my-thesis

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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