

11

, g

12

= g

21

, and g

22

), down by a factor of

100/5 = 20 from what one had before (now 3 tons of data, or half a

truckload).

Third distillation: Metric coeﬃcients expressed as analytical

functions of coordinates

Instead of giving the three metric coeﬃcients at each point of the

2 ×107 points of the surface, give them as functions of the two coor-

dinates x

1

, x

2

, in terms of power series or an expansion in spherical

harmonics or otherwise with some modest number, say 100, of ad-

justable coeﬃcients. Then the information about the geometry itself

(as distinct from the coordinates of the 2×107 points located on that

25

Box 2.1 (continued)

geometry) is caught up in these three hundred coeﬃcients, a single

page of printout. Goodbye to any truck! In brief, metric provides a

shorthand way of giving the distance between every point and every

other point -but its role, its justiﬁcation and its meaning lies in these

distances and only in these many distances.

2.2 Gravity Is Geometry

Gravity is the geometry of four-dimensional spacetime. That is the central idea of Ein-

stein’s 1915 general theory of relativity —the classical (nonquantum) theory of relativistic

gravitation. It is not diﬃcult to imagine a curved space. The curved surface of a sphere or

a car fender is two-dimensional examples. But gravitational eﬀects arise from the curvature

of four-dimensional spacetime with three space dimensions and one time dimension. It is

more diﬃcult to imagine a notion of curvature involving time, but the Global Positioning

System (described in Box 2.2) provides an everyday practical example of its implications.

2.3 The Principle of Equivalence

The term “mass” that appears in Newton’s equation for the gravitational force between

two interacting masses refers to “gravitational mass”; Newton’s law should indicate this

property of matter

F

G

= G

M

G

m

G

r

2

(2.4)

where M

G

and m

G

are the gravitational masses of the interacting objects, separated by a

distance r.

The term “mass” that appears in Newton’s equation of motion

F = ma (2.5)

26

refers to the “inertial mass”; Newton’s equation of motion should indicate this property of

matter:

F = m

I

a (2.6)

where m

I

is the inertial mass of the particle moving with an acceleration a(r) in the gravi-

tational ﬁeld of the mass M

G

.

Newton showed by experiment that the inertial mass of an object is equal to its gravita-

tional mass, m

I

= m

G

to an accuracy of 1 part in 103. Recent experiments have shown this

equality to be true to an accuracy of 1 part in 10

12

. Newton therefore took the equations

F = G

M

G

m

G

r

2

= m

I

a (2.7)

and used the condition m

G

= m

I

to obtain

a = G

M

G

r

2

(2.8)

Galileo had previously shown that objects made from diﬀerent materials fall with the same

acceleration in the gravitational ﬁeld at the surface of the Earth, a result that implies

m

G

∝ m

I

. This is the Newtonian Principle of Equivalence.

Einstein used this Principle as a basis for a new Theory of Gravitation. He extended

the axioms of Special Relativity, that apply to ﬁeld-free frames, to frames of reference in

“free fall” [12]. One of Einstein’s greatest insights was to recognize that special relativity

is valid not globally, but only locally, inside locally freely falling (inertial) reference frames.

Since, in the presence of gravity, inertial reference frames must be restricted to be local,

the inertial-frame version of the principle of relativity must similarly be restricted to say:

All the local, nongravitational laws of physics are the same in every local inertial frame,

everywhere and everywhen in the universe. Here, by “local” laws we mean those laws,

classical or quantum, which can be expressed entirely in terms of quantities conﬁned to

(measurable within) a local inertial frame; and the exclusion of gravitational laws from this

version of the principle of relativity is necessary because gravity is to be described by a

curvature of spacetime which (by deﬁnition, see below) cannot show up in a local inertial

frame. This version of the principle of relativity can be described in operational terms:

If two diﬀerent observers, in two diﬀerent local Lorentz frames, in diﬀerent (or the same)

regions of the universe and epochs of the universe, are given identical written instructions

27

for a self-contained physics experiment (an experiment that can be performed within the

conﬁnes of the local Lorentz frame), then their two experiments must yield the same results,

to within their experimental accuracies.

It is worth emphasizing that the principle of relativity is asserted to hold everywhere

and everywhen in the universe: the local laws of physics must have the same form in the

early universe, a fraction of a second after the big bang, as they have on earth today, and

as they have at the center of the sun or inside a black hole.

It is reasonable to expect that the speciﬁc forms that the local, nongravitational laws

of physics take in general relativistic local Lorentz frames are the same as they take in

the (global) Lorentz frames of special relativity. The assertion that this is so is a modern

version of Einstein’s equivalence principle (it is expressed because most of students know

special relativity).

The results of all experiments carried out in ideal freely falling frames are therefore fully

consistent with Special Relativity. All freely-falling observers measure the speed of light

to be c, its constant freespace value. It is not possible to carry out experiments in ideal

freely-falling frames that permit a distinction to be made between the acceleration of local,

freely-falling objects, and their motion in an equivalent external gravitational ﬁeld. As an

immediate consequence of the extended Principle of Equivalence, Einstein showed that a

beam of light would be observed to be deﬂected from its straight path in a close encounter

with a suﬃciently massive object. The observers would, themselves, be far removed from

the gravitational ﬁeld of the massive object causing the deﬂection.

Einstein’s original calculation of the deﬂection of light from a distant star, grazing the

Sun, as observed here on the Earth, included only those changes in time intervals that he

had predicted would occur in the near ﬁeld of the Sun. His result turned out to be in

error by exactly a factor of two. He later obtained the “correct” value for the deﬂection

by including in the calculation the changes in spatial intervals caused by the gravitational

ﬁeld [13].

28

Box 2.2

General Relativity and Daily Life

There is no better illustration of the unpredictable application of

fundamental science in daily life than the story of general relativity

and the Global Positioning System (GPS). Built at a cost of more

than $10 billion mainly for military navigation, the GPS has been

rapidly transformed into a thriving, multibillion-dollar commercial

industry. GPS is based on an array of 24 Earth-orbiting satellites,

each carrying a precise atomic clock. With a hand-held GPS receiver

that detects radio emissions from any of the satellites that happen to

be overhead, a user can determine latitude, longitude, and altitude

to an accuracy that currently can reach 50 feet, and local time to 50

billionths of a second. Apart from the obvious military uses, the GPS

is ﬁnding applications in airplane navigation, wilderness recreation,

sailing, and interstate trucking. Even Hollywood has met the GPS,

pitting James Bond in “Tomorrow Never Dies” against an evil genius

able to insert deliberate errors into the system and send British ships

into harm’s way.

figure Box2-2

Schematic illustration of segments used in operation of the Global

Positioning System. (Adapted from a ﬁgure courtesy of the Aerospace

Corporation)

29

Box 2.2 (continued)

Because the satellite clocks are moving in high-speed orbits and are

far from Earth, they tick at diﬀerent rates than clocks on the ground.

Gravity and speed contribute comparable amounts to the total dis-

crepancy. The oﬀset is so large that, if left uncompensated, it would

lead to navigational errors that would accumulate at a rate greater

than 6 miles per day. In GPS, the relativity is accounted for by

electronic adjustments to the rates of the satellite clocks, and by

mathematical corrections built into the computer chips that solve for

the user’s location [15].

2.4 The Spacetime Metric, and Gravity as a Curvature of

Spacetime

The Einstein equivalence principle guarantees that nongravitational physics within a local

Lorentz frame can be described using a spacetime metric g, which gives for the invariant

interval between neighboring events with separation vector

ξ = ∆x

α ∂

∂x

α

, the standard

special relativistic expression [14]

ξ

2

= g

αβ

ξ

α

ξ

β

= (∆s)

2

= −(∆t)

2

+ (∆x)

2

+ (∆y)

2

+ (∆z)

2

(2.9)

Correspondingly, in a local Lorentz frame the components of the spacetime metric take on

their standard special-relativity values

g

αβ

= η

αβ

≡ {−1 if α = β = 0, +1 if α = β = (x, or y, or z), 0 otherwise} (2.10)

Turn, now, to a ﬁrst look at the gravity-induced constraints on the size of a local Lorentz

frame: Above the earth set up, initially, a family of local Lorentz frames scattered over the

entire region from two earth radii out to four earth radii, with all the frames initially at rest

with respect to the earth [Fig. 2.1(a)]. From experience —or, if you prefer, from Newton’s

theory of gravity which after all is quite accurate near earth— we know that as time passes

30

(b) (a)

Figure 2.1: (a) A family of local Lorentz frames, all momentarily at rest above the earth’s

surface. (b) A family of local, 2-dimensional Euclidean coordinate systems on the earth’s

surface. The nonmeshing of Lorentz frames in (a) is analogous to the nonmeshing of Eu-

clidean coordinates in (b) and motivates attributing gravity to a curvature of spacetime.

these frames will all fall toward the earth. If (as a pedagogical aid) we drill holes through

the earth to let the frames continue falling after reaching the earth’s surface, the frames

will all pass through the earth’s center and ﬂy out the earth’s opposite side.

Obviously, two adjacent frames, which initially were at rest with respect to each other,

acquire a relative velocity during their fall, which causes them to interpenetrate and pass

through each other as they cross the earth’s center. Gravity is the cause of this relative

velocity.

If these two adjacent frames could be meshed to form a larger Lorentz frame, then as

time passes they would always remain at rest relative to each other. Thus, a meshing to form

a larger Lorentz frame is impossible. The gravity-induced relative velocity prevents it. In

brief: Gravity prevents the meshing of local Lorentz frames to form global Lorentz frames.

This situation is closely analogous to the nonmeshing of local, 2-dimensional, Euclidean

coordinate systems on the surface of the earth [Figure 2.1(b)]: The curvature of the earth

prevents a Euclidean mesh -thereby giving grief to map makers and surveyors. This analogy

suggested to Einstein, in 1912, a powerful new viewpoint on gravity: Just as the curvature

of space prevents the meshing of local Euclidean coordinates on the earth’s surface, so it

must be that a curvature of spacetime prevents the meshing of local Lorentz frames in the

spacetime above the earth -or anywhere else in spacetime, for that matter. And since it is

already known that gravity is the cause of the nonmeshing of Lorentz frames, it must be

that gravity is a manifestation of spacetime curvature.

31

2.5 Free-fall Motion and Geodesics of Spacetime

In order to make more precise the concept of spacetime curvature, we will need to study

quantitatively the relative acceleration of neighboring, freely falling particles. Before we

can carry out such a study, however, we must understand quantitatively the motion of a

single freely falling particle in curved spacetime. That is the objective of this section.

In a global Lorentz frame of ﬂat, special relativistic spacetime a free particle moves

along a straight world line, i.e., a world line with the form

(t, x, y, z) = (t

0

, x

0

, y

0

, z

0

) + (p

0

, p

x

, P

y

, P

z

)ζ; i.e., x

α

= x

α

0

+ p

α

ζ (2.11)

here p

α

are the Lorentz-frame components of the particle’s 4-momentum; ζ is the aﬃne

parameter such that p =

d

dζ

, i.e. p

α

= dx

α

/dζ; and x

α

0

are the coordinates of the particle

when its aﬃne parameter is ζ = 0. The straight-line motion (24.17) can be described equally

well by the statement that the Lorentz-frame components p

α

of the particle’s 4-momentum

are constant, i.e., are independent of ζ

dp

α

dζ

= 0 (2.12)

Even nicer is the frame-independent description, which says that as the particle moves it

parallel-transports its tangent vector p along its world line

∇

p

p = 0, or, equivalently p

α

;β

p

β

= 0 (2.13)

For a particle of nonzero rest mass m, which has p = ma and ζ = τ/m with u = d/dτ its

4-velocity and τ its proper time, Eq. (2.13) is equivalent to ∇

u

u = 0.

This description of the motion is readily carried over into curved spacetime using the

equivalence principle: Let P(ζ) be the world line of a freely moving particle in curved

spacetime. At a speciﬁc P

0

= P(ζ

0

) event on that world line introduce a local Lorentz

frame (so the frame’s spatial origin, like the particle, passes through P

0

as time progresses).

Then the equivalence principle tells us that the particle’s law of motion must be the same

in this local Lorentz frame as it is in the global Lorentz frame of special relativity:

(

dp

α

dζ

)

ζ=ζ

0

= 0 (2.14)

More powerful than this local-Lorentz-frame description of the motion is a description that

is frame-independent. We can easily deduce such a description from Eq. (2.14) Since the

32

connection coeﬃcients vanish at the origin of the local Lorentz frame where Eq. (2.14) is

being evaluated, Eq. (2.14) can be written equally well, in our local Lorentz frame, as

0 = (

dp

α

dζ

+ Γ

α

βγ

p

β

dx

γ

dζ

)

ζ=ζ

0

= ((p

α

,γ

+ Γ

α

βγ

p

β

)

dx

γ

dζ

)

ζ=ζ

0

= (p

α

;γ

p

γ

)

ζ=ζ

0

(2.15)

Thus, as the particle passes through the spatial origin of our local Lorentz coordinate system,

the components of the directional derivative of its 4-momentum along itself vanishes. Now, if

two 4-vectors have components that are equal in one basis, their components are guaranteed

[by the tensorial transformation law] to be equal in all bases, and correspondingly the two

vectors, viewed as frame-independent, geometric objects, must be equal. Thus, since Eq.

(2.15) says that the components of the 4-vector ∇

p

p and the zero vector are equal in our

chosen local Lorentz frame, it must be true that

∇

p

p = 0 (2.16)

at the moment when the particle passes through the point P

0

= P(ζ

0

). Moreover, since P

0

is an arbitrary point (event) along the particle’s world line, it must be that Eq. (2.16) is

a geometric, frame-independent equation of motion for the particle, valid everywhere along

its world line. Notice that this geometric, frame-independent equation of motion ∇

p

p = 0

in curved spacetime is precisely the same as that [Eq. (2.13)] for at spacetime.

Our equation of motion , Eq. (24.16), for a freely moving point particle says, in words,

that the particle parallel transports its 4-momentum along its world line. In any curved

manifold, not just in spacetime, the relation is called the geodesic equation, and the curve

to which p is the tangent vector is called a geodesic. On the surface of a sphere such as

the earth, the geodesics are the great circles; they are the unique curves along which local

Euclidean coordinates can be meshed, keeping one of the two Euclidean coordinates constant

along the curve, and they are the trajectories generated by an airplane’s inertial guidance

system, which tries to ﬂy the plane along the straightest trajectory it can. Similarly, in

spacetime the trajectories of freely falling particles are geodesics; they are the unique curves

along which local Lorentz coordinates can be meshed, keeping the three spatial coordinates

constant along the curve and letting the time vary, thereby producing a local Lorentz

reference frame, and they are also the spacetime trajectories along which inertial guidance

systems will guide a spacecraft.

33

The geodesic equation guarantees that the square of the 4-momentum will be conserved

along the particle’s world line

(g

αβ

p

α

p

β

)

;γ

p

γ

= 2g

αβ

p

α

p

β

;γ

p

γ

= 0 (2.17)

(It also can be deduced in a local Lorentz frame where Γ

α

βγ

so each gradient with a “;”

reduces to a partial derivative with a “,”.) Also in Eq. (2.17) the term involving the

gradient of the metric has been discarded since it vanishes, and the two terms involving

derivatives of p

α

and p

β

, being equal, have been combined. In index-free notation the frame

independent relation Eq. (2.17) says

∇

p

( p. p) = 2 p.∇

p

p = 0 (2.18)

This is a pleasing result, since the square of the 4-momentum is the negative of the particle’s

squared rest mass, p. p = −m

2

, which surely should be conserved along the particle’s free-fall

world line! Note that, as in at spacetime, so also in curved, for a particle of ﬁnite rest mass

the free-fall trajectory (the geodesic world line) is timelike, p. p = −m

2

< 0, while for a

zero-rest-mass particle it is null, p. p = 0. Spacetime also supports spacelike geodesics, i.e.,

curves with tangent vectors p that satisfy the geodesic equation (24.22) and are spacelike,

p. p > 0. Such curves can be thought of as the world lines of freely falling “tachyons,”

i.e., faster-than-light particles -though it seems unlikely that such particles really exist in

Nature. Note that the constancy of p. p along a geodesic implies that a geodesic can never

change its character: if initially timelike, it will always remain timelike; if initially null, it

will remain null; if initially spacelike, it will remain spacelike.

When studying the motion of a particle with ﬁnite rest mass, one often uses as the tan-

gent vector to the geodesic the particle’s 4-velocity u = p/m rather than the 4-momentum,

and correspondingly one uses as the parameter along the geodesic the particle’s proper time

τ rather than ζ (recall: u = d/dτ; p = d/dζ). In this case the geodesic equation becomes

∇

u

u = 0 (2.19)

Similarly, for spacelike geodesics, one often uses as the tangent vector u = d/ds, where s is

proper distance (square root of the invariant interval) along the geodesic; and the geodesic

equation then assumes the same form (2.19) as for a timelike geodesic.

The geodesic world line of a freely moving particle has three very important properties:

34

1. When written in a coordinate basis, the geodesic equation ∇

p

p = 0 becomes the

Following diﬀerential equation for the particle’s world line x

α

(ζ) in the coordinate

system

d

2

x

α

dζ2

= −Γ

α

µν

dx

µ

dζ

dx

ν

dζ

(2.20)

here Γ

α

µν

is the connection coeﬃcients of the coordinate system’s coordinate basis.

Note that these are four coupled equations (α= 0; 1; 2; 3) for the four coordinates

x

α

as functions of aﬃne parameter ζ along the geodesic. If the initial position, x

α

at

ζ = 0, and initial tangent vector (particle momentum), p

α

= dx

α

/dζ at ζ = 0, are

speciﬁed, then these four equations will determine uniquely the coordinates x

α

(ζ) as

a function of ζ along the geodesic.

2. Consider a spacetime that possesses a symmetry, which is embodied in the fact that the

metric coeﬃcients in some coordinate system are independent of one of the coordinates

x

A

. Associated with that symmetry there will be a conserved quantity P

A

= p.∂/∂x

A

associated with free-particle motion.

3. Among all timelike curves linking two events P

0

and P

1

in spacetime, those whose

proper time lapse (timelike length) is stationary under small variations of the curve are

timelike geodesics. Now, one can always send a photon from P

0

to P

1

by bouncing it

oﬀ a set of strategically located mirrors, and that photon path is the limit of a timelike

curve as the curve becomes null. Therefore, there exist timelike curves from P

0

to

P

1

with vanishingly small length, so the geodesics cannot minimize the proper time

lapse. This means that the curve of maximal proper time lapse (length) is a geodesic,

and that any other geodesics will have a length that is a “saddle point” (stationary

under variations of the path but not a maximum or a minimum) [16].

2.6 The Einstein Field Equation

One crucial issue remains to be studied in this overview of the foundations of general

relativity: What is the physical law that determines the curvature of spacetime? Einstein’s

search for that law, his Einstein ﬁeld equation, occupied a large fraction of his eﬀorts during

the years 1913, 1914, and 1915. Several times he thought he had found it, but each time his

35

proposed law turned out to be fatally awed; for some favor of his struggle see the excerpts

from his writings in Sec. 17.7 of Ref. [9].

In this section we shall brieﬂy examine one segment of Einstein’s route toward his ﬁeld

equation: the segment motivated by contact with Newtonian gravity.

The Newtonian potential Φ is a close analog of the general relativistic spacetime metric

g: From Φ we can deduce everything about Newtonian gravity, and fromg we can deduce ev-

erything about spacetime curvature. In particular, by diﬀerentiating Φ twice we can obtain

the Newtonian tidal ﬁeld E, and by diﬀerentiating the components of g twice we can obtain

the components of the relativistic generalization of E: the components of the Riemann cur-

vature tensor R

α

βγδ

(it is possible to obtain this by attend to relativistic description of tidal

gravity, however Riemann curvature tensor deﬁned as R

α

βγδ

= ∂

γ

Γ

α

βδ

∂

δ

Γ

α

βγ

+Γ

λ

βδ

Γ

α

λγ

−Γ

λ

βγ

Γ

α

λδ

,

and it is connected with the curvature of the spacetime -when it vanishes the manifold is

ﬂat).

In Newtonian gravity Φ is determined by Newton’s ﬁeld equation

∇

2

Φ = 4πGρ (2.21)

which can be rewritten in terms of the tidal ﬁeld ε

jk

= ∂

2

Φ/∂x

j

∂x

k

as

ε

j

j

= 4πGρ (2.22)

Note that this equates a piece of the tidal ﬁeld, its trace, to the density of mass. By analogy

we can expect the Einstein ﬁeld equation to equate a piece of the Riemann curvature tensor

(the analog of the Newtonian tidal ﬁeld) to some tensor analog of the Newtonian mass

density. Further guidance comes from the demand that in nearly Newtonian situations, e.g.,

in the solar system, the Einstein ﬁeld equation should reduce to Newton’s ﬁeld equation.

To exploit that guidance, we can

1. Write the Newtonian tidal ﬁeld for nearly Newtonian situations in terms of general

relativity’s Riemann tensor, ε

jk

= R

j0k0

.

2. Then take the trace and note that by its symmetries R

0

000

= 0 so that ε

j

j

= R

α

0α0

= R

00

.

3. Thereby infer that the Newtonian limit of the Einstein equation should read, in a

local Lorentz frame,

R

00

= 4πGρ (2.23)

36

here R

00

is the time-time component of the Ricci curvature tensor -which can be

regarded as a piece of the Riemann tensor.

An attractive proposal for the Einstein ﬁeld equation should now be obvious: Since the

equation should be geometric and frame-independent, and since it must have the Newtonian

limit , Eq.(2.23), it presumably should say

R

αβ

= 4πGρ×a second-rank symmetric tensor that generalizes the Newtonian mass

density ρ)

The obvious required generalization of ρ is the stress-energy tensor T

αβ

, so

R

αβ

= aπGT

αβ

(2.24)

Einstein flirted extensively with this proposal for the ﬁeld equation during 1913-1915. How-

ever, it, like several others he studied, was fatally awed. When expressed in a coordinate

system in terms of derivatives of the metric components g

µν

, it becomes (because R

αβ

and T

αβ

both have ten independent components) ten independent diﬀerential equations

for the ten g

µν

. This is too many equations: By an arbitrary change of coordinates,

x

α

new

= F

α

(x

0

old

, x

1

old

, x

2

old

, x

3

old

) involving four arbitrary functions F

0

, F

1

, F

2

, F

3

, one

should be able to impose on the metric components four arbitrary conditions, analogous to

gauge conditions in electromagnetism (for example, one should be able to set g

00

= −1 and

g

0j

= 0 everywhere); and correspondingly, the ﬁeld equations should constrain only six, not

ten of the components of the metric (the six g

ij

in our example).

In November 1915 Einstein (1915), and independently Hilbert (1915) [who was familiar

with Einstein’s struggle as a result of private conversations and correspondence] discovered

the resolution of this dilemma: Because the local law of 4-momentum conservation guar-

antees T

αβ

;β

= 0 independent of the ﬁeld equation, if we replace the Ricci tensor in Eq.

(2.24) by a constant (to be determined) times some new curvature tensor G

αβ

that is also

automatically divergence free independent of the ﬁeld equation (G

αβ

= 0), then the new

ﬁeld equation G

αβ

= κT

αβ

(with κ = constant) will not constrain all ten components of

the metric. Rather, in a coordinate system the four equations [G

αβ

− κT

αβ

]

;β

= 0 with

α = 0; 1; 2; 3 will automatically be satisﬁed; they will not constrain the metric components

in any way, and there will remain in the ﬁeld equation only six independent constraints on

the metric components, precisely the desired number.

37

It turns out, in fact, that from the Ricci tensor and the scalar curvature one can construct

a curvature tensor G

αβ

with the desired property:

G

αβ

≡ R

αβ

−

1

2

Rg

αβ

(2.25)

Today we call this the Einstein curvature tensor (when R

αβ

is The Ricci tensor, deﬁned

by the contraction R

αβ

= R

γ

αγβ

= g

γδ

R

δαγβ

; R is the Ricci scalar, deﬁned by contraction

R = R

β

β

= g

αβ

R

αβ

). That it has vanishing divergence, independently of how one chooses

the metric,

∇.G ≡ 0 (2.26)

is called the contracted Bianchi identity (for more information see section 13.5 of Ref. [9]).

The Einstein ﬁeld equation, then, should equate a multiple of T

αβ

to the Einstein tensor

G

αβ

:

G

αβ

= κT

αβ

(2.27)

The proportionality factor κ is determined from the Newtonian limit: By rewriting the ﬁeld

equation (2.27) in terms of the Ricci tensor

R

αβ

−

1

2

g

αβ

R = κT

αβ

(2.28)

then taking the trace to obtain R = −κg

µν

T

µν

, then inserting this back into (2.28), we

obtain

R

αβ

= κ(T

αβ

−

1

2

g

αβ

g

µν

T

µν

) (2.29)

In nearly Newtonian situations and in a local Lorentz frame, the mass-energy density T

00 ∼

=

ρ is far greater than the momentum density T

j0

and also far greater than the stress T

jk

;

and correspondingly, the time-time component of the ﬁeld equation (2.29) becomes

R

00

= κ(T

00

−

1

2

η

00

η

00

T

00

) =

1

2

κT

00

=

1

2

κρ , where η

ij

is the metric of SR (2.30)

By comparing with the correct Newtonian limit (2.23) and noting that in a local Lorentz

frame R

00

= R

00

, we see that

κ = 8πG (2.31)

Up to now we use of geometrized units in which the speed of light is unity. Just as that

has simpliﬁed greatly the mathematical notation in this Chapter, so also future notation

38

will be greatly simpliﬁed if we set Newton’s gravitation constant to unity. This further

geometrization of our units corresponds to equating mass units to length units via the

relation

1 =

G

c

2

= 7.42 ×10

−28

m

kg

; i.e. 1 kg = 7.42 ×10

−28

m (2.32)

Any equation can readily be converted from conventional units to geometrized units by

removing all factors of c and G; and it can readily be converted back by inserting whatever

factors of c and G one needs in order to make both sides of the equation dimensionally

correct. Preface to Table 2.1 lists a few important numerical quantities in both conventional

units and geometrized units.

In geometrized units the Einstein ﬁeld equation (2.27), with κ = 8πG = 8π [Eq. (2.31)],

assumes the following standard form

G

µν

= 8πT

µν

; i.e., G = 8πT (2.33)

2.7 An Another Way to Learning General Relativity

In this chapter GR foundations are presented in usual way, but recently another way is

invented that an overview on it is presented below (for more about it refer to chapter III of

Ref. [17])

General relativity is easy. Nowadays, it can be made as intuitive as universal gravity

and its inverse square law - by using the right approach. The main ideas of general rel-

ativity, like those of special relativity, are accessible to secondary-school students. Black

holes, gravitational waves, space-time curvature and the limits of the universe can then be

understood with as easily as the Doppler eﬀect or the twins paradox.

It is that, just as special relativity is based on a maximum speed c, general relativity

is based on a maximum force c

4

/4G or on a maximum power c

5

/4G. The maximum force

and the maximum power are achieved only on insurmountable limit surfaces; these limit

surfaces are called horizons. It is possible to deduce the ﬁeld equations of general relativity.

In particular, the existence of a maximum for force or power implies that space-time is

curved. It explains why the sky is dark at night, and it shows that the universe is of ﬁnite

size.

39

Table 2.1: Some useful quantities in conventional and geometrized units. Note: 1 Mpc =

10

6

parsecs (pc), 1 pc = 3.026 light year (“lt y”), 1 lt yr = 0.946×10

16

m, 1 AU = 1.49×10

11

m. For other useful astronomical constants see C. W. Allen, Astrophysical Quantities.

Quantity Conventional Units Geometries Unites

speed of light 2.998 ×10

8

m sec

−1

one

Newton’s gravita-

tion constant, G

6.673×10

−11

m

3

kg

−1

sec

−2

one

G/c

2

7.425 ×10

−28

m kg

−1

one

c

5

/G 3.629 ×10

52

W one

c

2

/

√

G 3.479 ×10

24

gauss cm

= 1.160 ×10

24

volts

one

Planck’s reduced

constant

1.055 ×10

−34

kg m

2

s

−1

(1.616 ×10

−35

m)

2

sun’s mass, M

1.989 ×10

30

kg 1.477 km

sun’s radius, R

6.960 ×10

8

m 6.960 ×10

8

m

earth’s mass, M

⊕

5.977 ×10

24

kg 4.438 mm

earth’s radius, R

⊕

6.371 ×10

6

m 6.371 ×10

6

m

Hubble constant

H

0

65±25 km sec

−1

Mpc

−1

[(12±5)×10

9

lt yr]

−1

density to close

universe, ρ

crit

9

+11

−5

×10

−27

kg m

−3

7

+8

−3

×10

−54

m

−2

40

The theory of special relativity appears when we recognize the speed limit c in nature

and take this limit as a basic principle. At the end of the twentieth century it was shown

that general relativity can be approached by using a similar basic principle:

• There is in nature a maximum force:

F ≤

c

4

4G

= 3.0 · 10

43

N (2.34)

In nature, no force in any muscle, machine or system can exceed this value. For

the curious, the value of the force limit is the energy of a (Schwarzschild) black hole

divided by twice its radius. The force limit can be understood intuitively by noting

that (Schwarzschild) black holes are the densest bodies possible for a given mass. Since

there is a limit to how much a body can be compressed, forces - whether gravitational,

electric, centripetal or of any other type —cannot be arbitrary large.

Alternatively, it is possible to use another, equivalent statement as a basic principle:

• There is a maximum power in nature:

P ≤

c

5

4G

= 9.1 · 10

51

W (2.35)

No power of any lamp, engine or explosion can exceed this value. The maximum

power is realized when a (Schwarzschild) black hole is radiated away in the time that

light takes to travel along a length corresponding to its diameter.

The existence of a maximum force or power implies the full theory of general relativity. In

order to prove the correctness and usefulness of this approach, a sequence of arguments is

required. The sequence is the same as for the establishment of the limit speed in special

relativity. First of all, we have to gather all observational evidence for the claimed limit.

Secondly, in order to establish the limit as a principle of nature, we have to show that

general relativity follows from it. Finally, we have to show that the limit applies in all

possible and imaginable situations. Any apparent paradoxes will need to be resolved.

The maximum force principle does make sense, provided that we visualize it by means

of the useful deﬁnition: force is the ﬂow of momentum per unit time. Momentum cannot

be created or destroyed. We use the term “ﬂow” to remind us that momentum, being

a conserved quantity, can only change by inﬂow or outﬂow. In other words, change of

41

momentum always takes place through some boundary surface. This fact is of central

importance. Whenever we think about force at a point, we mean the momentum “ﬂowing”

through a surface at that point. The maximum force principle thus boils down to the

following: if we imagine any physical surface (and cover it with observers), the integral

of momentum ﬂow through the surface (measured by all those observers) never exceeds a

certain value. It does not matter how the surface is chosen, as long as it is physical, i.e., as

long as we can ﬁx observers onto it.

This principle imposes a limit on muscles, the eﬀect of hammers, the ﬂow of material,

the acceleration of massive bodies, and much more. No system can create, measure or

experience a force above the limit. No particle, no galaxy and no bulldozer can exceed

it. The existence of a force limit has an appealing consequence. In nature, forces can be

measured. Every measurement is a comparison with a standard. The force limit provides

a natural unit of force which ﬁts into the system of natural units (When Planck discovered

the quantum of action, he had also noticed the possibility to deﬁne natural units. On a

walk with his seven-year-old son in the forest around Berlin, he told him that he had made

a discovery as important as the discovery of universal gravity) that Max Planck derived

from c, G and h (or ). The maximum force thus provides a standard of force valid in every

place and at every instant of time.

The expression for the maximum force involves the speed of light c and the gravitational

constant G; it thus qualiﬁes as a statement on relativistic gravitation. The fundamental

principle of special relativity states that speed v obeys v ≤ c for all observers. Analogously,

the basic principle of general relativity states that in all cases force F and power P obey

F ≤ c

4

/G and P ≤ c

5

/G. It does not matter whether the observer measures the force

or power while moving with high velocity relative to the system under observation, during

free fall, or while being strongly accelerated. It is essential that the observer records values

measured at his own location and that the observer is realistic, i.e., made of matter and

not separated from the system by a horizon. These conditions are the same that must be

obeyed by observers measuring velocity in special relativity.

Since physical power is force times speed, and since nature provides a speed limit, the

force bound and the power bound are equivalent. We have already seen that force and power

appear together in the deﬁnition of 4-force; we can thus say that the upper bound is valid

42

for every component of a force, as well as for its magnitude. The power bound limits the

output of car and motorcycle engines, lamps, lasers, stars, gravitational radiation sources

and galaxies. It is equivalent to 1.2 × 10

49

horsepower. The maximum power principle

states that there is no way to move or get rid of energy more quickly than that.

The power limit can be understood intuitively by noting that every engine produces

exhausts, i.e. some matter or energy that is left behind. For a lamp, a star or an evaporating

black hole, the exhausts are the emitted radiation; for a car or jet engine they are hot gases;

for a water turbine the exhaust is the slowly moving water leaving the turbine; for a rocket

it is the matter ejected at its back end; for a photon rocket or an electric motor it is

electromagnetic energy. Whenever the power of an engine gets close to the limit value,

the exhausts increase dramatically in mass–energy. For extremely high exhaust masses,

the gravitational attraction from these exhausts –even if they are only radiation prevents

further acceleration of the engine with respect to them. The maximum power principle thus

expresses that there is a built-in braking mechanism in nature; this braking mechanism is

gravity.

Yet another, equivalent limit appears when the maximum power is divided by c

2

.

• There is a maximum rate of mass change in nature:

dm

dt

≤

c

3

4G

= 1.0 · 10

35

kg/s (2.36)

This bound imposes a limit on pumps, jet engines and fast eaters. Indeed, the rate

of ﬂow of water or any other material through tubes is limited. The mass ﬂow limit

is obviously equivalent to either the force or the power limit.

2.7.1 The Experimental Evidence

Like the maximum speed principle, the maximum force principle must ﬁrst of all be checked

experimentally. No one has yet dedicated so much eﬀort to testing the maximum force or

power. However, it is straightforward to conﬁrm that no experiment, whether microscopic,

macroscopic or astronomical, has ever measured force values larger than the stated limit.

Many people have claimed to have produced speeds larger than that of light. So far, nobody

has ever claimed to have produced a force larger than the limit value.

43

The large accelerations that particles undergo in collisions inside the Sun, in the most

powerful accelerators or in reactions due to cosmic rays correspond to force values much

smaller than the force limit. The same is true for neutrons in neutron stars, for quarks inside

protons, and for all matter that has been observed to fall towards black holes. Furthermore,

the search for space-time singularities, which would allow forces to achieve or exceed the

force limit, has been fruitless.

In the astronomical domain, all forces between stars or galaxies are below the limit value,

as are the forces in their interior. Not even the interactions between any two halves of the

universe exceed the limit, whatever physically sensible division between the two halves is

taken. (The meaning of “physically sensible division” will be deﬁned below; for divisions

that are not sensible, exceptions to the maximum force claim can be constructed.)

Astronomers have also failed to ﬁnd any region of space-time whose curvature is large

enough to allow forces to exceed the force limit. Indeed, none of the numerous recent

observations of black holes has brought to light forces larger than the limit value or objects

smaller than the corresponding black hole radii. Observations have also failed to ﬁnd a

situation that would allow a rapid observer to observe a force value that exceeds the limit

due to the relativistic boost factor.

The power limit can also be checked experimentally. It turns out that the power or

luminosity —of stars, quasars, binary pulsars, gamma ray bursters, galaxies or galaxy clus-

ters can indeed be close to the power limit. However, no violation of the limit has ever been

observed. Even the sum of all light output from all stars in the universe does not exceed

the limit. Similarly, even the brightest sources of gravitational waves, merging black holes,

do not exceed the power limit. Only the brightness of evaporating black holes in their ﬁnal

phase could equal the limit. But so far, none has ever been observed.

Similarly, all observed mass ﬂow rates are orders of magnitude below the corresponding

limit. Even physical systems that are mathematical analogues of black holes —for example,

silent acoustical black holes or optical black holes— do not invalidate the force and power

limits that hold in the corresponding systems.

The experimental situation is somewhat disappointing. Experiments do not contradict

the limit values. But neither do the data do much to conﬁrm them. The reason is the lack

of horizons in everyday life and in experimentally accessible systems. The maximum speed

44

at the basis of special relativity is found almost everywhere; maximum force and maximum

power are found almost nowhere. For more information about this topic refer to chapter

III of Ref. [17].

Chapter 3

Special Cases of Spacetime

3.1 Introduction

Every thing in GR is predicted by metric (g), but by what metric is determined? It is

Einstein ﬁeld equation that according to the physical properties of spacetime determines

metric (geometry of spacetime).

When Einstein formulate his equation, he said that no exact solution is possible for

it but, on January 13, 1916, just seven weeks after formulating the ﬁnal version of his

ﬁeld equation, G = 8πT, Albert Einstein read to a meeting of the Prussian Academy of

Sciences in Berlin a letter from the eminent German astrophysicist Karl Schwarzschild.

Schwarzschild, as a member of the German army, had written from the World-War-One

Russian front to tell Einstein of a mathematical discovery he had made: he had found the

world’s ﬁrst exact solution to the Einstein ﬁeld equation [18].

In this chapter we want to analyze this solution and some other (Kerr and Kerr-Newman

metrics) brieﬂy. First we try to obtain Schwarzschild metric from Einstein ﬁeld equation

by using its physical properties, then this metric and some other similar metrics (Kerr and

Kerr-Newman) will be analyzed.

3.2 Schwarzschild Solution

Schwarzschild spacetime geometry is the vacuum Einstein ﬁeld equation G = 0. Schwarzschild

consider some simpliﬁcations for him solution

45

46

1. Spherically symmetric solution: it means that there exists a privileged point, called

the origin O, such that system is invariant under spatial rotation about O.

2. Static solution: metric should be time independent, and if metric is static, we expect

cross terms to be absent (consider the interval between two events (x

0

, x

1

, x

2

, x

3

) and

(x

0

+dx

0

, x

1

+dx

1

, x

2

, x

3

), then ds

2

= g

00

(dx

0

)

2

+2g

01

dx

0

dx

1

+g

11

(dx

1

)

2

, because

spacetime is static the interval is invariant under a time inverse x

0

→x

0

= −x

0

, thus

interval become ds

2

= g

00

(dx

0

)

2

−2g

01

dx

0

dx

1

+ g

11

(dx

1

)

2

and thus g

01

= 0).

3. Hypersurface-orthogonal vector ﬁelds.

3.2.1 Coordinates and Metric for a Static, Spherical System

To deduce the gravitational ﬁeld for a static spherical star-or for any other static, spherical

system-begin with the metric of special relativity (no gravity) in the spherically symmetric

form [19]

ds

2

= −dt

2

+ dr

2

+ r

2

dΩ

2

(3.1)

where

dΩ

2

= dθ

2

+ sin

2

(θ)dφ

2

(3.2)

Try to modify this metric to allow for curvature due to the gravitational inﬂuence of the

star, while preserving spherical symmetry. The simplest and most obvious guess is to allow

those metric components that are already non-zero in equation (3.1) to assume diﬀerent

values:

ds

2

= −e

2Φ

dt

2

+ e

2Λ

dr

2

+ R

2

dΩ

2

(3.3)

where Φ, Λ, and R are functions of r only. (The static assumption demands ∂g

µν

/∂t = 0.)

To verify that this guess is good, use it in constructing stellar models, and check that the

resulting models have the same generality (same set of quantities freely speciﬁable) as in

Newtonian theory and as expected from general physical considerations. An apparently

more general metric

ds

2

= −a

2

dt

2

−2a b dr dt + c

2

dr

2

+ R

2

dΩ

2

(3.4)

47

actually is not more general in any physical sense. One can perform a coordinate transfor-

mation to a new time coordinate t

deﬁned by

e

Φ

dt

= a dt + b dr (3.5)

By inserting this in equation (3.4), and by deﬁning e

2Λ

= b

2

+c

2

, one obtains the postulated

line element (3.3), apart from a prime on the t.

The necessity to allow for arbitrary coordinates in general relativity may appear bur-

densome when one is formulating the theory; but it gives an added ﬂexibility, something

one should always try to turn to one’s advantage when formulating and solving problems.

The g

rt

= 0 simpliﬁcation (called a coordinate condition) in equation (3.3) results from an

advantageous choice of the t coordinate. The r coordinate, however, is also at one’s disposal

(as long as one chooses it in a way that respects spherical symmetry; thus not r

= r +cos θ

). One can turn this freedom to advantage by introducing a new coordinate r

(r) deﬁned

by

r

= R(r) (3.6)

With this choice of the radial coordinate, and with the primes dropped, Eq. (3.3) reduces

to

ds

2

= −e

2Φ

dt

2

+ e

2Λ

dr

2

+ r

2

dΩ

2

(3.7)

a line element with just two unknown functions, Φ(r) and Λ(r). This coordinate sys-

tem and metric have been used in most theoretical models for relativistic stars since the

pioneering work of Schwarzschild (1916), Tolman (1939), and Oppenheimer and Volkoﬀ

(1939). These particular coordinates are sometimes called “curvature coordinates” and

sometimes “Schwarzschild coordinates.” The central idea of these coordinates, in a nut-

shell, is (Schwarzschild r-coordinate) = (proper circumference)/ 2π [19]. {Consider ν = 2Φ

and λ = 2Λ, hence

ds

2

= −e

ν

dt

2

+ e

λ

dr

2

+ r

2

(dθ

2

+ sin

2

θ dφ

2

) , ν = ν(r) , λ = λ(r) (3.8)

thus

g

µν

=

_

_

_

_

_

_

_

−e

−ν

0 0 0

0 e

λ

0 0

0 0 r

2

0

0 0 0 r

2

sin

2

θ

_

_

_

_

_

_

_

(3.9)

48

we have G

µν

= R

µν

−

1

2

Rg

µν

, thus

G

0

0

= g

00

G

00

= −e

−λ

_

∂

r

λ

r

−

1

r

_

−

1

r

2

G

1

0

=

e

−λ

r

∂

t

λ = e

λ−ν

G

0

1

G

1

1

= e

−λ

_

∂

r

ν

r

−

1

r

2

_

−

1

r

2

_

¸

¸

_

¸

¸

_

eqs(*) (3.10)

G

2

2

= G

3

3

= −

1

2

e

−λ

_

∂

r

ν∂

r

λ

2

+

∂

r

λ

r

−

∂

r

ν

r

−

(∂

r

ν)

2

2

−∂

r

∂

r

ν

_

(3.11)

otherwise component of G vanishes. The contracted Bianchi identity show that G

2

2

= G

3

3

vanishes automatically, if the Eqs(*) vanishes, thus we have three independent equations

G

0

0

= 0

G

0

1

= 0

G

1

1

= 0

_

¸

¸

_

¸

¸

_

⇒e

λ

=

_

1 −

2M

r

_

−1

, λ + ν = h(t) = 0 (3.12)

because ν, λ are time (t) independent. Thus

g

µν

=

_

_

_

_

_

_

_

−

_

1 −

2M

r

_

0 0 0

0

_

1 −

2M

r

_

−1

0 0

0 0 r

2

0

0 0 0 r

2

sin

2

θ

_

_

_

_

_

_

_

(3.13)

Is the Schwarzschild metric (M is a constant of integration). Now it is possible to see the

properties of Schwarzschild spacetime in its metric

1. From the metric, we have g

µν,0

= 0, it means metric is stationery.

2. From the metric, we have g

µν

= 0, µ = ν, it means metric is static.

3. The coordinates are adapted to the killing vector ﬁeld X

α

= δ

α

0

, such that this time

killing vector ﬁeld, X

α

, is hypersurface-orthogonal to the family of hypersurface t =

constant and so the solution is static and t is a worldline.

When this metric Written as a line element in a special coordinate system that Schwarzschild

invented for the purpose, Schwarzschild’s solution takes the form

ds

2

= −(1 −2M/r)dt

2

+

dr

2

(1 −2M/r)

+ r

2

(dθ

2

+ sin

2

θ dφ

2

) (3.14)

49

where M is a constant of integration. The connection coeﬃcients, Riemann tensor, and

Ricci and Einstein tensors for this metric can be computed and results are tabulated in Box

3.1. The key bottom line is that the Einstein tensor vanishes. Therefore, the Schwarzschild

metric (3.14) is a solution of the Einstein ﬁeld equations with vanishing stress-energy tensor.

Many readers know already the lore of this subject: The Schwarzschild spacetime is

reputed to represent the vacuum exterior of a nonrotating, spherical star; and also the

exterior of a spherical star as it implodes to form a black hole; and also the exterior and

interior of a nonrotating, spherical black hole; and also a wormhole that connects two

diﬀerent universes or two widely separated regions of our own universe.

Returning to the Schwarzschild line element (3.14), let us examine several speciﬁc regions

of spacetime: At “radii” r large compared to the integration constant M, the line element

(3.14) takes the form

ds

2

= −dt

2

+ dr

2

+ r

2

(dθ

2

+ sin

2

θ dφ

2

) (3.15)

This is the line element of at spacetime, ds

2

= −dt

2

+dx

2

+dy

2

+dz

2

written in spherical

polar coordinates [x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ].Thus, Schwarzschild

spacetime is asymptotically at in the region of large radii r/M → ∞. This is just what

one might expect physically when one gets far away from all sources of gravity. Thus, it is

reasonable to presume that the Schwarzschild spacetime geometry is that of some sort of

isolated, gravitating body which is located in the region r ≈ M.

The large-r line element (3.15) not only reveals that Schwarzschild spacetime is asymp-

totically at; it also shows that in the asymptotically at region the Schwarzschild t is the

time coordinate of a Lorentz reference frame. Notice that the region of strong spacetime

curvature has a boundary (say, r ≈ 100M) that remains forever ﬁxed relative to the asymp-

totically Lorentz spatial coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. This

means that the asymptotic Lorentz frame can be regarded as the body’s asymptotic rest

frame. We conclude, then, that far from the body the Schwarzschild t coordinate becomes

the Lorentz time of the body’s asymptotic rest frame, and the Schwarzschild r, θ, φ coordi-

nates become spherical polar coordinates in the body’s asymptotic rest frame. As we move

inward from r = ∞, we gradually begin to see spacetime curvature.

50

Box 3.1

Connection Coefficients and Curvature Tensors for

Schwarzschild

The coordinate basis vectors for the Schwarzschild solution are

e

t

=

∂

∂t

, e

r

=

∂

∂r

, e

θ

=

∂

∂θ

, e

φ

=

∂

∂φ

e

t

=

∇t, e

r

=

∇r, e

θ

=

∇θ, e

φ

=

∇φ

The covariant metric coefficients in this coordinate basis are [cf. Eq.

(3.14)]

g

tt

= −

_

1 −

2M

r

_

, g

rr

= −

_

1 −

2M

r

_

−1

g

θθ

= r

2

, g

φφ

= r

2

sin

2

θ

and the contravariant metric coefficients are the inverse of these

g

tt

= −

_

1 −

2M

r

_

−1

, g

rr

= −

_

1 −

2M

r

_

,

g

θθ

=

1

r

2

, g

φφ

=

1

r

2

sin

2

θ

Γ

t

rt

= Γ

t

tr

=

M

r

2

_

1 −

2M

r

_

−1

,

Γ

r

tt

=

M

r

2

_

1 −

2M

r

_

,

Γ

r

rr

= −

M

r

2

_

1 −

2M

r

_

−1

,

Γ

r

θθ

= −r

_

1 −

2M

r

_

,

Γ

θ

rθ

= Γ

θ

θr

= Γ

φ

rφ

= Γ

φ

φr

=

1

r

,

Γ

r

φφ

= −r sin

2

θ

_

1 −

2M

r

_

,

Γ

θ

φφ

= −sin θ cos θ

Γ

φ

θφ

= Γ

φ

φθ

= cot θ

51

Box 3.1 (continued)

The orthonormal basis associated with the above coordinate basis is

e

ˆ

0

=

∂

t

_

1 −2M/r

, e

ˆ r

=

_

1 −

2M

r

∂

r

,

e

ˆ

θ

=

1

r

∂

θ

, e

ˆ

φ

=

1

r sin θ

∂

φ

The nonzero connection coeﬃcients in this orthonormal basis are

Γ

ˆ r

ˆ

t

ˆ

t

= Γ

ˆ

t

ˆ r

ˆ

t

=

M

r

2

_

1 −2M/r

, Γ

ˆ

φ

ˆ

θ

ˆ

φ

= −Γ

ˆ

θ

ˆ

φ

ˆ

φ

=

cot θ

r

,

Γ

ˆ

θ

ˆ r

ˆ

θ

= Γ

ˆ

φ

ˆ r

ˆ

φ

= Γ

ˆ r

ˆ

θ

ˆ

θ

= Γ

ˆ r

ˆ

φ

ˆ

φ

=

_

1 −2M/r

r

The nonzero components of the Riemann tensor in this orthonormal

basis are

R

ˆ r

ˆ

tˆ r

ˆ

t

= −R

ˆ

θ

ˆ

φ

ˆ

θ

ˆ

φ

= −

2M

r

3

,

R

ˆ

θ

ˆ

t

ˆ

θ

ˆ

t

= R

ˆ

φ

ˆ

t

ˆ

φ

ˆ

t

= −R

ˆ r

ˆ

φˆ r

ˆ

φ

= −R

ˆ r

ˆ

thetaˆ r

ˆ

θ

=

M

r

3

The Ricci tensor, curvature scalar, and Einstein tensor all vanish

—which implies that the Schwarzschild metric is a solution of the

vacuum Einstein ﬁeld equations.

To ﬁnd out the precise mean of M should more work on metric, If the Schwarzschild metric

is written asymptotically (when r/M → ∞), and slight deviations of the Schwarzschild

metric coeﬃcients from those of a Lorentz frame, to ﬁrst order in M/r is calculated, Direct

comparison of it by Newtonian limit in GR (that obtain by assume g

µν

= ηµν +h

µν

, when

η

µν

is SR —ﬂat Minkowski spacetime- metric and h

µν

is little respect to η

µν

and do some

approximations), shows that a Newtonian description of the body’s distant gravitational

ﬁeld will entail a Newtonian potential given by

Φ = −

M

r

(3.16)

This, of course, is the external Newtonian ﬁeld of a body with mass M. Thus, the integration

52

constant M in the Schwarzschild line element is the mass which characterizes the body’s

distant, nearly Newtonian gravitational ﬁeld.

We can describe the physical interpretation of M as the body’s mass in operational

terms as follows: Suppose that a test particle (e.g., a small planet) moves around our central

body in a circular orbit with radius r M. A Newtonian analysis of the orbit predicts

that, as measured using Newtonian time, the period of the orbit will be P = 2π(r

3

/M)

1

2

.

Moreover, since Newtonian time is very nearly equal to the time t of the nearly Lorentz

coordinates, and since that t is Lorentz time in the body’s relativistic, asymptotic rest

frame, the orbital period as measured by observers at rest in the asymptotic rest frame

must be P = 2π(r

3

/M)

1

2

. Thus, M is the mass that appears in Kepler’s laws for the orbits

of test particles far from the central body. This quantity is often called the body’s “active

gravitational mass”, since it is the mass that characterizes the body’s gravitational pull. It

is also called the body’s “total mass-energy” because it turns out to include all forms of

mass and energy that the body possesses (rest mass, internal kinetic energy, and all forms

of internal binding energy including gravitational).

Next we shall move inward, from the asymptotically at region of Schwarzschild space-

time, toward smaller and smaller radii. As we do so, the spacetime geometry becomes more

and more strongly curved, and the Schwarzschild coordinate system becomes less and less

Lorentz. As an indication of extreme deviations from Lorentz, notice that the signs of the

metric coeﬃcients

g

tt

= −

_

1 −

2M

r

_

, g

rr

=

_

1 −

2M

r

_

−1

(3.17)

get reversed as one moves from r > 2M through r = 2M and into the region r < 2M.

Correspondingly, outside r = 2M world lines of changing t but constant r, θ, φ are timelike,

while inside r = 2M those world lines are spacelike; and similarly outside r = 2M world

lines of changing r but constant t, θ, φ are spacelike, while inside they are timelike. In

this sense, outside r = 2M, t plays the role of a time coordinate and r the role of a space

coordinate; while inside r = 2M, t plays the role of a space coordinate and r the role

of a time coordinate. Moreover, this role reversal occurs without any change in the role

of r as 1/2π times the circumference of circles around the center. Historically this role

reversal presented for many decades severe conceptual problems, even to the best experts

on general relativity. Henceforth we shall refer to the location of role reversal, r = 2M, as

53

the gravitational radius of the Schwarzschild spacetime.

When one moves from r > 2M through r = 2M and arrive to r = 2M , g

rr

will be

inﬁnite (singularity) and g

tt

will be vanished, it means that, when r = 2M then t → ∞,

and if we want to study spacetime at r = 2M or in region r < 2M, another metric (i.e.

another coordinate system, not another spacetime) should be used, for example Eddington-

Fiklesteine coordinate, in other words singularity at r = 2Mis a coordinate singularity that

will be removed by some coordinate transformation, but singularity at r = 0 is an intrinsic

singularity that will not be removed by any coordinate transformation. The horizon is the

surface which one when using the coordinates of a distant observer in ﬂat space will never

see anything pass. In Schwarzschild spacetime it is r = 2M, it is not a real singularity, but

merely a coordinate singularity.

Finklestein, a postdoctoral fellow at the Stevens Institute of Technology in Hoboken,

New Jersey, found the following simple transformation which moves the region t → ∞,

r = 2M of Schwarzschild coordinates in to a ﬁnite location. His transformation involves

introducing a new time coordinate(for a more complete discussion refer to [20])

˜

t = t + 2M ln |(r/2M) −1| (3.18)

but leaving unchanged the radial and angular coordinates. By inserting the coordinate

transformation (3.18) into the Schwarzschild line element (3.14) we obtain the following

line element for Schwarzschild spacetime written in Eddington-Finklestein coordinates:

ds

2

= −

_

1 −

2M

r

_

d

˜

t

2

+

4M

r

d

˜

t dr +

_

1 −

2M

r

_

dr

2

+ r

2

(dθ

2

+ sin

2

θ dφ

2

) (3.19)

Notice that, by contrast with the line element in Schwarzschild coordinates, none of the

metric coeﬃcients diverge as r approaches 2M.

3.2.2 Birkhoﬀ’s Theorem

In 1923, George Birkhoﬀ, a professor of mathematics at Harvard, proved a remarkable the-

orem: The Schwarzschild spacetime geometry is the unique spherically symmetric solution

of the vacuum Einstein ﬁeld equation G = 0 [21]. This Birkhoﬀ theorem can be restated in

more operational terms as follows: Suppose that you ﬁnd a solution of the vacuum Einstein

ﬁeld equation, written as a set of metric coeﬃcients g

¯ α

¯

β

in some coordinate system {x

¯ µ

}.

54

Suppose, further, that these g

¯ α

¯

β

(x

¯ µ

) exhibit spherical symmetry, but do not coincide with

the Schwarzschild expressions [Box 3.1]. Then Birkhoﬀ guarantees the existence of a coor-

dinate transformation from your coordinates x

¯ µ

to Schwarzschild’s coordinates such that,

when that transformation is performed, the resulting new metric components g

αβ

(x

µ

) have

precisely the Schwarzschild form. This implies that, thought of as a coordinate-independent

spacetime geometry, the Schwarzschild solution is completely unique.

3.3 Other Spacetimes

Up to now, simplest case of spacetime —Schwarzschild— was analyzed but in real world,

objects have electric charge, spin and other possible properties that aﬀect spacetime, and for

a more realistic description of nature, more complete metrics that contain these properties

are necessary. In this section two metrics that contain these properties is presented.

3.3.1 Kerr Geometry

If one want to interpret the Schwarzschild solution, it is possible to do this by an example;

this is the static spherical symmetry star (or neutron star or black hole) that Schwarzschild

solution described exterior spacetime of it. Now consider a spinning object, what metric

describe spacetime of it? Can Schwarzschild metric do it? Answer is no, the metric that

can describe this spacetime is Kerr metric:

Another solution to Einstein’s ﬁeld equations R

ab

= 0 [22] was discovered in 1963 by

Roy Kerr, a New Zealand mathematician. This solution describes the gravitational ﬁeld

of a rotating mass. The Kerr metric, in what are called Boyer-Lindquist [23] coordinates,

(t, r, θ, φ —generalization of Schwarzschild coordinate; object rotates in φ direction) is [24]

ds

2

=−(∆/ρ

2

)[dt −a sin

2

θ dφ]

2

+ (sin

2

θ/ρ)[(r

2

+ a

2

)dφ −a dt]

2

+ (ρ

2

/∆)dr

2

+ ρ

2

dθ

2

;

∆ ≡ r

2

−2Mr + a

2

ρ ≡ r

2

+ a

2

cos

2

θ

(3.20)

The parameter a should be interpreted as the angular momentum per unit mass of the

rotating central body of mass M (a = S/M, where S is angular momentum of body).

The interpretation of the Boyer-Lindquist coordinates is not straightforward. It is cer-

tainly true that as r → ∞, the Riemannian curvature of the Kerr metric approaches zero,

55

and so “at inﬁnity” the Boyer-Lindquist coordinates can be thought of as spherical coordi-

nates on the Euclidean space that we get by setting t = constant [22]. Notice that if we let

the angular momentum S be zero, this solution collapses to the Schwarzschild solution.

The Kerr geometry has a horizon (and therefore describes a black hole) if and only if

M ≥ a. It seems likely that in any collapsing body which violates this constraint, centrifugal

forces will halt the collapse before a size ≈ M is reached, in other words an object with

higher angular momentum than corresponding to a = M cannot stick together. The horizon

is located at

r

H

= M +

_

M

2

−a

2

(3.21)

As with the Schwarzschild horizon of a nonrotating black hole, so also here, particles and

photons can fall inward through the horizon; but no particle or photon can emerge outward

through it; as the Schwarzschild geometry it is a coordinate singularity (It requires an inﬁnite

coordinate time for any particle or photon to fall inward through the horizon, t → ∞ as

r →r

H

) that can be removed by a coordinate transformation. The horizon is “generated”

by outgoing null geodesics (outgoing photon world lines).

Same as Schwarzschild geometry that coordinate transformation was done to remove

singularity, in Kerr geometry to remove the coordinate singularity, one must perform an

inﬁnite compression of coordinate time, and an inﬁnite untwisting in the neighborhood of

the horizon. Kerr coordinates achieve this by replacing t with a null coordinate

˜

V , and φ

with an untwisted angular coordinate

˜

φ: (

˜

V , r, θ,

˜

φ —generalization of ingoing Eddington-

Finkelstein coordinates; (

˜

V , θ,

˜

φ) = constant is an ingoing, “radial,” null geodesic; object

rotates in

˜

φ direction) [25]:

Relationship to Boyer-Lindquist:

d

˜

V = dt + (r

2

+ a

2

)(dr/∆)

d

˜

φ = dφ + a(dr/∆)

ds

2

=−[1 −ρ

−2

(2Mr)]d

˜

V

2

+ 2 dr d

˜

V

+ ρ

2

dθ + ρ

−2

[(r

2

+ a

2

) −∆a

2

sin

2

θ] sin

2

θ d

˜

φ

2

(3.22)

But about symmetries, the metric coeﬃcients in Boyer-Lindquist coordinates are in-

dependent of t and φ, and in Kerr coordinates are independent of

˜

V and d

˜

φ. Thus the

56

spacetime geometry is “time-independent” (stationary) and axially symmetric (that is ex-

pected).

A question is remaining that, in these metrics only two chrematistics of object is appear,

mass M and angular momentum S, but is these enough to describe spacetime completely?

There is a theory about this, Brandon Carter (1970): “All uncharged, stationary, axially

symmetric black holes with event horizons of spherical topology fall into disjoint families

not deformable into each other. The black holes in each family have external gravitational

ﬁelds determined uniquely by two parameters: the mass M and the angular momentum S.”

(Note: the “Kerr solutions”; form one such family; it is very likely that there are no others,

but this has not been proved as of December 1972).

Further works on Kerr geometry is out of goals of this text and only in end of this

text some numeric calculations on this metric will be done (but Kerr–Newman metric,

generalization of Kerr metric will analyzed more completely).

3.3.2 Kerr-Newman Geometry

When the real events (nonspherical star, collapse with small but nonzero net charge of

one sign or the other) are studied, for example the gravitational collapse of a realistic star

produces a black hole, it is impossible to use simple Schwarzschild metric or Kerr metric.

For case with small charge and small asymmetries, perturbation-theory calculations predict

an object (ﬁnal black hole) with external ﬁeld determined entirely by the mass M, charge Q,

and intrinsic angular momentum S of the object (collapsing star). For fully relativistic event

(collapse), with large asymmetries and possibly a large charge, the ﬁnal black hole (if one

forms) is also characterized uniquely by M, Q, and S. This is the conclusion that strongly

suggests itself in 1972 from a set of powerful theorems described in Box 3.2. Why M, Q,

and S should be the complete governors of the ﬁnal external ﬁeld of the object (black hole),

one can understand heuristically as follows. Of all quantities intrinsic to any isolated source

of gravity and electromagnetism, only M, Q, and S possess (and are deﬁned in terms of)

unique, conserved imprints in the distant external ﬁelds of the source. When a star collapses

to form a black hole, its distant external ﬁelds are forced to maintain unchanged the imprints

of M, Q, and S. In eﬀect, M, Q, and S provide anchors or constraints on the forms of the

ﬁelds. Initially other constraints are produced by the distributions of mass, momentum,

57

stress, charge, and current inside the star. But ultimately the star plunges through a

horizon, cutting itself oﬀ causally from the external universe. (The nonpropagation of long-

wavelength waves through curved spacetime plays a key role in this cutoﬀ) Subsequently,

the only anchors remaining for the external ﬁelds are the conserved imprints of M, Q, and

S. Consequently, the external ﬁelds quickly settle down into unique shapes corresponding to

the given M, Q, and S. Of course, the settling down involves dynamic changes of the ﬁelds

and an associated outﬂow of gravitational and electromagnetic waves. And, of course, the

outﬂowing waves carry oﬀ mass and angular momentum (but not charge), thereby leaving

M and S changed. And, of course, the external ﬁelds must then readjust themselves to the

new M and S. But the process will quickly converge, producing a black hole with speciﬁc

ﬁnal values of M, Q, and S and with external ﬁelds determined uniquely by those values.

The derivation of the solution and the proof of its uniqueness are much too complex to

be given here. However, the solution turns out to be the “Kerr-Newman geometry” and its

associated electromagnetic ﬁeld (we only work with geometry).

Written in the t, r, θ, φ coordinates of Boyer and Lindquist (generalization of Schwarzschild

coordinates and object rotates in φ direction), the Kerr-Newman geometry has the form [26]

ds

2

= −

∆

ρ

2

[dt −a sin

2

θ dφ]

2

+

sin

2

θ

ρ

2

[(r

2

+ a

2

) dφ −a dt]

2

+

ρ

2

∆

dr

2

+ ρ

2

dθ

2

(3.23)

where

∆ ≡ r

2

−2Mr + a

2

+ Q

2

, ρ

2

≡ r

2

+ a

2

cos

2

θ

a ≡ S/M ≡ angular momentum per unit mass

(3.24)

Same as Kerr geometry it is possible to write the Kerr-Newman metric in Kerr coordinates

(

˜

V , r, θ,

˜

φ —generalization of ingoing Eddington-Finkelstein coordinates; (

˜

V , θ,

˜

φ) =

constant is an ingoing, “radial,” null geodesic; black hole rotates in

˜

φ direction) [25]:

ds

2

=−[1 −ρ

−2

(2Mr −Q

2

)]d

˜

V

2

+ 2 dr d

˜

φ + ρ

2

dθ

2

+ ρ

2

[(r

2

+ a

2

)

2

−∆ a

2

sin

2

θ] sin

2

θ d

˜

φ

2

−2a sin

2

θ d

˜

φ dr

−2aρ

−2

(2Mr −Q

2

) sin

2

θ d

˜

φ d

˜

V

(3.25)

Relationship to Boyer-Lindquist:

d

˜

V = dt + (r

2

+ a

2

)(dr/∆), d

˜

φ = dφ + a(dr/∆) (3.26)

58

If these metric describe a black hole, a constraint on parameters is exist: the Kerr-Newman

geometry has a horizon, and therefore describes a black hole, if and only if M

2

≥ Q

2

+ a

2

.

It seems likely that in any collapsing body which violates this constraint, centrifugal forces

and/or electrostatic repulsion will halt the collapse before a size ∼ M is reached.

Both Kerr and Schwarzschild metrics are limiting cases of Kerr-Newman metric

Q = 0 Kerr geometry

S = 0 Reissner-Nordstrem geometry

Q = S = 0 Schwarzschild geometry

M

2

= Q

2

+ a

2

“Extreme Kerr-Newman geometry”

Kerr-Newman geometry has symmetries same as Kerr geometry; the metric coeﬃcients

in Boyer-Lindquist coordinates are independent of t and φ , and in Kerr coordinates are

independent of

˜

V and

˜

φ. Thus the spacetime geometry is “time-independent” (stationary)

and axially symmetric. The horizon is located at

r = r

+

≡ M +

_

M

2

−Q

2

−a

2

(3.27)

As the Schwarzschild horizon of a nonrotating black hole, so also here, particles and photons

can fall inward through the horizon; but no particle or photon can emerge outward through

it. The horizon is “generated” by outgoing null geodesics (outgoing photon world lines).

And a singularity exist in horizon, in Kerr-Newman geometry a coordinate singularity

exists in horizon (in Boyer-Lindquist coordinates) and a coordinate transformation can be

done to remove this singularity, to remove this coordinate singularity, one must perform

an inﬁnite compression of coordinate time, and an inﬁnite untwisting in the neighborhood

of the horizon. Kerr coordinates achieve this by replacing t with a null coordinate

˜

φ, and

φ with an untwisted angular coordinate

˜

φ (Kerr-Newman metric in Kerr coordinate was

written above).

Same as Kerr geometry further works on Kerr-Newman geometry is out of goals of this

text and only in end of this text some numeric calculations on this metric will be done.

3.4 The Many-Fingered Nature of Time

We conclude this chapter with a discussion of a concept which John Archibald Wheeler (the

person who has most clariﬁed the conceptual underpinnings of general relativity) calls the

59

many-ﬁngered nature of time.

In the ﬂat spacetime of special relativity there are preferred families of observers: Each

such family lives in a global Lorentz reference frame and uses that frame to split spacetime

into space plus time. The hypersurfaces of constant time (“slices of simultaneity”) which

result from that split are at hypersurfaces which slice through all of spacetime [Fig. 3.1(a)].

Of course, diﬀerent preferred families live in diﬀerent global Lorentz frames and thus split

up spacetime into space plus time in diﬀerent manners [e.g., the dotted slices of constant

time in Fig. 3.1(a) as contrasted to the dashed ones]. As a result, there is no universal

concept of time in special relativity; but, at least, there are some strong restrictions on

time: Each family of observers will agree that another family’s slices of simultaneity are at

slices.

In general relativity, i.e., in curved spacetime, even this restriction is gone: In a generic

curved spacetime there are no at hypersurfaces, and hence no candidates for at slices of

simultaneity. Hand in hand with this goes the fact that, in a generic curved spacetime

there are no global Lorentz frames, and thus no preferred families of observers. A family of

observers who are all initially at rest with respect to each other, and each of whom moves

freely (inertially), will soon acquire relative motion because of tidal forces. As a result,

their slices of simultaneity (deﬁned locally by Einstein light-ray synchronization, and then

deﬁned globally by patching together the little local bits of slices) may soon become rather

contorted. Correspondingly, as is shown in Fig. 3.1(b), diﬀerent families of observers will

slice spacetime up into space plus time in manners that can be quite distorted, relative

to each other -with “ﬁngers” of one family’s time slices pushing forward, ahead of the

other family’s here, and lagging behind there, and pushing ahead in some other place. In

curved spacetime it is best to not even restrict oneself to inertial (freely falling) observers.

For example, in the spacetime of a static star, or of the exterior of a Schwarzschild black

hole, the family of static observers [observers whose world lines are { (r, θ , φ) = const, t

varying}] are particularly simple; their world lines mold themselves to the static structure

of spacetime in a simple, static manner. However, these observers are not inertial; they do

not fall freely. This need not prevent us from using them to split up spacetime into space

plus time, however. Their proper reference frames produce a perfectly good split; and when

one uses that split, in the case of a black hole, one obtains a 3–dimensional-space version

60

o

b

s

e

r

v

e

r

o

b

s

e

r

v

e

r

o

b

s

e

r

v

e

r

(b) (a) (c)

x

t

t

=

3

t =2

t =

1

t =

0

Figure 3.1: Spacetime diagrams showing the slices of simultaneity as deﬁned by various

families of observers. Diagram (a) is in at spacetime, and the three families (those with

solid slices, those with dashed, and those with dotted) are inertial, so their slices of constant

time are those of global Lorentz frames. Diagram (b) is in curved spacetime, and the two

families’ slices of simultaneity illustrate the “many ﬁngered” nature of time. Diagram (c)

illustrates the selection of an arbitrary foliation of spacelike hypersurfaces of simultaneity,

and the subsequent construction of the world lines of observers who move orthogonal to

those hypersurfaces, i.e., for whom light-ray synchronization will deﬁne those hypersurfaces

as simultaneities.

of the laws of black-hole physics which is a useful tool in astrophysical research.

For any family of observers, accelerated or inertial, the slices of simultaneity as deﬁned

by Einstein light-ray synchronization (or equivalently by the space slices of the observer’s

proper reference frames) are the 3-surfaces orthogonal to the observers’ world lines; cf. Fig.

3.1(c). To see this most easily, pick a speciﬁc event along a speciﬁc observer’s world line, and

study the slice of simultaneity there from the viewpoint of a local Lorentz frame in which

the observer is momentarily at rest. Light-ray synchronization guarantees that, locally, the

observer’s slice of simultaneity will be the same as that of this local Lorentz frame; and,

since the frame’s slice is orthogonal to its own time direction and that time direction is the

same as the direction of the observer’s world line, the slice is orthogonal to the observer’s

world line. If the observers’ relative motions are suﬃciently contorted (in curved spacetime

or in at), it may not be possible to mesh their local slices of simultaneity, deﬁned in this

manner, into global slices of simultaneity; i.e., there may not be any global 3–dimensional

hypersurfaces orthogonal to their world lines. We can protect against this eventuality,

however, by choosing the slices ﬁrst: Select any foliation of spacelike slices through the

curved spacetime [Fig. 3.1(c)]. Then there will be a family of timelike world lines that are

61

everywhere orthogonal to these hypersurfaces. A family of observers who move along those

world lines and who deﬁne their 3-spaces of simultaneity by local light-ray synchronization

will thereby identify the orthogonal hypersurfaces as their simultaneities [27].

Chapter 4

Lagrangian and Hamiltonian

Formalism

4.1 Overview

Until now, we try to explain the problem; learn GR and obtain Geodesic equation, i.e. a

method to obtain equation (or equations) of motion, then speak about some special cases

of spacetime geometry and conﬁrm ourselves to them. But are there any other methods

to obtain equation (or equations) of motion? To explain this question remember that

in classical mechanics there are some ways to obtain equation (or equations) of motion

-second Newton’s law of motion, Lagrange’s method, Hamilton’s method- that according

to the situation one of them is selected. What about GR? Does Lagrange’s or Hamilton’s

method applicable in GR? If it is possible we can compare results of diﬀerent methods and

evaluate their correctness.

In this chapter, ﬁrst Lagrange’s and Hamilton’s method in classical mechanics are re-

viewed and then try to generalize them to GR.

4.2 Lagrangian Method

Assume a system that (q

1

, q

2

, . . . , q

N

) coordinates is needed to describe it. The coordinates

necessarily are not Cartesian and can be length, angle or any other things that are necessary

62

63

to describe the system. Generalized velocities are denoted by ( ˙ q

1

, ˙ q

2

, . . . , ˙ q

N

) where

˙ q

i

=

dq

i

dt

(4.1)

(q, ˙ q), where q denote all of coordinates —henceforth abbreviation symbol (q, ˙ q) is used

stead of (q

1

−, q

2

, . . . , q

N

, ˙ q

1

, ˙ q

2

, . . . , ˙ q

N

)— is called a conﬁguration of the system and

then there are 2N degrees of freedom. Collection of all of imaginable conﬁgurations of a

system is called conﬁguration space.

Nature have an important intrinsic properties, that coordinates, q

i

, and velocities, ˙ q

i

, of

a system at a moment determine acceleration of it at that moment, i.e. at the inﬁnitesimal

near point (respect to time) blow equations can be written

q

i

(t + ) = q

i

+ ˙ q

i

(t)

˙ q

i

(t + ) = ˙ q

i

+ ¨ q

i

(t)

(4.2)

Thus if conﬁguration of the system, (q(0), ˙ q(0)), is known at the moment, then conﬁguration

of the system can be determined for every time, (q(t), ˙ q(t)), uniquely. In other words

trajectory is determined uniquely (in nonquantum physics).

It is important to note, which is expressed above is an important intrinsic characteristic

of the nature and it is impossible to explain it more fundamentally, e.g. it is possible that

in nature coordinates at a moment can determine velocities only at that moment; but our

world behaves in such way that coordinates and velocities at a moment are independent

and if coordinates are known at a moment, velocities can not be determined at that mo-

ment. Fundamental question that should be answered is, how is trajectory of the system

in conﬁguration space determined? It is answered by an important principle of mechanics

that is called principle of least action.

4.2.1 Principle of Least Action

Consider a system is in conﬁguration (q

a

, ˙ q

a

) at t

1

and its dynamic put it in conﬁguration

(q

b

, ˙ q

b

) at t

2

. Then one asks, which trajectory was traversed by system form initial to

ﬁnal conﬁguration? In other words what was the trajectory from (q

a

, ˙ q

a

) to (q

b

, ˙ q

b

)?

According to the principle of least action there is a function is named Lagrangian, L(q, ˙ q),

and trajectory is determined in such a way that its integral along the trajectory —that is

64

named action, S— become a local extremum in all of possible trajectories, i.e. ﬁrst degree

deviation of this quantity respect to inﬁnitesimal deviation around it vanishes.

S =

_

t

2

t

1

L(q(t), ˙ q(t)) dt (4.3)

Now results of principle of least action are explained. Consider an arbitrary trajectory that

is q

a

at t

1

and q

b

at t

2

. In other words a trajectory is considered that connect q

a

to q

b

, for

this trajectory action is

S[q] :=

_

t

2

t

1

L(q

i

, ˙ q

i

) dt (4.4)

If this trajectory is correct trajectory (that selected by nature) then

δS

δq

= 0 (4.5)

It means that if another trajectory, q

(t) = q(t)+χ(t), is considered that χ(t) is inﬁnitesimal

then, ﬁrst degree deviation of action should vanish, i.e. up to ﬁrst degree we should have

S[q + χ] −S[q] = 0 (4.6)

From equations 3 and 5 obtain

S[q + χ] =

_

t

2

t

1

L(q

i

+ χ

i

, ˙ q

i

+ ˙ χ

i

, t) dt

=

_

t

2

t

1

_

L(q

i

, ˙ q

i

, t) +

∂L

∂q

i

χ

i

+

∂L

∂ ˙ q

i

˙ χ

i

_

dt

= S +

_

t

2

t

1

_

∂L

∂q

i

χ

i

+

d

dt

(

∂L

∂ ˙ q

i

χ

i

) −χ

i

d

dt

(

∂L

∂q

i

)

_

dt

= S +

_

t

2

t

1

_

∂L

∂q

i

−

d

dt

(

∂L

∂q

i

)

_

dt

(4.7)

because S deviations should vanish for any inﬁnitesimal variation of trajectory then, below

constraint should be not violated.

∂L

∂q

i

−

d

dt

∂L

∂ ˙ q

i

= 0 (4.8)

this equation is called Euler-Lagrange equation, that produces equation (or equations) of

motion. Last question is: what is the Lagrangian?

In classical mechanics we can obtain it by some caculations [28]

L = T(q, ˙ q) −V (q

1

, q

2

, . . . , q

N

) (4.9)

65

where T is the kinetic energy of particles of the system and V is potential function that

depend on interactions between particles of the system.

In GR, the square of the spacetime interval between two inﬁnitesimally near events

deﬁnes the metric,

ds

2

= g

µν

x

µ

x

ν

(4.10)

The motion of a particle subject to gravity only is a worldline such that its proper time τ

is a maximum [29], namely the geodesic on the four dimensional curved spacetime,

δ

_

ds = 0 (4.11)

This condition is equivalent to [30]

δ

_

L dτ = 0 , where L =

1

2

g

µν

˙ x

µ

˙ x

ν

(4.12)

where the dot denotes diﬀerentiation with respect to proper time τ (c dτ = ds), and L is

the Lagrangian.

4.3 Hamiltonian Mechanics

For a system with N degree of freedom, the Lagrange equations provide a set of N diﬀer-

ential equations of second order in the time for the position coordinates. The Hamiltonian

formalism yields 2N coupled diﬀerential equations of ﬁrst order for the momentum and po-

sition coordinates. In any case, there are 2N integration constants when solving the system

of equations that are determined by initial conditions (Hamiltonian method is suitable to

generalize classical mechanics to quantum mechanics).

To describe it, ﬁrst it is necessary to deﬁne generalize momentum, conjugate momentum

of q

i

coordinate is

p

i

=

∂L

∂ ˙ q

i

(4.13)

This momentum can diﬀer from linear momentum. Hamiltonian is [31]

H =

N

i=1

˙ q

i

p

i

−L (4.14)

66

It is important to note that in Hamiltonian formalism q

i

and p

i

are independent variables

to understand this point, we write

dH =

N

i=1

d ˙ q

i

p

i

+ ˙ q

i

dp

i

−dL =

N

i=1

d ˙ q

i

p

i

˙ q

i

dp

i

−

∂L

∂q

i

dq

i

−

∂L

∂ ˙ q

i

˙ q

i

(4.15)

but according to deﬁnition of conjugate momentum, p

i

=

∂L

∂ ˙ q

i

, ﬁrst and last terms cancel

out each other, thus

dH =

N

i=1

˙ q

i

dp

i

−

∂L

∂q

i

dq

i

= ˙ q

i

dp

i

− ˙ p

i

dq

i

(4.16)

where Euler-Lagrange equation was used in last expression. This equation shows that

1. Independent variables of H are q

i

and p

i

.

2. We have

˙ q

i

=

∂H

∂p

i

˙ p

i

= −

∂H

∂q

i

(4.17)

they are 2N ﬁrst order coupled diﬀerential equations that called Hamilton equations —

these equations produce equation (or equations) of motion in nonquantum mechanics. If

initial values, (q(t

0

), p(t

0

)), are known, these equations give an unique trajectory. Hamilton

equations are valid in GR and only Lagrangian that deﬁned in GR should be used.

Chapter 5

Computer Algebra in General

Relativity

5.1 Overview

Numerical analysis is one of the most important aspects of GR, because in GR complicated

equations should be solved that it is impossible to solve them analytically. In this chapter

equations of motion of a lightlike particle will be solved numerically; to do this work it is

necessary:

1. First, spacetime (metric) should be determined; in this chapter 3 types of it are

examined: schwarzschild, Kerr and Kerr-Newman spacetime.

2. A method should be selected to obtain equation/equations of motion; Lagrange’s

method, Hamiltonian method and Geodesic equation are used in this chapter.

3. Equation/Equations of motion are be calculated.

4. Equation/Equations of motion are solved numerically.

In this chapter we obtain equations of motion of a lightlike particle around a black hole of

mass M, spin angular momentum S and charge Q (Of course we use geometrized units in

this chapter, in calculations we consider M = 1, S = 0.37 and Q = 0.5) from three diﬀerent

ways and Maple 10 software (Maple codes are putted in appendices) is used to solve these

67

68

equations respect to initial conditions (t

0

,

˙

t

0

, r

0

, ˙ r

0

, θ

0

,

˙

θ

0

, φ

0

,

˙

φ

0

) and plot the trajectory of

particle, last point is about initial conditions that their values must comfort to the invariant

condition [32]

g

µν

˙ x

µ

˙ x

ν

= −c

2

(5.1)

This condition is saying that the particle is a light like particle. Seven of initial conditions

can be choose arbitrary and the last one must earn by using the constraint and seven

arbitrary initial conditions. At last results of diﬀerent methods will be compared and old

physics law, that diﬀerent methods should produced same results, will be evaluated.

5.2 Lagrangeian Method

According to the last chapter, in GR Lagrangian is deﬁned

L =

1

2

g

µν

˙ x

µ

˙ x

ν

(5.2)

it is possible to deﬁned Lagrangian diﬀerently

L = (g

µν

˙ x

µ

˙ x

ν

)

1

2

(5.3)

but we use ﬁrst deﬁnition of it because it is simpler (it is not square root), and calculate

the lagrangian for diﬀerent metrics, then Euler-Lagrange equation produces equations of

motion.

d

dτ

(

∂L

∂ ˙ x

α

) −

∂L

∂x

α

= 0 (5.4)

5.2.1 Schwarzschild Geometry

Schwarzschild metric is

ds

2

= −(1 −2M/r)dt

2

+

dr

2

(1 −2M/r)

+ r

2

(dθ

2

+ sin

2

θ dφ

2

) (5.5)

If lagrangian is calculated from eq 5.1 and 5.5, hence

L =

1

2

_

−

_

1 −

2M

r

_

˙

t

2

+

_

1 −

2M

r

_

−1

˙ r

2

+ r

2

˙

θ

2

+ r

2

sin

2

(θ)

˙

φ

2

_

(5.6)

69

−50 25

25

0

0

−25

−50

−25

Figure 5.1: Polar coordinate plotting of the trajectory of the lightlike particle around a

Schwarzschild black hole (or any massive object)with initial conditions of eq 5.8, that obtain

by Lagrangian method.

equations of motion is obtained by using of Euler-Lagrange equation (eq 5.4)(by execution

of Maple codes in appendix A they will be obtained and you will see them).

0 =−

(2r −2) ( ˙ r

˙

t)

r

2

+

2(r

2

−2r)

˙

t ˙ r

r

3

−

(r

2

−2r)

_

¨

t

_

r

2

0 =

1

r

2

(r −2)

2

_

−r

2

˙ r

2

+ (r

4

−2r

3

)¨ r + (r

2

−4r + 4)

˙

t

2

+(−r

5

+ 4r

4

−4r

3

)

˙

θ

2

+ (−r

5

+ 4r

4

−4r

3

)

_

sin

2

(θ)

˙

φ

__

0 =r

2

¨

θ −r

2

cos(θ) sin(θ)

˙

φ

2

+ 2

˙

θ ˙ r

0 =r

2

sin

2

(θ)

¨

φ + 2r

2

sin(θ) cos(θ)

˙

φ

˙

θ + 2r sin

2

(θ)

˙

φ ˙ r

(5.7)

Now by using of proper initial conditions these equations numerically solved and trajectory

of the light like particle is obtained. Initial conditions that is used here are (satisfy null

constraint, eq 5.1)

r(0) = 26, dr(0) = 0, φ(0) = 0, dφ(0) = 0.00714,

θ(0) =

π

2

, dθ(0) = 0.006193053228, t(0) = 0, dt(0) = 1.0718

(5.8)

after determination of the initial conditions equations of motion are solved and trajectory

of the lightlike particle is obtained and can be plotted in polar coordinate(Fig 5.1).

70

5.2.2 Kerr Geometry

Kerr metric is

ds

2

=−(∆/ρ

2

)[dt −a sin

2

θ dφ]

2

+ (sin

2

θ/ρ)[(r

2

+ a

2

)dφ −a dt]

2

+ (ρ

2

/∆)dr

2

+ ρ

2

dθ

2

;

∆ ≡ r

2

−2Mr + a

2

ρ ≡ r

2

+ a

2

cos

2

θ

(5.9)

Same as the last section Lagrangian is calculated (from eq 5.2) and then equations of motion

are obtained by using of Euler-Lagrange Equation (eq 5.4), but only lagrangian is written

here because equations of motion are very large (by execution of Maple codes in appendix

A they will be obtained and you will see them).

L =

1

2(r

2

+ a

2

cos

2

(θ))

_

r

2

+ a

2

−2Mr + a

2

sin

2

(θ)

_

˙

t

2

−8a sin

2

(θ)M r

˙

t

˙

φ +

_

r

2

+ a

2

cos

2

(θ)

_

2

˙ r

2

r

2

+ a

2

−2Mr

+

_

r

2

+ a

2

cos

2

(θ)

_

2

˙

θ

2

+ sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr)

_

˙

φ

(5.10)

Initial conditions that satisﬁed null constraint (eq 5.1) are

r(0) = 26, dr(0) = 0, φ(0) = 0, dφ(0) = 0.00714,

θ(0) =

π

2

, dθ(0) = 0.006349792778, t(0) = 0, dt(0) = 1.0718

(5.11)

Trajectory of the lightlike particle around a Kerr black hole (or any massive body) with

initial conditions of eq 5.11 is plotted in Figure 5.2.

5.2.3 Kerr-Newman Geometry

Kerr-NewMan spacetime is the most realistic spacetime that is studied in this text (analyzed

in chapter 3), Kerr-Newman metric is

ds

2

= −

∆

ρ

2

[dt −a sin

2

θ dφ]

2

+

sin

2

θ

ρ

2

[(r

2

+ a

2

) dφ −a dt]

2

+

ρ

2

∆

dr

2

+ ρ

2

dθ

2

∆ ≡ r

2

−2Mr + a

2

+ Q

2

, ρ

2

≡ r

2

+ a

2

cos

2

θ

a ≡ S/M ≡ angular momentum per unit mass

(5.12)

71

25 0 −25

25

0

−50

−25

−50

−75

Figure 5.2: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr black hole (or any massive object) with initial conditions of eq 5.11, that obtain by

Lagrangian method.

Lagrangian is

L =

1

2(r

2

+ a

2

cos

2

(θ))

_

_

r

2

+ a

2

−2Mr + Q

2

+ a

2

sin

2

(θ)

_

˙

t

2

+ 2a sin

2

(θ)(−2Mr + Q

2

)

˙

t

˙

φ

(r

2

+ a

2

cos

2

(θ))

2

˙ r

2

r

2

+ a

2

−2Mr + Q

2

+ (r

2

+ a

2

cos

2

(θ))

2

˙

θ

2

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr + Q

2

)

_

˙

φ

2

_

(5.13)

Same as Kerr geometry equations of motion are not written (by execution of Maple codes

in appendix A they will be obtained and you will see them). Initial conditions that satisﬁed

null constraint (eq 5.1) are

r(0) = 26, dr(0) = 0, φ(0) = 0, dφ(0) = 0.00714,

θ(0) =

π

2

, dθ(0) = 0.006398603884, t(0) = 0, dt(0) = 1.0718

(5.14)

Trajectory of the lightlike particle in Kerr-Newman geometry with initial conditions of eq

5.14 is plotted in Fig. 5.3.

72

25 0 −25

25

−50

0

−25

−75

−50

Figure 5.3: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr-Newman black hole (or any massive object) with initial conditions of eq 5.14, that

obtain by Lagrangian method.

5.3 Hamiltonian Method

First step in Hamiltonian method is calculation of Lagrangian that was done in last section,

then Hamiltonian (eq 4.14) is calculated

H =

˙ x

a

∂L

∂ ˙ x

a

−L (5.15)

and then equations of motion are calculated by using of Hamilton equations

P

α

=

∂L

∂ ˙ x

α

˙ x

α

=

∂H

∂P

α

˙

P

α

= −

∂H

∂x

α

(5.16)

5.3.1 Schwarzschild Geometry

According to the lagrangian of the Schwarzschild metric and deﬁnition of Hamiltonian

eq(4.14), Hamiltonian of this metric is

H = −

1

2

_

(r

2

−2Mr)

˙

t

2

r

2

+

r

2

˙ r

2

r

2

−2Mr

+ r

2

˙

θ

2

+ sin

2

(θ)r

2

˙

φ

2

_

(5.17)

73

Now equations of motion are found by using of eq 4.17 (by execution of Maple codes in

appendix B they will be obtained and you will see them).

0 =

¨

t +

˙

t

_

˙ r(2r −2M)

r

2

−2Mr

−

2 ˙ r

r

_

0 = ¨ r −

r

2

−2Mr

r

2

_

˙

t

2

r

2

(−Mr) + r

˙

θ

2

+ r sin

2

(θ)

˙

φ

2

+ ˙ r

2

_

r

r

2

−2Mr

−

1

2

r

2

(2r −2M)

(r

2

−2Mr)

2

__

+

2 ˙ r

2

r

−

˙ r

2

(2r −2M)

r

2

−2Mr

0 =

¨

θ +

˙

θ

2 ˙ r

r

−

˙

φ

2

sin(θ) cos(θ)

0 =

¨

φ +

˙

φ

_

2 cos(θ)

˙

θ

sin(θ)

+

2 ˙ r

r

_

(5.18)

Same as Lagrangian method Initial conditions are needed to solve these equations that

satisﬁed null constraint (eq 5.1) (initial conditions that used for Schwarzschild geometry in

Lagrangian method)

r(0) = 26, dr(0) = 0, φ(0) = 0, dφ(0) = 0.00714,

θ(0) =

π

2

, dθ(0) = 0.006193053228, t(0) = 0, dt(0) = 1.0718

(5.19)

Trajectory of the lightlike particle around a Schwarzschild black hole (or any massive body)

with initial conditions of eq 5.8 is plotted in Fig. 5.4.

5.3.2 Kerr Geometry

For Kerr metric Hamiltonian is

H =

˙

t

_

−

1

2

(r

2

+ a

2

−2Mr + a

2

sin

2

(θ))

˙

t + 2a sin

2

(θ)(−2Mr)

˙

φ

r

2

+ a

2

cos

2

(θ)

_

+

1

2

(r

2

+ a

2

cos

2

(θ))

2

˙ r

2

r

2

+ a

2

−2Mr

+

1

2

(r

2

+ a

2

cos

2

(θ))

˙

θ

2

+

˙

φ

_

+

1

2

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr)

_

r

2

+ a

2

cos

2

(θ)

_

(5.20)

Equations of motion are very large and are not written here (by execution of Maple codes in

appendix B they will be obtained and you will see them). Initial conditions that were used

in Lagrangian method for Kerr metric (eq 5.11), are used here (they satisfy null constraint,

eq 5.1), result is plotted in Figure 5.5.

74

−50 25

25

0

0

−25

−50

−25

Figure 5.4: Polar coordinate plotting of the trajectory of the lightlike particle around a

Schwarzschild black hole (or any massive object) with initial conditions of eq 5.18, that

obtain by Hamiltonian method.

25 0 −25

25

0

−50

−25

−50

−75

Figure 5.5: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr black hole (or any massive object) with initial conditions of eq 5.11, that obtain by

Hamiltonian method.

75

25 0 −25

25

−50

0

−25

−75

−50

Figure 5.6: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr-Newman black hole (or any massive object) with initial conditions of eq 5.11, that

obtain by Hamiltonian method.

5.3.3 Kerr-Newman Geometry

For Kerr-Newman metric Hamiltonian is

H =

˙

t

_

−

1

2

(r

2

+ a

2

−2Mr + Q

2

+ a

2

sin

2

(θ))

˙

t + 2a sin

2

(θ)(−2Mr + Q

2

)

˙

φ

r

2

+ a

2

cos

2

(θ)

_

+

1

2

(r

2

+ a

2

cos

2

(θ))

2

˙ r

2

r

2

+ a

2

−2Mr + Q

2

+

1

2

(r

2

+ a

2

cos

2

(θ))

˙

θ

2

+

˙

φ

_

+

1

2

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr + Q

2

)

_

r

2

+ a

2

cos

2

(θ)

_

(5.21)

Equations of motion are very large and are not written here (by execution of Maple codes

in appendix B they will be obtained and you will see them). Initial conditions that were

used in Lagrangian method for Kerr-Newman metric (eq 5.11), are used here (they satisfy

null constraint, eq 5.1), result is plotted in Figure 5.6.

5.4 Geodesic Equation

Geodesic equation is last method that is used to obtain equations of motion of a lightlike

particle in three type spacetime (Schwarzschild, Kerr and Kerr-Newman spacetime). In this

76

−50 25

25

0

0

−25

−50

−25

Figure 5.7: Polar coordinate plotting of the trajectory of the lightlike particle around a

Schwarzschild black hole (or any massive object) with initial conditions of eq 5.11, that

obtain by Geodesic equation.

method geodesic equation is uesed

d

2

x

α

dτ

2

+ Γ

α

βγ

dx

β

dτ

x

γ

dτ

= 0 (5.22)

Therefore we have 4 second-order diﬀerential equations. To ﬁnd equations of motion using

Geodesic equation, we ﬁnd Γ

α

βγ

,

Γ

α

βγ

=

1

2

g

αδ

(∂

γ

g

βδ

+ ∂

β

g

γδ

−∂δg

βγ

) when g

ab

is metric (5.23)

determine initial conditions and then solve 4 coupled diﬀerential equations. Initial con-

ditions of before methods will be used for this method too (in this way it is possible to

compare results of diﬀerent methods).

77

5.4.1 Schwarzschild Geometry

Christoffel symbols of Schwarzschild metric was written in Box 3.1, then according to the

geodesic equation (eq 5.20) equations of motion are (by execution of Maple codes in ap-

pendix C they will be obtained and you will see them)

0 =

¨

t +

˙

t

2m˙ r

r

2

−2Mr

0 = ¨ r − ˙ r

2

M

r

2

−2Mr

+ (r −2M)

_

M

˙

t

2

r

3

−

˙

θ

2

−sin

2

(θ)

˙

φ

2

_

0 =

¨

θ +

˙

θ

2 ˙ r

r

−sin(θ) cos(θ)

˙

φ

2

0 =

¨

φ +

˙

φ

_

2(r −2rM) ˙ r

r

2

−2Mr

+

2 cos(θ)

˙

θ

sin(θ)

_

(5.24)

Initial conditions are same as Schwarzschild metric in Hamiltonian method. result is plotted

in Figure 5.7.

5.4.2 Kerr and Kerr-Newman Geometry

Christoffel symbols of Kerr and Kerr-Newman metrics are written in Box 5.1, then according

to the geodesic equation (eq 5.20) equations of motion are calculated (they are very long and

are not written here but by execution of Maple codes in appendix C they will be obtained

and you will see them) and by using of initial conditions of Kerr and Kerr-Newman metrics

in Hamiltonian method, equations can be solved numerically, results are plotted in Figure

5.8 (Kerr metric) and Figure 5.9 (Kerr-Newman metric).

78

Box 5.1

Connection Coeﬃcients of Kerr and Kerr-Newman metrics

Christoffel symbols of Kerr-Newman metric in Boyer-Lindquist cor-

rdinates

Γ

r

tt

=

(r

2

−2Mr + a

2

+ Q

2

)(Mr

2

−rQ

2

−Ma

2

+ a

2

sin

2

(θ))

(r

2

+ a

2

cos

2

(θ))

3

Γ

θ

tt

=−

a

2

sin(θ) cos(θ)(2Mr −Q

2

)

(r

2

+ a

2

cos

2

(θ))

3

Γ

t

tr

=

(r

2

+ a

2

)(Mr

2

−rQ

2

−Ma

2

cos

2

(θ))

(r

2

−2Mr + a

2

+ q

2

)(r

2

+ a

2

cos

2

(θ))

2

Γ

φ

tr

=

a(Mr

2

−rQ

2

−Ma

2

cos

2

(θ))

(r

2

−2Mr + a

2

+ q

2

)(r

2

+ a

2

cos

2

(θ))

2

Γ

t

tθ

=−

a

2

sin(θ) cos(θ)(2Mr −Q

2

)

(r

2

+ a

2

cos

2

(θ))

2

Γ

φ

tθ

=−

a cos(θ)(2Mr −Q

2

)

sin(θ)(r

2

+ a

2

cos

2

(θ))

2

Γ

r

tφ

=−

a sin

2

(θ)(r

2

−2Mr + a

2

+ q

2

)(Mr

2

−rQ

2

−Ma

2

cos

2

(θ))

(r

2

+ a

2

cos

2

(θ))

3

Γ

θ

tφ

=

a sin(θ) cos(θ)(r

2

+ a

2

)(2Mr −Q

2

)

(r

2

+ a

2

cos

2

(θ))

3

Γ

r

rr

=−

Mr

2

−rQ

2

−ra

2

sin

2

(θ) −Ma

2

cos

2

(θ)

(r

2

−2Mr + a

2

+ Q

2

)(r

2

+ a

2

cos

2

(θ))

Γ

θ

rr

=

a

2

sin(θ) cos(θ)

(r

2

−2Mr + a

2

+ Q

2

)(r

2

+ a

2

cos

2

(θ))

Γ

r

rθ

=−

a

2

sin(θ) cos(θ)

r

2

+ a

2

cos

2

(θ)

Γ

θ

rθ

=

r

r

2

+ a

2

cos

2

(θ)

Γ

t

rφ

=−

a sin

2

(θ)(Mr

2

a

2

(1 + cos

2

(θ)) −rQ

2

a

2

(1 + cos

2

(θ))

(r

2

−2Mr + a

2

+ Q

2

)(r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ))

−

rQ

2

a

2

(3Mr

4

−2Q

2

r

3

−Ma

4

cos

2

(θ))

(r

2

−2Mr + a

2

+ Q

2

)(r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ))

79

Box 5.1 (continued)

Γ

φ

rφ

=

r

5

−2Mr

4

+ Q

2

r

3

−Mr

2

a

2

(1 + cos

2

(θ)) + rQ

2

a

2

(r

2

−2Mr + a

2

+ Q

2

)(r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ))

+

(Ma

4

+ 2r

3

a

2

) cos

2

(θ) + (r −M)a

4

cos

4

(θ)

(r

2

−2Mr + a

2

+ Q

2

)(r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ))

Γ

r

θθ

=−

(r

2

−2Mr + a

2

+ Q

2

)r

r

2

+ a

2

cos

2

(θ)

Γ

θ

θθ

=−

a

2

sin(θ) cos(θ)

r

2

+ a

2

cos

2

(θ)

Γ

t

θφ

=

a

3

sin

3

(θ) cos(θ)(2Mr −Q

2

)

r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ)

Γ

φ

θφ

=

r

4

+ (2Mra

2

−Q

2

a

2

) sin

2

(θ) + 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ)

(r

4

+ 2r

2

a

2

cos

2

(θ) + a

4

cos

4

(θ)) tan(θ)

Γ

r

φφ

=−

1

r

6

+ 3r

4

a

2

cos

2

(θ) + 3r

2

a

4

cos

4

(θ) + a

6

cos

6

(θ)

(sin

2

(θ)(r

2

−2Mr + a

2

+ Q

2

)(r

5

+ 2r

3

a

2

cos

2

(θ) + (−Mr

2

a

2

+ rQ

2

a

2

+ Ma

4

cos

2

(θ)) sin

2

(θ) + ra

4

cos

4

(θ))

Γ

θ

φφ

=−

1

r

6

+ 3r

4

a

2

cos

2

(θ) + 3r

2

a

4

cos

4

(θ) + a

6

cos

6

(θ)

_

sin(θ) cos(θ)(r

6

+ r

4

a

2

(1 + 2 cos

2

(θ) + cos

4

(θ))

+ (4Mr

3

a

2

−2r

2

a

2

Q

2

) sin

2

(θ) + 2Mra

4

−a

4

Q

2

+ 2r

2

a

4

cos

2

(θ) + (a

6

+ a

4

Q

2

−2Mra

4

) cos

4

(θ)

_

If taking Q = 0 in Kerr-Newman Christoffel symbols, Christoffel

symbols of Kerr metric will be obtained.

5.5 Compare Diﬀerent Methods

If results of diﬀerent methods for a special space time are compared, it is clear that they are

indistinguishable and then diﬀerent methods make unique results. This is the old physics

law that said: if a problem is solved by diﬀerent methods, all methods should answer unique.

80

25 0 −25

25

0

−50

−25

−50

−75

Figure 5.8: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr black hole (or any massive object) with initial conditions of eq 5.11, that obtain by

Geodesic equation.

25 0 −25

25

−50

0

−25

−75

−50

Figure 5.9: Polar coordinate plotting of the trajectory of the lightlike particle around a

Kerr-Newman black hole (or any massive object) with initial conditions of eq 5.14, that

obtain by Geodesic equation.

81

5.6 Compare Diﬀerent Spacetimes

What are the diﬀerences of result trajectories of diﬀerent spacetime? To know this by

studying the apoastrons of diﬀerent trajectories (that are the answer of the diﬀerent space-

times). In Fig 5.10 you can see apostrons of three trajectories of three black holes that have

seven unique initial conditions (last of them is detrmined by null constraint Eq 5.1), results

are (Maple codes for maximum ﬁnding are written in appendix A with Lagrangian Maple

codes):

• Schwarzschild metric

the apoastron (circular dots •) advances by ∼ 29

◦

per period;

• Kerr metric

the apoastron (square dots ) advances by ∼ 23

◦

per period;

• Kerr-Newman metric

the apoastron (circles ◦) advances by ∼ 21

◦

per period;

82

25 0 −25 −50

25

0

−75

−25

−50

Figure 5.10: For a massive object the diﬀerence between the Schwarzschild metric (green

line), the Kerr metric (red line) and the Kerr-Newman metric is signiﬁcant. The initial

conditions, spin angular momentum and charge of the black hole are speciﬁed in Sec. 5. 2.

For the Schwarzschild metric, the apoastron (circular dots •) advances by ∼ 29

◦

per period;

for the Kerr metric of corrotating orbit, the apoastron (square dots ) advances by ∼ 23

◦

per period, and for the Kerr-Newman metric of corrotating orbit, the apoastron (circles ◦)

advances by ∼ 21

◦

per period. The unit for each tick is M, and the circle signiﬁes a neutron

star, assuming that M = 1.4M

and R = 12 km(= 5.8M).

Appendix A

Maple Codes for Lagrangian

Method

The kerr-Newman none zero metric coeﬃcients, which describes a metric for a spherical

object of mass M, spin angular momentum J and charge Q are (in geometrical units i. e.

G = c = 1):

g

00

= −

r

2

+ a

2

−2Mr + +Q

2

+ a

2

sin

2

(θ)

r

2

+ a

2

cos

2

(θ)

g

11

=

r

2

+ a

2

cos

2

(θ)

r

2

+ a

2

−2Mr + +Q

2

g

22

= r

2

+ a

2

cos

2

(θ)

g

33

=

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr + +Q

2

)

_

r

2

+ a

2

cos

2

(θ)

g

03

= g

30

=

2a sin

2

(θ)(−2Mr + Q

2

)

r

2

+ a

2

cos

2

(θ)

wherea = J/M.

If Q = 0 , the Kerr-Newman solution becomes Kerr solution and if Q = 0 and J = 0

the Kerr-Newman solution becomes Schwarzschild solution.

We can use Lagrangian to ﬁnd equations of motion.In general relativity Lagrangian is:

L =

1

2

g

αβ

dx

α

dτ

dx

β

dτ

83

84

The equations of motion are obtainable by employing the Euler–lagrange equation:

d

dτ

_

∂L

∂v

α

_

= 0, where v

α

=

dx

α

dτ

Therefore we have 4 second-order diﬀerential equations.

We apply a constraint which is used to determine the initial conditions for a nulllike

particle such as a light particle (i. e. initial conditions must satisfy this equation)

g

αβ

dx

α

dτ

dx

β

dτ

= −c

2

To ﬁnd equations of motion, we must solve 4 diﬀerential equations.

Below we use Lagrangian to ﬁnd equation of motion for a type of black holes (Schwarzschild,

Q = J = 0). We can ﬁnd equation of motion for other type of black holes by changing J

and Q.

> restart: with(plots): with(plottools):with(tensor):

Lagrangian method

Deﬁning coordinates,

> coord:=[t,r,theta, phi];

Deﬁning Kerr-Newman metric, Use Boyer-Lindquist (1967) coordinates,

> Eq11:=Delta=(r)^2+a^2-2*M*r+Q^2;

Eq12:=rho^2=(r^2)+(a^2)*(cos(theta))^2;

The metric is:

> g_compts:=array(symmetric, sparse, 1..4, 1..4, [

(1,1)=-((rhs(Eq11))+(a^2)*(sin(theta))^2)

/(rhs(Eq12)),

(2,2)=rhs(Eq12)/rhs(Eq11),

(1,4)=((2*a*(sin(theta))^2)*(rhs(Eq11)-(r^2

+a^2)))/rhs(Eq12),

(3,3)=rhs(Eq12),

(4,4)=(((sin(theta))^2)*(((r^2+a^2)^2)-(a^2)*(

(sin(theta))^2)*rhs(Eq11)))/rhs(Eq12)]):

85

Applying Q = 0 and J = 0 to equations for Schwarzschild solution; G, M and c are made

uniﬁed, i. e. geometrical units are used.

> G:=1;M:=1;c:=1;J:=0;Q:=0;a:=J/M;

Finding Lagrangian by using metric,

> x:=array(1..4):

> x[1]:=var1:x[2]:=var3:x[3]:=var5:x[4]:=var7:

> x_d := array(1..4):

> x_d[1]:= var2: x_d[2]:=var4: x_d[3]:=var6:

x_d[4]:=var8:

> L00:= 0:

> for i from 1 by 1 while i < 5 do

for j from 1 by 1 while j < 5 do

L00 := L00 + (1/2)*g_compts[i,j]*x_d[i]*x_d[j];

end do;

> end do;

> L0 := subs({t=var1, r=var3, theta=var5, phi(tau)=var7}, L00):

> U:=array(1..4):

> U[1]:=0:U[2]:=0:U[3]:=0:u[4]:=0:

> V:=array(1..4):

> V[1]:=0:V[2]:=0:V[3]:=0:V[4]:=0:

> equ:=array(1..4):

> equ[1]:=0:equ[2]:=0:equ[3]:=0:equ[4]:=0:

Finding

∂L

∂v

α

> for k from 1 by 1 while k < 5 do

> equ[k] := diff(L0,x_d[k])=U[k]:

end do:

> eqv:=array(1..4):

> eqv[1]:=0:eqv[2]:=0:eqv[3]:=0:eqv[4]:=0:

86

Finding

∂L

∂x

α

> for i from 1 by 1 while i < 5 do

> eqv[i] := diff(L0,x[i])=V[i]:

end do:

> for f from 1 by 1 while f < 5 do

equ[f] := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)}, equ[f]):

eqv[f] := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)}, eqv[f]):

> end do:

Finding Euler–Lagrange equation for each component,

> eq:=array(1..4):

> eq[1]:=0:eq[2]:=0:eq[3]:=0:eq[4]:=0:

> for h from 1 by 1 while h < 5 do

equ[h]:=diff(lhs(equ[h]),tau):

eq[h]:= equ[h]-lhs(eqv[h])=0:

> end do:

Initial values

Provide the initial values that satisfy nulllikke particle constraint, and use Maple to solve

these diﬀerential equations numerically and to form plots based on numerical solutions,

> xd := array(1..4):

> xd[1]:= td3: xd[2]:=rd3: xd[3]:=thd3:

xd[4]:=phd3:

> con:= 0:

87

> for w from 1 by 1 while w < 5 do

for q from 1 by 1 while q < 5 do

> con := con + g_compts[w,q]*xd[w]*xd[q];

end do;

> end do;

> const := subs({t=var13, r=var33, theta=var53,

phi(tau)=var73}, con):

> constraint:=const=-c^2:

> ini1 := r(0) = 26:

> ini2 := D(r)(0) = 0:

> ini3 := phi(0) = 0:

> ini4 := D(phi)(0) = 0.00714:

> ini5:=t(0)=0:

> ini6:=D(t)(0)=1.0718:

> ini7:=theta(0)=Pi/2:

> var13:=rhs(ini5):var33:=rhs(ini1):var53:=rhs(ini7):

var73:=rhs(ini3):rd3:=rhs(ini2):td3:=rhs(ini6):

phd3:=rhs(ini4):

> th_d3:=isolate(constraint, thd3):

> ini8:=D(theta)(0)=rhs(th_d3):

> ini := ini1,ini2,ini3,ini4,ini5,ini6,ini7,ini8:

equations of motion are solved by above initial conditions,

> Eq000:=dsolve({eq[1],eq[2],eq[3],eq[4], ini},

{t(tau), r(tau), theta(tau),phi(tau)}, numeric,

output=listprocedure):

Finding maximums

> rd11:=rhs(Eq000[5]):

> Digits:=10:

Finding where maximums occur? by solving

dr

dτ

= 0

88

> tau1:=fsolve(rd11(tau)=0,tau=1..7800):

> tau2:=fsolve(rd11(tau)=0,tau=1..1800):

How much is maximum of r?

> max_r:=eval(rhs(Eq000(tau)[4]),tau=tau1):

Finding φ where r is maximum,

> phi1:=eval(rhs(Eq000(tau)[2]),tau=tau1):

> phi2:=eval(rhs(Eq000(tau)[2]),tau=tau2):

How many degrees is the distance of the ﬁrst max and the second max?

> del:=phi1-phi2:

> degrees:=(del*180)/3.14-360:

Polar coordinate plotting (The radius is the ﬁrst parameter and φ is the second parameter),

> Schwplot:=polarplot([rhs(Eq000(tau)[4]),

rhs(Eq000(tau)[2]), tau=0..5300],

axesfont=[TIMES, ROMAN, 12],

legend="Schwarzschild"):

> pns:=disk([0,0],5.8,color=gray):

> pschw2 := disk([max_r*cos(phi1), max_r*sin(phi1)], 1, color=black):

> pschw3 := disk([max_r*cos(phi2), max_r*sin(phi2)], 1, color=black):

> display([pns,Schwplot,pschw2,pschw3],scaling=constrained);

Plotting a space curve that the radius is the ﬁrst parameter, the θ is the second parameter

and the φ is the third parameter.

> traj:=spacecurve([rhs(Eq000(tau)[4]),

rhs(Eq000(tau)[8]), rhs(Eq000(tau)[2]),

tau=0..4000], coords=spherical,

numpoints=1000, color=black):

> pnm:=sphere([0,0,0],5.8):

> display([traj,pnm],scaling=constrained):

89

Showing motion of a light like particle around a black hole.

> B :=sphereplot(5.8,theta=0..2*Pi,phi=0..Pi):

> animate(spacecurve,[[rhs(Eq000(tau)[4]),

rhs(Eq000(tau)[8]), rhs(Eq000(tau)[2])],

tau=0..d, coords=spherical,numpoints=1000,

color=black],d=0..2000,background=B,

scaling=constrained):

Appendix B

Maple Codes for Hamiltonian

method

The kerr-Newman none zero metric coeﬃcients, which describes a metric for a spherical

object of mass M, spin angular momentum J and charge Q are (in geometrical units i. e.

G = c = 1):

g

00

= −

r

2

+ a

2

−2Mr + +Q

2

+ a

2

sin

2

(θ)

r

2

+ a

2

cos

2

(θ)

g

11

=

r

2

+ a

2

cos

2

(θ)

r

2

+ a

2

−2Mr + +Q

2

g

22

= r

2

+ a

2

cos

2

(θ)

g

33

=

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr + +Q

2

)

_

r

2

+ a

2

cos

2

(θ)

g

03

= g

30

=

2a sin

2

(θ)(−2Mr + Q

2

)

r

2

+ a

2

cos

2

(θ)

wherea = J/M.

If Q = 0 , the Kerr-Newman solution becomes Kerr solution and if Q = 0 and J = 0

the Kerr-Newman solution becomes Schwarzschild solution.

We can use Hamiltonian to ﬁnd equations of motion, and we can obtain Hamiltonian

90

91

by using Lagrangian:

L =

1

2

g

αβ

dx

α

dτ

dx

β

dτ

H =

3

α=0

v

α

∂L

∂v

α

−L, where v

α

=

dx

α

dτ

We have

P

α

=

∂L

∂v

α

,

dx

α

dτ

=

∂H

∂P

α

,

dP

α

dτ

= −

∂H

∂x

α

We can use these terms to ﬁnd 4 second-order diﬀerential equations that by solving them

equations of motion can be found.

We apply a constraint which is used to determine the initial conditions for a nulllike

particle such as a light particle (i.e. initial conditions must satisfy this equation)

g

αβ

dx

α

dτ

dx

β

dτ

= −c

2

Below we use Hamiltonian to ﬁnd equation of motion for a type of black holes (Kerr). We

can ﬁnd equation of motion for other type of black holes by changing J and Q.

> restart: with(plots): with(plottools):with(tensor):

Hamiltonian method

Deﬁning coordinates,

> coord:=[t,r,theta, phi];

Use Boyer-Lindquist (1967) coordinates,

> Eq11:=Delta=(r)^2+a^2-2*M*r+Q^2;

Eq12:=rho^2=(r^2)+(a^2)*(cos(theta))^2;

The metric is:

> g_compts:=array(symmetric, sparse, 1..4, 1..4, [

(1,1)=-((rhs(Eq11))+(a^2)*(sin(theta))^2)

/(rhs(Eq12)), (2,2)=rhs(Eq12)/rhs(Eq11),

(1,4)=((2*a*(sin(theta))^2)*(rhs(Eq11)

-(r^2+a^2)))/rhs(Eq12), (3,3)=rhs(Eq12),

(4,4)=(((sin(theta))^2)*(((r^2+a^2)^2)

-(a^2)*((sin(theta))^2)*rhs(Eq11)))/rhs(Eq12)]):

92

Applying Q = 0 and J = 0.37 to equations for Kerr solution; G, M and c are made uniﬁed,

i.e.geometrical units are used.

> M:=1;G:=1;J:=0.37;Q:=0;c:=1;a:=J/M;

Finding Lagrangian,

> x:=array(1..4):

> x[1]:=var1:x[2]:=var3:x[3]:=var5:x[4]:=var7:

> x_d := array(1..4):

> x_d[1]:= var2: x_d[2]:=var4: x_d[3]:=var6: x_d[4]:=var8:

> L00:= 0:

> for i from 1 by 1 while i < 5 do

for j from 1 by 1 while j < 5 do

L00 := L00 + (1/2)*g_compts[i,j]*x_d[i]*x_d[j];

end do;

> end do;

> L0 := subs({t=var1, r=var3, theta=var5, phi(tau)=var7}, L00):

Finding Hamiltonian,

> H0:=0:

> for k from 1 by 1 while k < 5 do

H0 := H0 + x_d[k]*diff(L0,x_d[k]):

end do:

> H0:=H0-L0:

> eq00:= p0 = diff(L0,var2):

> eq01:= p1 = diff(L0,var4):

> eq02:= p2 = diff(L0,var6):

> eq03:= p3 = diff(L0,var8):

Hamiltonian is a function of

dx

α

dτ

, if we want to ﬁnd

dH

dP

α

we must make it a function of P

components (i. e. we must replace variables(e.g. var8) by functions of P components). But

in some equations we have two variables and isolating one of them is not adequate. So we

must replace a variable in an equation by its equivalent in other equation. Therefore well

93

have one variable in each equation that we can isolate it. Below we replace var8 in the eq00

by its equivalent in the eq03.

> corr01:=isolate(eq03,var8):

> corr02:=subs({corr01},eq00):

Replacing var2 in the eq03 by its equivalent in the eq00.

> corr11:=isolate(eq00,var2):

> corr12:=subs({corr11},eq03):

> t_d:= isolate(corr02, var2):

> r_d:= isolate(eq01, var4):

> theta_d:= isolate(eq02, var6):

> phi_d:= isolate(corr12,var8):

> H:= subs({t_d,r_d,phi_d,theta_d},H0):

Finding

dx

α

dτ

=

∂H

∂P

α

> eq10:= q00d = diff(H,p0):

> eq11:= q01d = diff(H,p1):

> eq12:= q02d = diff(H,p2):

> eq13:= q03d = diff(H,p3):

Finding

dP

α

dτ

=

∂H

∂x

α

> eq20:= p0d = -diff(H,var1):

> eq21:= p1d = -diff(H,var3):

> eq22:= p2d = -diff(H,var5):

> eq23:= p3d = -diff(H,var7):

Finding

d

2

x

α

dτ2

94

> q10d:= subs({var1=t(tau), var3=r(tau),var5=theta(tau),

var7=phi(tau), p0=p0(tau),p1=p1(tau),

p2=p2(tau), p3=p3(tau)}, eq10):

> q11d:= subs({var1=t(tau), var3=r(tau), var5=theta(tau),

var7=phi(tau), p0=p0(tau), p1=p1(tau),

p2=p2(tau), p3=p3(tau)}, eq11):

> q12d:= subs({var1=t(tau), var3=r(tau), var5=theta(tau),

var7=phi(tau), p0=p0(tau), p1=p1(tau),

p2=p2(tau), p3=p3(tau)}, eq12):

> q13d:= subs({var1=t(tau), var3=r(tau), var5=theta(tau),

var7=phi(tau), p0=p0(tau), p1=p1(tau),

p2=p2(tau), p3=p3(tau)}, eq13):

> eq30:= q00dd = diff(rhs(q10d),tau):

> eq31:= q01dd = diff(rhs(q11d),tau):

> eq32:= q02dd = diff(rhs(q12d),tau):

> eq33:= q03dd = diff(rhs(q13d),tau):

Finding

d

2

x

α

dτ2

as a function of r(τ), φ(τ), θ(τ) and t(τ) by replacing

dP

α

dτ

=

∂H

∂x

α

and P

α

=

∂L

∂v

α

into

d

2

x

α

dτ

2

equation.

> q10dd:= subs({t(tau)=var1, diff(t(tau), tau)=var2,

r(tau)=var3, diff(r(tau),tau)=var4,

theta(tau)=var5, diff(theta(tau),tau)=var6,

phi(tau)=var7, diff(phi(tau),tau)=var8,

p0(tau)=p0,p1(tau)=p1,p2(tau)=p2, p3(tau)=p3,

diff(p0(tau),tau)=p0d,diff(p1(tau),tau)=p1d,

diff(p2(tau),tau)=p2d, diff(p3(tau),tau)=p3d},

eq30):

> q11dd:= subs({t(tau)=var1, diff(t(tau),tau)=var2, r(tau)=var3,

diff(r(tau),tau)=var4, theta(tau)=var5,

diff(theta(tau),tau)=var6, phi(tau)=var7,

diff(phi(tau),tau)=var8, p0(tau)=p0,

p1(tau)=p1,p2(tau)=p2, p3(tau)=p3,

95

diff(p0(tau),tau)=p0d,diff(p1(tau),tau)=p1d,

diff(p2(tau),tau)=p2d, diff(p3(tau),tau)=p3d},

eq31):

> q12dd:= subs({t(tau)=var1, diff(t(tau),tau)=var2,

r(tau)=var3, diff(r(tau),tau)=var4,

theta(tau)=var5, diff(theta(tau),tau)=var6,

phi(tau)=var7, diff(phi(tau),tau)=var8,

p0(tau)=p0,p1(tau)=p1,p2(tau)=p2, p3(tau)=p3,

diff(p0(tau),tau)=p0d,diff(p1(tau),tau)=p1d,

diff(p2(tau),tau)=p2d, diff(p3(tau),tau)=p3d},

eq32):

> q13dd:=subs({t(tau)=var1, diff(t(tau),tau)=var2, r(tau)=var3,

diff(r(tau),tau)=var4, theta(tau)=var5,

diff(theta(tau),tau)=var6, phi(tau)=var7,

diff(phi(tau),tau)=var8, p0(tau)=p0,

p1(tau)=p1,p2(tau)=p2, p3(tau)=p3,

diff(p0(tau),tau)=p0d,diff(p1(tau),tau)=p1d,

diff(p2(tau),tau)=p2d, diff(p3(tau),tau)=p3d},

eq33):

Replacing

dP

α

dτ

by −

∂H

∂x

α

(that is a function of variables) in

d

2

x

α

dτ

equation.

> q20dd:=subs({eq20,eq21,eq22,eq23},q10dd):

> q21dd:=subs({eq20,eq21,eq22,eq23},q11dd):

> q22dd:=subs({eq20,eq21,eq22,eq23},q12dd):

> q23dd:=subs({eq20,eq21,eq22,eq23},q13dd):

Replacing P

α

by

∂L

∂v

α

(that is a function of variables) in

d

2

x

α

dτ

equation.

> q30dd:=subs({eq00,eq01,eq02,eq03},q20dd):

> q31dd:=subs({eq00,eq01,eq02,eq03},q21dd):

> q32dd:=subs({eq00,eq01,eq02,eq03},q22dd):

> q33dd:=subs({eq00,eq01,eq02,eq03},q23dd):

Replacing variables by r(τ), φ(τ), θ(τ), t(τ) and their derivatives with respect to τ.

96

> q40dd := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)},

q30dd):

> q41dd := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)},

q31dd):

> q42dd := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)},

q32dd):

> q43dd := subs({var1=t(tau), var2=diff(t(tau),tau),

var3=r(tau), var4=diff(r(tau),tau),

var5=theta(tau), var6=diff(theta(tau),tau),

var7=phi(tau), var8=diff(phi(tau),tau)},

q33dd):

Finding 4 diﬀerential equations,

> eqt := diff(t(tau),tau,tau) - rhs(q40dd)=0:

> eqr := diff(r(tau),tau,tau) - rhs(q41dd)=0:

> eqthete:= diff(theta(tau),tau,tau) - rhs(q42dd)=0:

> eqphi := diff(phi(tau),tau,tau) - rhs(q43dd)=0:

Initial conditions

Provide the initial values that satisfy nulllikke particle constraint, and use Maple to solve

these diﬀerential equations numerically and to form plots based on numerical solutions,

> x_d := array(1..4):

> x_d[1]:= td3: x_d[2]:=rd3: x_d[3]:=thd3:

97

x_d[4]:=phd3:

>con:= 0:

> for i from 1 by 1 while i < 5 do

for j from 1 by 1 while j < 5 do

con := con + g_compts[i,j]*x_d[i]*x_d[j];

end do;

> end do;

> cond := subs({t=var13, r=var33, theta=var53,

phi(tau)=var73}, con):

> cond1:=cond=-c^2:

> th_d:=isolate(cond1, thd3):

> ini1 := r(0) = 26:

> ini2 := D(r)(0) = 0:

> ini3 := phi(0) = 0:

> ini4 := D(phi)(0) = 0.00714:

> ini5:=t(0)=0:

> ini6:=D(t)(0)=1.0718:

> ini7:=theta(0)=Pi/2:

> var13:=rhs(ini5):var33:=rhs(ini1):var53:=rhs(ini7):

var73:=rhs(ini3):rd3:=rhs(ini2):td3:=rhs(ini6):

phd3:=rhs(ini4):

> th_d3:=isolate(cond1, thd3):

> ini8:=D(theta)(0)=rhs(th_d3):

> ini := ini1,ini2,ini3,ini4,ini5,ini6,ini7,ini8:

equations of motion are solved by above initial conditions,

> Eq000:=dsolve({eqt,eqr,eqthete,eqphi, ini},

{t(tau), r(tau), theta(tau),phi(tau)},

numeric, output=listprocedure):

Polar coordinate plotting (The radius is the ﬁrst parameter and φ is the second parameter)

> kerrplot:=polarplot([rhs(Eq000(tau)[4]),

98

rhs(Eq000(tau)[2]), tau=0..5300],

axesfont=[TIMES, ROMAN, 12],

legend="Kerr-Newman"):

> pns:=disk([0,0],5.8,color=gray):

> display([pns,kerrplot],scaling=constrained);

Plotting a space curve that the radius is the ﬁrst parameter, the θ is the second parameter

and the φ is the third parameter.

> traj:=spacecurve([rhs(Eq000(tau)[4]),

rhs(Eq000(tau)[8]), rhs(Eq000(tau)[2]),

tau=0..2000], coords=spherical,

numpoints=1000, color=black):

> pnm:=sphere([0,0,0],5.8):

> display([traj,pnm],scaling=constrained);

Showing motion of a light like particle around a black hole.

> B :=sphereplot(5.8,theta=0..2*Pi,phi=0..Pi):

> animate(spacecurve,[[rhs(Eq000(tau)[4]),

rhs(Eq000(tau)[8]), rhs(Eq000(tau)[2])],

tau=0..d, coords=spherical,numpoints=1000,

color=black],d=0..2000,background=B,

scaling=constrained);

Appendix C

Maple Codes For Geodesic

Equation

The kerr-Newman none zero metric coeﬃcients, which describes a metric for a spherical

object of mass M, spin angular momentum J and charge Q are (in geometrical units i. e.

G = c = 1):

g

00

= −

r

2

+ a

2

−2Mr + +Q

2

+ a

2

sin

2

(θ)

r

2

+ a

2

cos

2

(θ)

g

11

=

r

2

+ a

2

cos

2

(θ)

r

2

+ a

2

−2Mr + +Q

2

g

22

= r

2

+ a

2

cos

2

(θ)

g

33

=

sin

2

(θ)

_

(r

2

+ a

2

)

2

−a

2

sin

2

(θ)(r

2

+ a

2

−2Mr + +Q

2

)

_

r

2

+ a

2

cos

2

(θ)

g

03

= g

30

=

2a sin

2

(θ)(−2Mr + Q

2

)

r

2

+ a

2

cos

2

(θ)

wherea = J/M.

If Q = 0 , the Kerr-Newman solution becomes Kerr solution and if Q = 0 and J =

0 the Kerr-Newman solution becomes Schwarzschild solution. For one of the spacetime

components, Geodesic equation is

d

2

x

α

dτ

2

+ Γ

α

βγ

dx

β

dτ

dx

γ

dτ

= 0

99

100

Therefore we have 4 second-order diﬀerential equations.

We apply a constraint which is used to determine the initial conditions for a nulllike

particle such as a light particle (i. e. initial conditions must satisfy this equation)

g

αβ

dx

α

dτ

dx

β

dτ

= −c

2

To ﬁnd equations of motion using the Geodesic equation, we must ﬁnd Γ

α

βγ

and then solve

4 diﬀerential equations.

Below we use geodesic equation to ﬁnd equation of motion for a type of black holes

(Kerr-Newman). We can ﬁnd equation of motion for other type of black holes by changing

J and Q.

> restart; with(plots); with(plottools); with(tensor);

Kerr-Newman solution

Deﬁning the coordinates:

> coord := [t, r, theta, phi]

Deﬁning Kerr-Newman metric, Use Boyer-Lindquist (1967) coordinates,

>Eq11:=Delta=(r)^2+a^2-2*M*r+Q^2;

Eq12:=rho^2=(r^2)+(a^2)*(cos(theta))^2;

> g_compts:=array(symmetric, sparse, 1..4, 1..4, [

(1,1)=-((rhs(eq11))+(a^2)*(sin(theta))^2)

/(rhs(eq12)), (2,2)=rhs(eq12)/rhs(eq11),

(1,4)=((2*a*(sin(theta))^2)*(rhs(eq11)

-(r^2+a^2)))/rhs(eq12), (3,3)=rhs(eq12),

(4,4)=(((sin(theta))^2)*(((r^2+a^2)^2)

-(a^2)*((sin(theta))^2)*rhs(eq11)))/rhs(eq12)]):

> g:=create([-1,-1],eval(g_compts)):

Finding Christoﬀel symbols

Γ

α

βγ

101

> ginv:=invert(g,’detg’):

> d1g:= d1metric( g, coord ): d2g:= d2metric( d1g, coord ):

> Cf1:= Christoffel1( d1g ):

> Cf2:= Christoffel2( ginv, Cf1 ):

Finding Geodesic equation

> eqns0 := geodesic_eqns(coord, tau, Cf2):

Applying Q = 0.5 and J = 0.37 to equations for Kerr-Newman solution; G, M and c are

made uniﬁed, i. e. geometrical units are used.

> M := 1;G:=1; c:=1; J :=0.37;Q:=0.5;a:=J/M;

Deﬁning eqns3 as a function of r(τ), θ(τ), φ(τ), t(τ) and their ﬁrst and second derivatives

with respect to τ.

> eqns11:=subs({diff(theta(tau),tau,tau)

=var10,diff(r(tau),tau,tau)=var11,

diff(phi(tau),tau,tau)=var12,

diff(t(tau),tau,tau)=var13}, eqns0):

> eqns12:=subs({diff(theta(tau),tau)=var20,

diff(r(tau),tau)=var21, diff(phi(tau),

tau)=var22,diff(t(tau),tau)=var23}, eqns11):

> eqns13:=subs({theta=var30,r=var31}, eqns12):

> eqns21:=subs({var30=theta(tau),var31=r(tau)}, eqns13):

> eqns22:=subs({var20=diff(theta(tau),tau),

var21=diff(r(tau),tau), var22=diff(phi(tau),tau),

var23=diff(t(tau),tau)}, eqns21):

> eqns23:=subs({var10=diff(theta(tau),tau,tau),

var11=diff(r(tau),tau,tau), var12=diff(phi(tau),

tau,tau), var13=diff(t(tau),tau,tau)}, eqns22):

> eqns3:=subs({var10=diff(theta(tau),tau,tau),

var11=diff(r(tau),tau,tau),var12=diff(phi(tau),

tau,tau),var13=diff(t(tau),tau,tau)}, eqns22):

102

Initial value

Provide the initial values that satisfy nulllikke particle constraint, and use Maple to solve

these diﬀerential equations numerically and to form plots based on numerical solutions,

> x_d := array(1..4):

> x_d[1]:= td3: x_d[2]:=rd3: x_d[3]:=thd3: x_d[4]:=phd3:

>con:= 0:

> for i from 1 by 1 while i < 5 do

for j from 1 by 1 while j < 5 do

con := con + g_compts[i,j]*x_d[i]*x_d[j];

end do;

> end do;

> const := subs({t=var13, r=var33, theta=var53,

phi(tau)=var73}, con):

> constraint:=const=-c^2:

> ini1 := r(0) = 26:

> ini2 := D(r)(0) = 0:

> ini3 := phi(0) = 0:

> ini4 := D(phi)(0) = 0.00714:

> ini5:=t(0)=0:

> ini6:=D(t)(0)=1.0718:

> ini7:=theta(0)=Pi/2:

> var13:=rhs(ini5);var33:=rhs(ini1);var53:=rhs(ini7);

var73:=rhs(ini3); rd3:=rhs(ini2);td3:=rhs(ini6);

phd3:=rhs(ini4):

> th_d3:=isolate(constraint, thd3):

> ini8:=D(theta)(0)=rhs(th_d3):

> ini := ini1,ini2,ini3,ini4,ini5,ini6,ini7,ini8:

equations of motion are solved by above initial conditions,

> Eq51:=dsolve({eqns3[1],eqns3[2],eqns3[3],eqns3[4],

ini}, {theta(tau), phi(tau), t(tau),r(tau)},

numeric, output=listprocedure):

103

Polar coordinate plotting (The radius is the ﬁrst parameter and φ is the second parameter)

> Kerrplot:=polarplot([rhs(Eq51(tau)[4]),

rhs(Eq51(tau)[2]), tau=0..4300],

axesfont=[TIMES, ROMAN, 12],

legend="Kerr-Newman"):

> pns:=disk([0,0],5.8,color=gray):

> display([pns,Kerrplot],scaling=constrained);

Plotting a space curve that the radius is the ﬁrst parameter, the θ is the second parameter

and the φ is the third parameter.

> traj:=spacecurve([rhs(Eq51(tau)[4]),

rhs(Eq51(tau)[8]), rhs(Eq51(tau)[2]),

tau=0..2000], coords=spherical,

numpoints=1000, color=black):

> pnm:=sphere([0,0,0],5.8):

> display([traj,pnm],scaling=constrained):

Showing motion of a light like particle around a black hole.

> B :=sphereplot(5.8,theta=0..2*Pi,phi=0..Pi):

> animate(spacecurve,[[rhs(Eq51(tau)[4]),

rhs(Eq51(tau)[8]), rhs(Eq51(tau)[2])],

tau=0..d, coords=spherical,

numpoints=1000, color=black],d=0..2000,

background=B,scaling=constrained):

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[32] Ref. 30, 1041.

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