Trajectory of a Lightlike Particle Around a Black Hole

Seyyed Mohammad Reza Taheri (smrt.qbs@gmail.com)

Mohammad Saber Naderi (sbrnaderi@gmail.com)

Supervisor: Dr. Hossein Farajollahi

September-2007

Abstract

This text is about motion of lightlike particles around massive objects in areas 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. KerrNewman 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 dierent methods and dierent spacetimes are compared.

1

Acknowledgements

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

in our research.

Were also thankful for funding provided by the Guilan University. And nally, wed like

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

In addition wed 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 Newtons 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

43 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 Birkhos 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 Dierent Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6 Compare Dierent Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A Maple Codes for Lagrangian Method 83

5B Maple Codes for Hamiltonian method 90

C Maple Codes For Geodesic Equation 99

Bibliography 104

List of Figures

1.1 Schematic of Physics Classication . . . . . . . . . . . . . . . . . . . . . . . 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 (Lagranges method) 69

5.2 Trajectory of a null particle in Kerr sapcetime (Lagranges method) . . . . 71

5.3 Trajectory of a null particle in Kerr-Newman sapcetime (Lagranges 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 dierent 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 Maxwells 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: Einsteins 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

9Figure 1.1: The frameworks and arenas for the laws of physics, and their relationship toeach 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 (g 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 briey, and in next chapters we focus on the

GR. In chapter 2 we conne 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 denition 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 dierences 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 dierent 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 simplied by unied constants, such as c (speed of light) and G (Newtons

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

we use them, but when it is not necessary, unied 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 DefinitionScalar c,G, kVector p, uSecond rank contravarianttensor

G , G = x

xxx G

Second rank covariant ten-sor

G , G = x

xx

x G

Tensor without indices G GPartial derivative of a ten-sor

Xa

xb= bXa = Xa,b

Einstein summation con-vention

Any index that is repeated in a productis automatically summed on

X =

3i=0

xiX i

Covariant derivative T = T; = T +T T.A = Aa;a

Connection coecients(Christoel symbols)

=12g

(g + g g)when gab is metric

12

t. Absolute, true, and mathematical time . . . ows 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

Newtons laws of motion (however Newtons laws of motion include law of inertia).

1.3.1 The Law of Inertia

Galileo (1544-1642) was the rst 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 inuences 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, dicult for us to appreciate the impact of Galileos 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 Galileos time. However, the fact that the object continues to move after the force

ceases to be applied caused considerable conceptual diculties 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 eect, 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 Newtons 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, rst published in the Principia in 1687, play

a fundamental role in Newtons 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 (Galileos 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 |FAB| on a body B, then B exerts a force ofequal magnitude |FBA| on A. The forces act in opposite directions so that

FAB = FBA (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 nite time to propagate between the bodies, the law must be

modied to include the properties of the eld 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 xj (called Galilean coordinates), and a Newtonian

gravitational potential .

2. The density of mass generates the Newtonian potential by

Poissons equation,

2 2t2

= 4

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

d2xj

dt2+

xj

= 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 rst real breakthrough came in 1900 when Max Plank introduced the concept of

the quantum of energy. In his eorts 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 h, which he called the energy of

quantum, where h is a fundamental constant called Planks 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

rst 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 Schrodinger (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 Schrodinger equation; instead of

the matrix eigenvalue problem of Heisenberg, Schrodinger obtained a dierential 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 (x; t) .

Thats 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

dont 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 (x; t) which

is a function of the two variables position x and time t. At this stage we dont know what

(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 Schrodinger

equation

( 2

2m2

x2+ U)(x; t) = i(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 Schrodinger equation. (Here we have written it for a

single particle of mass m in 1-dimension).

Postulate 4: (Born hypothesis): ||2 is the probability density. This postulate statesthat the wave function is actually related to a probability density. The basic postulate in

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

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

P = AA||2, dx (1.4)

P is the probability for nding the particle somewhere between A and A. This meansthat ||2dx = probability of nding a particle between position x and x + dx at time t (ofcourse when (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 dened quantities in classical

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

we shall see that the concept of the denite 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 particles wave property spread out over some

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

nding 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 Maxwells equations and general

relativity is based on the Einstein equations G = 8GT .

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 Maxwells 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 Schroodinger equation

( 2

2m2

x2+ U)(x; t) = i(x; t)

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

tial U in one dimension x. Its 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

nd 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,

Maxwells equations and the Schrodinger equation are postulates and

thats that. Nothing more. They are sort of like the denitions that

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

They cannot be derived from anything else.

The wave function (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 dierent for dierent 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) elds. These always depend

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

distribution. Dierent q and j produce dierent E and B. In gen-

eral relativity, the fundamental equations are the Einstein equations

(G = 8GT) and the fundamental quantity that we always want

is the metric tensor g , which tells us how spacetime is curved.

(g is buried inside G). The fundamental input is the energy-

momentum tensor T which describes the distribution of matter.

Dierent T produces dierent g .

Similarly the fundamental equation of quantum mechanics is the

Schrodinger equation and the fundamental input is the potential U .

(This is related to force via F = U . Dierent input potentials Ugive dierent values of the fundamental quantity which is the wave

function (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 dicult, primarily for two reasons: tensors

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

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

some denitions 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 o a mirror attached to another and then

returns, the time of bounce tb as measured by the clock that does the bouncing is the

average of the times of emission te and reception tr as measured by the emitting and

receiving clock: tb = 12(te + tr).

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 (Einsteins equation)

R 12Rg = 8GT (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 Newtons 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

Newtons laws of motion.

21

In Einsteins 1915 general theory of relativity, a mass curves the one time dimension

and three space dimensions of spacetime according to Einsteins 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. Einsteins theory thus expresses

both the gravitational eect of mass and the response of mass to that eect 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(x), such that the expression

s2 = 2 = g(x)xx (2.2)

or in modern style

ds2 = gdxdx (2.3)

gives the intervals between any event x and any near by event x +x (or x +dx). In

the language of abstract dierential geometry, metric is a bilinear machine, g (. . . . . . .),to produce a number [scalar product g(u,v) (u.v)] out of two tangent vectors, u andv [9](Box 2.1).

In relativity, mass and energy are the same thing according to Einsteins famousE = mc2

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 Einsteins theory keep track of this complex feedback interrelationship

between energy and curvature.

Newtons theory of gravity is not wrong. It is a correct approximation to Einsteins

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

than the velocity of light. The rst 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 identiable points: buoys, ships, icebergs, lighthouse,

peaks, and ags: points to a total of n = 2 107. The total numberof distances to be given is n(n1)/2 = 21014. With 200 distancesper page of printout, this means 1012 pages weighing 6 gram each,

or 6 106 metric ton of data. With 6 tons per truck this means106 truckloads of data; or with on truck passing by every 5 seconds,

5 106 seconds or 2 months of night and day trac 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 = 107pages 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 suciently

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. Dene

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 xk, 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

ds2 = gjkdxjdxk

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

distilled now ve numbers per point (two coordinates, x1, x2, and

three metric coecients, g11, g12 = g21, and g22), 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 coecients expressed as analytical

functions of coordinates

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

2 107 points of the surface, give them as functions of the two coor-dinates x1, x2, in terms of power series or an expansion in spherical

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

justable coecients. Then the information about the geometry itself

(as distinct from the coordinates of the 2107 points located on that

25

Box 2.1 (continued)

geometry) is caught up in these three hundred coecients, 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 justication 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-

steins 1915 general theory of relativity the classical (nonquantum) theory of relativistic

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

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

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

more dicult 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 Newtons equation for the gravitational force between

two interacting masses refers to gravitational mass; Newtons law should indicate this

property of matter

FG = GMGmG

r2(2.4)

where MG and mG are the gravitational masses of the interacting objects, separated by a

distance r.

The term mass that appears in Newtons equation of motion

F = ma (2.5)

26

refers to the inertial mass; Newtons equation of motion should indicate this property of

matter:F = mIa (2.6)

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

tational eld of the mass MG.

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

tional mass, mI = mG to an accuracy of 1 part in 103. Recent experiments have shown this

equality to be true to an accuracy of 1 part in 1012. Newton therefore took the equations

F = GMGmG

r2= mIa (2.7)

and used the condition mG = mI to obtain

a = GMGr2

(2.8)

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

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

mG mI . 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 eld-free frames, to frames of reference in

free fall [12]. One of Einsteins 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 conned 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 denition, 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 dierent observers, in two dierent local Lorentz frames, in dierent (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

connes 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 specic 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 Einsteins 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 eld. As an

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

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

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

the gravitational eld of the massive object causing the deection.

Einsteins original calculation of the deection 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 eld 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 deection

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

eld [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 nding 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 harms way.

figure Box2-2

Schematic illustration of segments used in operation of the Global

Positioning System. (Adapted from a gure 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 dierent rates than clocks on the ground.

Gravity and speed contribute comparable amounts to the total dis-

crepancy. The oset 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 users 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 = x x , the standard

special relativistic expression [14]

2 = g = (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 = {1 if = = 0,+1 if = = (x, or y, or z), 0 otherwise} (2.10)

Turn, now, to a rst 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 Newtons

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 earthssurface. (b) A family of local, 2-dimensional Euclidean coordinate systems on the earthssurface. 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 earths surface, the frames

will all pass through the earths center and y out the earths 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 earths 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 earths 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 at, special relativistic spacetime a free particle moves

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

(t, x, y, z) = (t0, x0, y0, z0) + (p0, px, P y, P z); i.e., x = x0 + p (2.11)

here p are the Lorentz-frame components of the particles 4-momentum; is the ane

parameter such that p = dd , i.e. p = dx/d; and x0 are the coordinates of the particle

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

well by the statement that the Lorentz-frame components p of the particles 4-momentum

are constant, i.e., are independent of

dp

d= 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

pp = 0, or, equivalently p;p = 0 (2.13)

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

4-velocity and its proper time, Eq. (2.13) is equivalent to uu = 0.This description of the motion is readily carried over into curved spacetime using the

equivalence principle: Let P() be the world line of a freely moving particle in curvedspacetime. At a specic P0 = P(0) event on that world line introduce a local Lorentzframe (so the frames spatial origin, like the particle, passes through P0 as time progresses).Then the equivalence principle tells us that the particles law of motion must be the same

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

(dp

d)=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 coecients 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

d+ p

dx

d)=0 = ((p

, +

p

)dx

d)=0 = (p

;p

)=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 pp and the zero vector are equal in ourchosen local Lorentz frame, it must be true that

pp = 0 (2.16)

at the moment when the particle passes through the point P0 = P(0). Moreover, since P0is an arbitrary point (event) along the particles 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 pp = 0in 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 airplanes inertial guidance

system, which tries to y 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 particles world line

(gpp);p = 2gpp;p = 0 (2.17)

(It also can be deduced in a local Lorentz frame where 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 and p , being equal, have been combined. In index-free notation the frame

independent relation Eq. (2.17) says

p(p.p) = 2p.pp = 0 (2.18)

This is a pleasing result, since the square of the 4-momentum is the negative of the particles

squared rest mass, p.p = m2, which surely should be conserved along the particles free-fallworld line! Note that, as in at spacetime, so also in curved, for a particle of nite rest mass

the free-fall trajectory (the geodesic world line) is timelike, p.p = m2 < 0, while for azero-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 nite rest mass, one often uses as the tan-

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

and correspondingly one uses as the parameter along the geodesic the particles proper time

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

uu = 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 pp = 0 becomes theFollowing dierential equation for the particles world line x() in the coordinate

systemd2x

d2=

dx

d

dx

d(2.20)

here is the connection coecients of the coordinate systems coordinate basis.

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

x as functions of ane parameter along the geodesic. If the initial position, x at

= 0, and initial tangent vector (particle momentum), p = dx/d at = 0, are

specied, then these four equations will determine uniquely the coordinates x() as

a function of along the geodesic.

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

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

xA. Associated with that symmetry there will be a conserved quantity PA = p./xA

associated with free-particle motion.

3. Among all timelike curves linking two events P0 and P1 in spacetime, those whoseproper time lapse (timelike length) is stationary under small variations of the curve are

timelike geodesics. Now, one can always send a photon from P0 to P1 by bouncing ito 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 P0 toP1 with vanishingly small length, so the geodesics cannot minimize the proper timelapse. 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? Einsteins

search for that law, his Einstein eld equation, occupied a large fraction of his eorts 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 briey examine one segment of Einsteins route toward his eld

equation: the segment motivated by contact with Newtonian gravity.

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

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

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

the Newtonian tidal eld E, and by dierentiating 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 (it is possible to obtain this by attend to relativistic description of tidal

gravity, however Riemann curvature tensor dened asR =

+

,

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

at).

In Newtonian gravity is determined by Newtons eld equation

2 = 4G (2.21)

which can be rewritten in terms of the tidal eld jk = 2/xjxk as

jj = 4G (2.22)

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

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

(the analog of the Newtonian tidal eld) 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 eld equation should reduce to Newtons eld equation.

To exploit that guidance, we can

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

relativitys Riemann tensor, jk = Rj0k0.

2. Then take the trace and note that by its symmetries R0000 = 0 so that jj = R

00 = R00.

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

local Lorentz frame,

R00 = 4G (2.23)

36

here R00 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 eld 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 = 4Ga second-rank symmetric tensor that generalizes the Newtonian massdensity )

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

R = aGT (2.24)

Einstein flirted extensively with this proposal for the eld 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 , it becomes (because R

and T both have ten independent components) ten independent dierential equations

for the ten g . This is too many equations: By an arbitrary change of coordinates,

xnew = F(x0old, x

1old, x

2old, x

3old) 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 g00 = 1 andg0j = 0 everywhere); and correspondingly, the eld equations should constrain only six, not

ten of the components of the metric (the six gij in our example).

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

with Einsteins 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; = 0 independent of the eld equation, if we replace the Ricci tensor in Eq.

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

automatically divergence free independent of the eld equation (G = 0), then the new

eld equation G = T (with = constant) will not constrain all ten components of

the metric. Rather, in a coordinate system the four equations [G T ]; = 0 with = 0; 1; 2; 3 will automatically be satised; they will not constrain the metric components

in any way, and there will remain in the eld 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 with the desired property:

G R 12Rg (2.25)

Today we call this the Einstein curvature tensor (when R is The Ricci tensor, dened

by the contraction R = R = g

R ; R is the Ricci scalar, dened by contraction

R = R = gR). 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 eld equation, then, should equate a multiple of T to the Einstein tensor

G :

G = T (2.27)

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

equation (2.27) in terms of the Ricci tensor

R 12gR = T (2.28)

then taking the trace to obtain R = gT , then inserting this back into (2.28), weobtain

R = (T 12ggT

) (2.29)

In nearly Newtonian situations and in a local Lorentz frame, the mass-energy density T 00 = 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 eld equation (2.29) becomes

R00 = (T 00 120000T

00) =12T 00 =

12 , where 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 R00 = R00, we see that

= 8G (2.31)

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

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

38

will be greatly simplied if we set Newtons gravitation constant to unity. This further

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

relation

1 =G

c2= 7.42 1028 m

kg; i.e. 1 kg = 7.42 1028 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 eld equation (2.27), with = 8G = 8 [Eq. (2.31)],

assumes the following standard form

G = 8T ; i.e., G = 8T (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 eect 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 c4/4G or on a maximum power c5/4G. 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 eld 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 nite

size.

39

Table 2.1: Some useful quantities in conventional and geometrized units. Note: 1 Mpc =106 parsecs (pc), 1 pc = 3.026 light year (lt y), 1 lt yr = 0.9461016 m, 1 AU = 1.491011m. For other useful astronomical constants see C. W. Allen, Astrophysical Quantities.

Quantity Conventional Units Geometries Unitesspeed of light 2.998 108 m sec1 oneNewtons gravita-tion constant, G

6.6731011 m3 kg1 sec2 one

G/c2 7.425 1028 m kg1onec5/G 3.629 1052 W onec2/G 3.479 1024 gauss cm

= 1.160 1024 voltsone

Plancks reducedconstant

1.055 1034 kg m2 s1 (1.616 1035 m)2

suns mass, M 1.989 1030 kg 1.477 kmsuns radius, R 6.960 108 m 6.960 108 mearths mass, M 5.977 1024 kg 4.438 mmearths radius, R 6.371 106 m 6.371 106 mHubble constantH0

6525 km sec1 Mpc1 [(125)109 lt yr]1

density to closeuniverse, crit

9+115 1027 kg m 3 7+83 1054 m2

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 c4

4G= 3.0 1043N (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 c5

4G= 9.1 1051W (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 denition: force is the ow of momentum per unit time. Momentum cannot

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

a conserved quantity, can only change by inow or outow. 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 owing

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 ow 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 x observers onto it.

This principle imposes a limit on muscles, the eect of hammers, the ow 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 ts into the system of natural units (When Planck discovered

the quantum of action, he had also noticed the possibility to dene 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 qualies 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 c4/G and P c5/G. It does not matter whether the observer measures the forceor 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 denition 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 1049 horsepower. The maximum power principlestates 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 massenergy. 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 c2.

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

dt c

3

4G= 1.0 1035kg/s (2.36)

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

of ow of water or any other material through tubes is limited. The mass ow 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 rst of all be checked

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

power. However, it is straightforward to conrm 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 dened below; for divisions

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

Astronomers have also failed to nd 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 nd 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 nal

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

Similarly, all observed mass ow 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 conrm 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 eld 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 nal version of his

eld equation, G = 8T, 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

worlds rst exact solution to the Einstein eld equation [18].

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

metrics) briey. First we try to obtain Schwarzschild metric from Einstein eld 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 eld equationG = 0. Schwarzschild

consider some simplications 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 (x0, x1, x2, x3) and

(x0 + dx0, x1 + dx1, x2, x3), then ds2 = g00(dx0)2 +2g01dx0dx1 + g11(dx1)2, because

spacetime is static the interval is invariant under a time inverse x0 x0 = x0, thusinterval become ds2 = g00(dx0)2 2g01dx0dx1 + g11(dx1)2 and thus g01 = 0).

3. Hypersurface-orthogonal vector elds.

3.2.1 Coordinates and Metric for a Static, Spherical System

To deduce the gravitational eld 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]

ds2 = dt2 + dr2 + r2d2 (3.1)

where

d2 = d2 + sin2()d2 (3.2)

Try to modify this metric to allow for curvature due to the gravitational inuence 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 dierent

values:

ds2 = e2dt2 + e2dr2 + R2d2 (3.3)

where , , and R are functions of r only. (The static assumption demands g/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 speciable) as in

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

more general metric

ds2 = a2 dt2 2a b dr dt + c2 dr2 + R2 d2 (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 dened by

edt = a dt + b dr (3.5)

By inserting this in equation (3.4), and by dening e2 = b2+c2, 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 exibility, something

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

The grt = 0 simplication (called a coordinate condition) in equation (3.3) results from an

advantageous choice of the t coordinate. The r coordinate, however, is also at ones disposal

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

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

by

r = R(r) (3.6)

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

to

ds2 = e2dt2 + e2dr2 + r2d2 (3.7)a line element with just two unknown functions, (r) and (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 Volko

(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)/ 2 [19]. {Consider = 2and = 2, hence

ds2 = edt2 + edr2 + r2(d2 + sin2 d2) , = (r) , = (r) (3.8)

thus

g =

e 0 0 00 e 0 0

0 0 r2 0

0 0 0 r2 sin2

(3.9)

48

we have G = R 12Rg , thus

G00 = g00G00 = e (rr 1r ) 1r2

G10 =er t = e

G01G11 = e

(rr 1r2

) 1r2

eqs(*) (3.10)

G22 = G33 =

12e

(rr

2+

r

r r

r (r)

2

2 rr

)(3.11)

otherwise component of G vanishes. The contracted Bianchi identity show that G22 = G33

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

G00 = 0

G01 = 0

G11 = 0

e

=(1 2M

r

)1, + = h(t) = 0 (3.12)

because , are time (t) independent. Thus

g =

(1 2Mr ) 0 0 0

0(1 2Mr

)1 0 00 0 r2 0

0 0 0 r2 sin2

(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,0 = 0, it means metric is stationery.

2. From the metric, we have g = 0, = , it means metric is static.

3. The coordinates are adapted to the killing vector eld X = 0 , such that this time

killing vector eld, X, 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, Schwarzschilds solution takes the form

ds2 = (1 2M/r)dt2 + dr2

(1 2M/r) + r2(d2 + sin2 d2) (3.14)

49

where M is a constant of integration. The connection coecients, 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 eld 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

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

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

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

(3.14) takes the form

ds2 = dt2 + dr2 + r2(d2 + sin2 d2) (3.15)

This is the line element of at spacetime, ds2 = dt2 + dx2 + dy2 + dz2 written in sphericalpolar coordinates [x = r sin cos, y = r sin sin, z = r cos ].Thus, Schwarzschild

spacetime is asymptotically at in the region of large radii r/M . This is just whatone 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 xed relative to the asymp-totically Lorentz spatial coordinates x = r sin cos, y = r sin sin, z = r cos . This

means that the asymptotic Lorentz frame can be regarded as the bodys asymptotic rest

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

the Lorentz time of the bodys asymptotic rest frame, and the Schwarzschild r, , coordi-

nates become spherical polar coordinates in the bodys 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

et =

t, er =

r, e =

, e =

et = t, er = r, e = , e =

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

(3.14)]

gtt = (1 2M

r

), grr =

(1 2M

r

)1g = r2, g = r2 sin2

and the contravariant metric coefficients are the inverse of these

gtt = (1 2M

r

)1, grr =

(1 2M

r

),

g =1r2

, g =1

r2 sin2

trt = ttr =

M

r2

(1 2M

r

)1,

rtt =M

r2

(1 2M

r

),

rrr = M

r2

(1 2M

r

)1,

r = r(1 2M

r

),

r = r =

r =

r =

1r

,

r = r sin2 (1 2M

r

),

= sin cos =

= cot

51

Box 3.1 (continued)

The orthonormal basis associated with the above coordinate basis is

e0 =t

1 2M/r , er =

1 2Mr

r ,

e =1r , e =

1r sin

The nonzero connection coecients in this orthonormal basis are

rtt= t

rt=

M

r2

1 2M/r ,

=

=

cot r

,

r

= r

= r

= r

=

1 2M/r

r

The nonzero components of the Riemann tensor in this orthonormal

basis are

Rrtrt = R = 2Mr3

,

Rtt = Rtt = Rrr = Rr thetar =M

r3

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

which implies that the Schwarzschild metric is a solution of the

vacuum Einstein eld equations.

To nd 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 Schwarzschildmetric coecients from those of a Lorentz frame, to rst order in M/r is calculated, Direct

comparison of it by Newtonian limit in GR (that obtain by assume g = + h , when

is SR at Minkowski spacetime- metric and h is little respect to and do some

approximations), shows that a Newtonian description of the bodys distant gravitational

eld will entail a Newtonian potential given by

= Mr

(3.16)