An Investigation of the Existence of Closed Timelike Curves

and the Feasibility of Time Travel

Han Weiding

Jani Hariom Kirit

Ng Xin Zhao

Student Mentor: Lim Yen Kheng

Junior Mentor: Tran Chieu Minh

April 24, 2009

Abstract

The research on time travel is important to understand the true nature of time and

causality. In this report,the topic of closed timelike curves and the chronology protection

conjecture are reviewed. The basic concepts of General Relativity are used to analyse

two solutions of the Einstein Field Equations, which are Godel universe and van Stockum

solution because they contain closed timelike curves. The analysis on Godel universe

includes the derivation of the Godel metric, the analysis of null geodesics and closed

timelike curves. Geodesic in Godel solution is derived using Chandrasekhar’s method,

interpreted by M. Novello. It is showed that the closed timelike Geodesics does not exist

in Godel universe. Then using limited concepts of quantum physics, the Chronology

Protection Conjecture by Stephen Hawking is introduced. The weak energy condition is

used to exclude time travel in finite regions of spacetime. A response of anti-chronology

protection conjecture by Li Xin Li et. al. is presented using arguments of quantum

cosmology. Currently the feasibility of time travel is an open question. It is suggested

that a model of time travelling be constructed in Godel universe to further explore the

properties of time travel.

Contents

1 Introduction 3

2 Introduction to General Relativity 5

2.1 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Godel Universe 7

3.1 The Line Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Null Geodesics - The Path of Light . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Closed Timelike Curves and Time Travel . . . . . . . . . . . . . . . . . . . . 13

3.4.1 Chandrasekhar’s method . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.2 M. Novello’s Interpretation . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.3 Closed Timelike Geodesics - Chandrasekhar’s method . . . . . . . . 20

4 The van Stockum Solution 21

5 The Problem of Chronology Protection 24

5.1 Chronology Protection Conjecture . . . . . . . . . . . . . . . . . . . . . . . 24

5.1.1 Hawking’s Chronology Protection Conjecture . . . . . . . . . . . . . 24

5.1.2 Matt Visser’s “The Quantum Physics of Chronology Protection” . . 25

5.2 Li Xin Li’s Anti-Chronology Protection Conjecture . . . . . . . . . . . . . . 26

6 Discussion and Conclusion 29

6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A Introductory General Relativity 33

A.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.2 Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1

A.2.1 The Metric Tensor, gµν . . . . . . . . . . . . . . . . . . . . . . . . . 35

A.2.2 Causal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.2.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.2.4 Stress-Energy Tensor, Tµν . . . . . . . . . . . . . . . . . . . . . . . . 37

B Solving the Line Element 39

C Geodesics 43

C.1 Closed Timelike Geodesics - Chandrasekhar’s method . . . . . . . . . . . . 46

D Definitions for the Problem of Chronology Protection 47

D.1 Cauchy Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

D.2 Quantum Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

E Acknowledgements 49

E.1 Acknowledgement for Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 49

2

Chapter 1

Introduction

Is it true that Time unravels

Does it have a plan or path?

Would it mean that we could travel,

Back into the Past?

— Greg Baker, “And About Time, Too”

Ever since H.G. Well’s “The Time Machine” has been published, the fantasy of time-

travelling has captivated the hearts of writers and scientists alike. During the time of

Newtonian determinism, the notion of absolute time prevented the slightest possibility

of going backwards in time. However, Einstein’s General Relativity allows the bending

of time through the existence of closed timelike curves (CTCs). These CTCs allow time

travel, in the sense that an observer who travels along these curves could return to where

and when he started. The best known solutions from General Relativity that allow such

CTCs to exist are the van Stockum solution, the Gott two-string time machine, Kerr black

hole and the Godel universe.

Looking at the sheer number of different solutions that contain CTCs, it might be that

the universe is trying to tell us that time travel might be inherently possible. Therefore

it is instructive to study the some of the solutions that contain CTCs and determine

whether one can travel back in time, at least theorectically. For this purpose, the Godel

solution and van Stockum solution are studied. Besides that, the Chronology Protection

Conjecture proposed by Hawking and the problem with it are explored. The scope of

this study is confined within limited concepts of quantum mechanics and more on basic

concepts of General Relativity to retain focus.

The implication of this study is, as it shall be seen, that it gives one more reason for the

pursuit of the full theory of quantum gravity in order to settle the problem of chronology

3

protection. It is also important to answer the basic questions of our reality such as “Is

time real?” and “What is the true nature of causality and time?”. This report can serve as

an introduction to those who are new to the field of CTC and the problem of chronology

protection.

In Chapter 2, the Introduction to General Relativity is given briefly to allow the reader

to followed a thorough study of Godel universe in Chapter 3. This universe comes from

a paper presented to Albert Einstein on his 70th birthday in July, 1949. Kurt Godel, a

logician at the Institute for Advanced Study in Princeton University, had produced this

legendary physics paper named “An Example of a New Type of Cosmological Solutions

of the Einstein’s Field Equations of Gravitation”. His universe is a simple one, it is just a

infinitely large rigid rotating dust. Nevertheless, Godel universe has twisted time, a deep

and a mysterious concept in the trinity of space-time-matter, with space and matter into

a mind boggling enigma. Godel had shown the existence of closed timelike world lines

in his universe, which could be used by an observer to travel into his past and causally

influence it.

The van Stockum solution has its gravitational field generated by an infinite dust

cylinder, surrounded by vacuum, rotating about its axis. This solution has been named

after Willem Jacob van Stockum, who rediscovered it in 1937, independently of an even

earlier discovery by Cornelius Lanczos in 1924. This was the first solution to the Einstein’s

Field Equations which allows closed timelike curves to exist and is studied in Chapter 4.

In Chapter 5, the problem of chronology protection is explored. It all started in 1992,

when Stephen Hawking published a paper titled “Chronology Protection Conjecture” [7]

which states that the laws of physics do not allow the appearance of closed timelike curves.

Hawking’s Chronlogy Protection Conjecture proposes that when quantum mechanical ef-

fects are taken into consideration, the classical General Relativity will fail when one tries

to cross from the region of normal space to the regions containing CTC. However, Li Xin

Li et.al. responded to his claim and used quantum cosmology, a partial theory of quantum

gravity, to show that CPC does not hold [10]. Thus, Matt Viener says that we need a full

theory of quantum gravity to settle this dispute for good [19].

4

Chapter 2

Introduction to General Relativity

“When you are courting a nice girl an hour seems like a second. When you sit

on a red-hot cinder a second seems like an hour. That’s relativity.”—Albert

Einstein

The General Theory of Relativity (General Relativity) describes the structure of the space-

time along with the distribution of the mass and energy in a universe. In this section,

some of the terminologies and concepts of General Relativity, will be introduced (from [15]

and [3]). However, the detailed discussion of the Einstein’s Field Equations can be found

in the appendix.

2.1 Einstein’s Field Equations

The Einstein’s Field Equations are the central equations in general relativity. It equates

the curvature of spacetime on the left side with the mass-energy that causes it on the right

side.

Rµν − 1

2Rgµν + Λgµν = 8πTµν . (2.1)

The Metric Tensor gµν defines the inner product of two vectors mathematically, and

captures the geometrical and causal structure of the spacetime physically.

g[dxµ, dxν ] ≡ gµνdxµdxν , (2.2)

where g is a function on the vectors defined above. A universe can be modelled by choosing

the metric. Given a choosen basis, the metric tensor can be represented by a 4×4 matrix.

If the metric is diagonal with terms (−1, 1, 1, 1), then the spacetime is flat, if they are

variables then the spacetime is curved. Metric signature used here is (−,+,+,+) and is

just a convention for writing a metric. The opposite signature of (+,−,−,−) can also be

5

Figure 2.1: Light Cone

used but it must be consistent within a system. The figure 2.1 is drawn in flat spacetime.

The horizontal plane is made up of two out of the three space dimensions and the vertical

axis is time with the third space dimension compressed. The cones are light paths tracing

from the origin of the axis to both the future and the past.

Causal Structure The physical meaning of timelike region (the line element has the

sign of time) is the region of space where an object can causally influence the universe

by moving at speeds less than the speed of light. Null lines (line element is zero) are the

curves that are traced out by the path of a particle moving at light speed. Since nothing

can travel faster than light, the other regions of spacetime or spacelike (line element has

the sign of space) regions of an object, are free from any causal influence of that object.

CTCs are those paths which are timelike and closed, thus one can travel on a CTC and

come back to were he started to create a causal influence.

Geodesics are the shortest paths between two points in spacetime. It describes the

motion of a free particle under no external forces, but under the influence of the curvature

of spacetime or gravity. The equation for geodesics is

d2xλ

dt2= Γλ

µν

dxµ

dt

dxν

dt. (2.3)

Stress-Energy Tensor, Tµν completely describes the distribution of mass, energy and

stress or pressure that produces the curvature of spacetime.

6

Chapter 3

Godel Universe

“Physicists believe the separation between past, present, and future is only an

illusion, although a convincing one.”—(Albert Einstein)

In his ground breaking paper in 1949, Kurt Godel showed that time travel is possible.

This section is aimed at introducing and discussing the key elements of Godel work. It

shall be discussed how the Closed Timelike Curves come about, by analysing the null

geodesics and the null cones. Finally, to complete the picture, the section will conclude

with a study of the closed timelike geodesics. However, before we venture into the details,

the line element and the physical properties of the Godel universe shall be reviewed.

3.1 The Line Element

In order to create a universe which allows time travel it should grant the existence of

CTCs. As the structure of the curves is determined by the line element, it has to be

modelled in a certain specific form. Solving the field equations is a very involving task,

therefore only the heuristic details shall be discussed here and the mathematical specifics

are given in the appendix. The metric of a universe which has CTCs exist, generally takes

the following form (in the cylindrical coordinates (t, r, φ, z))

ds2 = gµν dxµ dxν = [(dt− u0(r) dφ)2 − dr2 − dz2 + 2v0(r) dt dφ].

The reason for choosing the above configuration can be explained as follows:

The functions [u0(r) and v0(r)], play a role in determining the curvature and the properties

of various curves (as being timelike, spacelike or null). The elements of the metric tensor

are functions of r alone. This firstly ensures that the curvature of the spacetime is modified

only with respect to r, and secondly that the determination of the various curves as being

timelike, spacelike or null, is only dependent on r. The time t and the azimuth direction

7

(angle) φ are coupled with each other, this generates a rotational effect as the angle would

change with time; however, t is not united with the other coordinates (r and z) making

the universe static. The coordiante z has neither been made a part of any function nor

has it been coupled with any function, therefore it does not play any significant role in the

metric. As it will be seen later, this assumption will make it possible to reduce the metric

from a four dimensional case to a three dimensional problem. The directions t and r have

also been left disassociated with any functions, so that the calculations remain simple. It

should be noted here that this is not the only way to generate a metric for a spacetime

with closed timelike curves.

The above metric in the (t, x, y, z) coordinates, becomes

ds2 = −dt2 + dx2 + dz2 + u(x) dy2 + 2v(x) dt dy. (3.1)

As it can be seen, u(x) and v(x) are functions of x and the coupling of t and y is taking

place, which is parallel to the above metric. The above form of the metric shall be used to

derive the functions u(x) and v(x). Moreover, a dust solution will be considered to create

the Stress-Energy Tensor, because dust is the simplest possible non vanishing matter-

energy (in comparison to general or perfect fluids) particle. The Stress-Energy Tensor

then becomes:

Tµν = ρ Vµ Vν ,

with Vµ and Vν representing the four velocity of the dust, which is the coordinate distance

travelled per unit of proper time (details can be found in the appendix). Using Einstein

equation

Rµν − 1

2Rgµν + Λgµν = 8πTµν ,

it is found that the functions take the form u0(x) = −12 e

2√

a/2x and v0(x) = −e√

a/2x.

The technique was borrowed from [9]. Therefore the line element is

ds2 = −dt2 + dx2 + dz2 − 1

2e2√

a/2x dy2 − 2e√

a/2x dy dy.

such that a = 16πρ, with ρ as the mass density. Likewise, it is also found that the

cosmological constant is Λ = −4πρ. This is exactly the form of the original Godel metric.

3.2 Physical Properties

The Godel universe [4] has the following features:

8

1. It is an exact dust Solution of the Einstein Field Equations: It satisfies the

Einstein Field Equations producing a physically realistic Stress-Energy Tensor. This

solution can be obtained analytically and it corresponds to a universe consisting of

rotating dust particles.

2. Homogeneous: All the particles making up the universe are uniformly distributed

and therefore all the particles are equivalent. It implies conservation of momentum

and energy.

3. Non-Isotropic: The universe is not equivalent in all directions as some properties

are directionally dependent.

4. Rotationally Symmetric (Cylindrical): The universe has symmetry under ro-

tational transformation about the axis of a cylinder.

5. Has Closed Timelike Curves : The universe allows the existence of closed timelike

curves in the space-time that are traversable.

6. Does not have Closed Timelike Geodesics: The closed curves mentioned above,

however, are not geodesics.

7. Has no Global Time: There can be no definition of a global time, because time

travel is possible.

We will try to see how these properties come about, as the paper proceeds.

The Godel universe is an exact solution to the Einstein Field Equations (as the metric

can be derived completely analytically) and its Stress-Energy Tensor takes the form of

a pressure free perfect fluid or just dust (property 1). Furthermore, any point on the

manifold M can be mapped on to any other point on M, thus it is taken for granted

that the space-time is completely homogeneous (property 2). In fact, the Godel rotating

solution and the Einstein static solution are the only spatially homogeneous cosmological

solutions with non-vanishing density of matter and equidistant world lines of matter (the

rigorous proof can be found in [4]).

The original metric g in a manifold M (a word manifold has been used in the paper to

refer to the overall spacetime of a universe) can also be given as an output of two smaller

metrics given by the metric g1 in a manifold M1 defined by the coordinates (t, x, y) [it

should be noted that throughout this paper the units (G = c = 1)],

ds21 = a2[dt20 − dx2 +e2x

2dy2 + 2ex dt0 dy], (3.2)

9

and the metric g2 [in a manifold M2 defined by the coordinate z] given by ds22 = dz2.

In order to describe the properties of this solution, it is sufficient to consider g1 because

the metric is neither a function of z, nor are any functions tied up with z (as mentioned

earlier). By certain complicated transformations found in [8] the metric g1 can take a

more favourable form in the cylindrical coordinates as

ds21 = 4a2[(dt2 − dr2 + (sinh4 r − sinh2 r) dφ2 + 2√

2 sinh2 r dt dφ]. (3.3)

Here, −∞ < t < ∞, 0 ≤ r < ∞ and 0 ≤ φ ≤ 2π. This form exhibits rotational sym-

metry of the solution about the axis r = 0 (property 4, thus property 3 entails naturally).

The parameter a describes the mass density and the cosmological constant of the universe:

1

a2= 8πGρ and λ = − 1

2a2, (3.4)

where a is the scaling factor of the metric, therefore we can consider it to be 1 (for

simplicity) (it is different from the a used in the previous section).

The behaviour of light cones in the Godel universe has been portrayed in the Fig.(3.1).

The light cones that are on the axis r = 0 behave like the ones in flat spacetime, but as

the r increases, the light cones gradually open out and tilt in the φ-direction. At the

critical radius (defined when r = ln(1 +√

2)), the light cones touch the t = 0 plane,

creating a closed null curve. As the value of the r increases, the light cones tilt to such

a great extent that they contain regions lying in the negative t direction. This enables

us to create closed timelike curves can exist through every point of the Godel universe

(property 5). The existence of CTCs gives way to the fact that there can be no global

time in M which increases along every future-directed timelike or null curve (property 7),

however every observer can still define local time. From figure 1, it can be seen that the

null geodesics arising from a point p on the r = 0 coordinate, firstly spiral out reaching a

maximal position at the Critical radius and then they finally reconverge to a point p′ some

time later. The Closed Timelike Geodesics (not shown) starting from p behave similarly;

they reach some maximum value less than the Critical radius and then reconverge back

to p′. Thus, a point outside the Critical radius can be connected to the point p only by a

closed timelike curves , not by a Closed Timelike Geodesics (property 6). Now an attempt

shall be made to understand some of these properties more rigorously. These properties

have been discussed in [8]

10

Figure 3.1: Godel universe in co-rotating cylindrical polar coordinates (t, r, φ), with the z coor-

diante suppressed. The light cones and the null geodesics have been indicated. Light cone opens

up and tips over as the radius increases, giving closed timelike curves beyond a critical radius. The

photons emitted at p, spiral out, reach the critical radius and reconverge back to p’.

3.3 Null Geodesics - The Path of Light

In this section the behaviour of Null (light) paths will be done. Engelbert Schucking and

Istvan Ozsvath in [14], have presented a detailed discussion of some of the properties of

the Godel. The discussions in this and the following section have been borrowed from

their paper.

In order to visually appreciate the properties better the radius r is transformed into

another parameter R, such that R = tanh r (as the metric is a function only of r), hence

the metric will now look like (following the structure from [14]).

ds2 =4

(1 −R2)2[((1 −R2) dt−

√2R2 dφ)2 − dR2 −R2 dφ2]. (3.5)

Geometrically speaking, this transformation maps the hyperbolic structure of our ear-

lier model into a flat unit disk. The new metric has the advantage that the entire mani-

fold extends only up to R = 1 (as hyperbolic tangent maps points from (−∞ ,+∞) into

11

(−1, 1)). This transformation is beneficial because the null geodesics and the character of

vectors (being time-like, null, space-like, or zero) remain unaffected, moreover all angles

and velocities also stay unchanged (Following the method suggested in [14]). We have ex-

tracted a common factor (2/(1−R2)2) from the metric which will always remain positive,

and hence it can be ignored.

To study the null geodesics, the shortest path between two points in the spacetime has

to found. This can be done by studying the variant problem for the above metric

δ

∫L dλ = 0, L =

1

2

([(1 −R2) t−

√2R2 φ]2 − R2 −R2 φ2

),

here t = dt/dλ, R = dR/dλ and φ = dφ/dλ with λ as a parameter. As the metric is

only a function of R, the rate of change of the Lagrangian with respect to the coordinates

t and φ is zero. Thus from the Euler-Lagrange equations the differentials of t and φ are

given by constants, the (∂L/∂φ) is given by:

∂L

∂φ= ((1 −R2)t−

√2R2φ)

√2R2 −R2φ = B = constant, (3.6)

as we are dealing with null geodesics, the extremal path is zero, thus we have

[(1 −R2)t−√

2R2φ]2 − R2 −R2φ2 = 0.

The simplest null geodesic (passing through the origin) can be characterized by having

B = 0 . Hence, from equation (3.6)

[(1 −R2)t−√

2R2φ] =φ√2, (3.7)

which would yield (with φ0 as an integration constant)

R =1√2

sin(φ− φ0)

In Cartesian coordinates (for φ = 0) the above equation can be given as

x2 +

(y − 1

2√

2

)2

=

(1

2√

2

)2

. (3.8)

Equation (3.8) is the equation of a circle with radius 1/2√

2 through the origin whose

center is displaced on the positive y axis. Light emitted at the origin in the direction of the

positive x axis will run in a circle reaching its maximum distance from the origin on the y

axis at y = 1/2√

2 at the top, and will return, coming in along the negative x axis. Thus,

Godel’s model provides an extreme example of light bending. If the restriction, φ = 0 is

dropped, circles of radius 1/2√

2 passing through the origin are obtained. Therefore when

a light ray is shot from the origin (in any direction), it would seem that it reaches some

12

distance equal to 1/√

2 and then bends, to such a great extent that it has to return to its

starting point. This limiting distance corresponds to the critical radius.

All the above discussion was limited to the domain where the time was kept as a

constant. Thus for variable time these circular light paths would look like helices. When

many such paths are combined they look like a cusp as shown in figure (3.1). The light ray

moves away from some point p, to reach out to the critical radius, finally reconverging to

the point p′ within some finite time. This explains why the null geodesics look like spindles

in the figure (3.1). Physically, this would imply that an observer (at point p′), including

other things can also see himself at some earlier point in time (point p), meaning if someone

is present in the Godel universe will be able to see himself (at some past moment), right

in front of him!

3.4 Closed Timelike Curves and Time Travel

The simplest possible Closed Timelike curve in the Godel universe, is a closed loop (a

circle). Let us consider it to have constant R and t. The tangent vector to this circle

is given by ∂∂φ and the square of the length of this vector is given by the inner product

defined below

l2 = g

[∂

∂φ,∂

∂φ

]=

[4R2

(1 −R2)2

](2R2 − 1), (3.9)

this circle is therefore spacelike, null or timelike if the value of R is smaller, equal to

or larger than 1/√

2. Thus if the R is greater than 1/√

2 the closed curve will be timelike.

Thus the curve will be a Closed Timelike Curve. Therefore if one travels beyond the

critical radius he will be able to time travel.

Furthermore, to realistically understand how time travel can take place, the behaviour

of the light cones shall be analysed below. The line element for light is given by

((1 −R2)dt−√

2R2 dφ)2 − dR2 −R2 dφ2 = 0.

Along, the azimuthal direction dR = 0 (as R is a constant), hence the rate of change

of time wrt. to the change in azimuth direction is given by

dt

Rdφ= −

√2R∓ 1

1 −R2. (3.10)

This indicates that the cones seem to be opening (and tilting) out, as the azimuth value

increases (shown in figure (3.2). From equation (3.10) it is evident that at R = 1/√

2, the

13

Figure 3.2: The Light cones open out and tip over with increasing Rdφ, venturing into

the region of negative t beyond R = 1/√

2

forward going null line will have slope zero in the azimuthal direction producing the null

geodesic (which can be closed as all the light cones along the circle can be connected).

Similarly, the null lines will tilt into the negative time zone, if the R goes beyond 1/√

2,

this would lead to the formation of the Timelike Curves that can be closed (if the light

cones along the circle are connected). This has been shown diagrammatically in the figure

(3.1) by the tipping of the cones, as the r increases. If an observer travels along one of the

CTCs, he will be able to time travel. One such path has been shown in the figure (3.4).

In this path an observer starts from s, goes beyond the critical radius, then moves on

a closed timelike path, and finally re-enters into the region inside the critical radius, and

arrives at h (which is earlier than s)!.

Time travel that is happening here is only a relative phenomenon, thus it is essential to

understand where does time travel “actually” take place. As the universe is homogeneous

and rotationally symmetric, each and every observer (O) in his own reference frame,

would see himself positioned in the centre, with the whole universe rotating around him.

Therefore he cannot assume for himself, a position beyond the critical radius (as he will

always be at the origin of his own reference frame). Thus an observer can never feel as if

he is travelling into the past. However, an observer O′ , who was co-moving with O till

the point s (before O began his journey), will observe O to be travelling into the past.

Moreover, it can be proven that Global time does not exist (by contradiction) through

the following method. Let us consider three observers Hariom, Wei Ding and Xin Zhao,

14

Figure 3.3: The path, in blue, that can be followed by an observer to travel into the past;

leaving at point s and returning at an earlier point h, to eventually witness his departure!

such that Hariom and Xin Zhao are at the centre of the universe, co-moving with each

other and Wei Ding is standing in the between these observers and the Critical radius

(from the reference frame of Hariom and Xin Zhao). Now Hariom accelerates and goes

beyond the critical radius of Xin Zhao, travels along a Closed Timelike path and returns

back to his earlier position. After this event, Xin Zhao will say that Hariom has travelled

into the past; however, Wei Ding will disagree because according to his reference frame

Hariom, never went beyond the critical radius and therefore Hariom never travelled into

the past at all.

There is a problem here because according to relativity different observers may disagree

on simultaneity of events, but they have to agree on the existence of events, at least. In

the above scenario Wei Ding and Xin Zhao seem to be disagreeing on the existence of the

event (of Hariom travelling into the past). Thus there seems to be a contradiction. The

underlying problem here is that we are trying to gauge both the observers (Wei Ding and

Xin Zhao), by a common time scale. If this condition is given up, then causality violation

can be accepted, which in our case happens to be true. Thus it can be concluded that

Global time doesn’t exist.

15

3.4.1 Chandrasekhar’s method

Chandrasekhar and Wright in [2], present their argument as to how Closed Timelike

Geodesics cannot exist in the Godel universe. Further detailed study of the Geodesics was

done by M. Novello et. al. [12]. In Godel’s paper, the line element can be written as

ds2 = dx02 − dx1

2 + (e2x1/2) dx22 − dx3

2 + 2ex1 dx0 dx2 (3.11)

the metric can be rewritten in the quadratic form

ds2 = (dx0 + ex1dx2)2 − dx1

2 − 1

2e2x1dx2

2 − dx32. (3.12)

The equation for geodesics is written as

d2xλ

ds2= −Γλ

µν

dxµ

ds

dxν

ds

If we let un be the four-velocity (the rate of change of both time and space coordinates

with respect to the time of the object), then

un =dxn

ds

and the geodesic equation can be written as

duλ

ds= −Γλ

µνuµuν , (3.13)

du0

ds= −2u0u1 − ex1u1u2, (3.14)

du1

ds= −ex1u0u2 − 1

2e2x1(u2)2, (3.15)

du2

ds= 2e−x1u1u0, (3.16)

du3

ds= 0. (3.17)

The four equations of motion are:

x0 = −√

2D2

D2 −B2σ + 2

√2 tan−1(

√α tanσ) + c0, (3.18)

x1 = log(cos2 σ + α sin2 σ

)+ c1, (3.19)

x2 = 2e−c1

√2B2

D2 −B2

tanσ

1 + α tan2 σ+ c2, (3.20)

x3 = Cs+ c3. (3.21)

With the 4 equations of motion, we can plot different graphs to illustrate the behaviour of

the different dimensions. Fig.(3.4) demonstrates the relations of all 4 xi with respect to the

invariant s, now transformed linearly into σ. Let α = 1/4,(D = 5, B = 3, C =√

15). Not

16

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

4

σ

Xi

Graph of Xi against σ

x0

x1x2x3

Figure 3.4: Graph of xi vs σ

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x

y

Graph of x vs. y

Figure 3.5: Geodesic of free particles in the XY plane

surprisingly, this is no different from what Chandrasekhar had in his paper. Secondly, we

look at how look at the path of free particles on the XY plane: This demonstrates closed

spacelike geodesics that result from the rotation of the universe about the origin. The

existence of a maximum radius from the origin for the geodesic agrees with M. Novello,

that the geodesic which can reach the origin (i.e. r = 0) are confined in the critical radius.

The change of x1 and x2 with σ is shown below. The spiral occurs because the cyclic

relation of x and y with the progression of σ. Indeed, if we project the spiral on the XY

plane we get Fig.(3.5) If we include x3 in the graph, we can expect a spiral, due to the

linear relation between x3 and σ:

17

00.5

11.5

22.5

3

−1.5

−1

−0.5

0−3

−2

−1

0

1

2

3

sigma

Graph of sigma vs. x vs. y

x

y

Figure 3.6: Geodesic of free particles in the XYσ space

−1.4−1.2

−1−0.8

−0.6−0.4

−0.20

−3−2

−10

12

30

5

10

15

20

25

x

Graph of x vs.y vs. z

y

z

Figure 3.7: Geodesic of free particles in the XYZ space

3.4.2 M. Novello’s Interpretation

Following the system of cylindrical coordinates mentioned before, M. Novello and col-

leagues used the geodesics and a newly defined ‘effective potential’ to interpret the pos-

sible motion of the particles. According to them, the partial solutions to the geodesics

equations in cylindrical coordinates are:

φ =

√2A0

cosh2 r− B0

sinh2 r cosh2 r, (3.22)

t = A0

(1 − 2 sinh2 r

cosh2 r

)+

√2B0

cosh2 r, (3.23)

z = C0, (3.24)

r2 = A20 −D2

0 −(√

2A0sinh r

cosh r− B0

sinh r cosh r

)2

, (3.25)

18

where x is the derivative of x with respect to s, A0,B0 and C0 are constants of integration,

and D0 is defined as

D20 = C2

0 +ǫ

a2,

with ǫ = 0 for null curves, and ǫ = 1 for timelike curves. The effective potential is defined

as

V (r) = D20 +

(√2A0

sinh r

cosh r− B0

sinh r cosh r

)2

, (3.26)

which allows Eqn.(3.25) to be expressed as

r2 = A02 − V (r). (3.27)

For simplicity of the later equations, we further define two parameters

γ =B0

A0, β2 =

D02

A02 . (3.28)

In his paper, M. Novello [12] had graphs of V (r) − β2A20 against r. Yet the vertical axis

can actually be expressed as

V (r) − β2A02 = V (r) −D0

2 (3.29)

=

(√2A0

sinh r

cosh r− B0

sinh r cosh r

)2

. (3.30)

There are 3 distinct cases for the behaviour of V (r), namely when B0 < 0, B0 = 0 and

B0 > 0. The third case is of special significance as we see in the following. The minimum

value of γ is given by

γmin =−√

2 +√

1 + β2

2√

1 + β2, (3.31)

which results in

cosh 2r =

√2√

1 + β2. (3.32)

From this, we find that all circular orbits radius r satisfies

r ≤ 1

2cosh−1

√2 < rc,

where rc is the critical radius, corresponding to the maximum radius of the null geodesics,

which agrees with Chandrasekhar.

19

3.4.3 Closed Timelike Geodesics - Chandrasekhar’s method

From the previous section on geodesics by Chandrasekhar, with a coordinates transforma-

tion to cylindrical coordinates, we can come up with an expression

cosh 2r =1

2

(1√α

+√α

), (3.33)

which indeed shows r’s independence of σ. Using the range of values of α in Eqn. (C.29),

we get

1 ≤ cosh 2r ≤√

2. (3.34)

At the maximum value of cosh 2r as denoted by the range above, dΣ2 = 0. This means

that the circular orbit of the maximum radius is the null geodesic. In other words, there

cannot be closed timelike geodesics, which must have a bigger radius than null geodesics.

However, this does not mean that there cannot be closed timelike curves, as they can be

curves other than geodesics.

Alternatively, the non existence of CTGs can be also proven by analysing the acceler-

ation vector of the Closed timelike circle mentioned in the beginning of the section (3.4).

The acceleration vector is given by:

ar =Γr

φφ

gφφ=

cosh r(2 sinh2 r − 1)

2 sinh r(sinh2 r − 1).

If we want to find Closed Timelike Geodesics, the above equation should have some

solution for ar = 0 for dt = dr = dz = 0, in the range of ’r’ where closed timelike curves

exists. As it can be seen ar does not vanish for sinh2 r > 1, hence Timelike Geodesics

cannot exist in the region of closed timelike curves, thus Closed Timelike Geodesics do not

exist. This means that there is no spontaneous time travel in Godel universe. One must

exert force to follow a closed timelike curve.

20

Chapter 4

The van Stockum Solution

“Once confined to fantasy and science fiction, time travel is now simply an

engineering problem.”—Michio Kaku

The van Stockum dust is an exact solution of the Einstein’s Field Equations in which

the gravitational field is generated by an infinite dust cylinder, surrounded by vacuum,

rotating about its axis. We shall be following the chain of thought developed by Steadman

[16] to analyse the van Stockum Solution.

If R is the radius of the cylinder, the whole universe can be divided into two regions,

one inside the cylindrical boundary (r < R) and the other outside it (r > R).

In the van Stockum metric can be shown in the same form as the one used while

deriving the Godel metric. Without going into the formal definitions it can ge given by:

ds2 = −Fdt2 +H(dr2 + dz2) + L dφ2 + 2M dt dφ. (4.1)

Here F , H, L and M are functions of radius only. For r < R , these functions take the

form:

F = 1, H = e−a2r2

, L = r2(1 − a2r2), M = ar2,

where a represents the angular velocity of the cylinder. To find out the simplest closed

timelike curves , we can assume a circle (just as we had done it earlier) with constant r,

t and z. The length (squared) of the tangent vector to this circle is given by the function

L(gφφ) = r2(1 − a2r2) , thus closed timelike curves can exist if and only if r > 1/a .

However this closed timelike curves can be eliminated if the cylinder is such that R < 1/a

. At the region beyond the radius R, the functions take the form:

F =r sinh(γ − β)

R sinh γ, H = e−a2R2

(r/R)−2a2R2

,

L =Rr sinh(3γ + β)

2 sinh 2γ cosh γ, M =

r sinh(γ + β)

sinh 2γ,

21

Figure 4.1: Van Stockum spacetime showing the existence of closed timelike curves due to

the tipping of the cones

such that γ = γ(r) = tanh−1(1 − 4a2R2)1/2 and β = β(r) = (1 − 4a2R2)1/2 ln(r/R).

The exterior domain can be considered in three cases. In case I, 0 < aR < 1/2, the

metric functions are hyperbolic, in case II, aR = 1/2 they are logarithmic and in case III,

aR > 1/2,the functions are trigonometric. It can be found that in case I and II, L cannot

be negative, thus the closed timelike curves does not exist. However, in case III does L

assumes a negative sign and therefore CTC may arise. The van Stockum spacetime is

shown in the following diagram: In fact, it turns out that Closed Timelike Geodesics also

exist for the case III. In order to see this, let us consider the acceleration vector, ar of the

Closed Timelike Circle for aR > 1/2, which is given by

ar =Γr

φφ

gφφ= −

[ea

2R2

(r/R)2a2R2

sin(4γ + β)

2r cos γ sin(3γ + β)

].

If Closed Timelike Geodesics have to be found, the above equation should have some

solution for ar = 0 for dt = dr = dz = 0, giving some timelike geodesics which should

exist in the range of a where closed timelike curves can exist. It is found that there are

infinitely many solutions, which are given by:

rn = Re(nπ−4α) cot α, such that n = 1, 2, 3 . . . (4.2)

This implies that, an observer can travel into his past without applying any force.

22

Thus one can create (and use) a time machine in the Van Stockum universe, for free,

without consuming any energy.

The only problem with this solution is that the universe it proposes is unphysical,

firstly because it applies to an infinitely long cylinder and secondly as it is asymptotically

non-flat, none of which are observed in our universe. However, Tipler [17] argues the case

for the possible occurrence of causality violation near a finite rotating cylinder and Bonnor

[1], has found the exact solution for a massless rotating rod of finite length and it contains

CTCs. Moreover, it appears that the addition of mass would not destroy the CTCs.

Before giving any concluding remarks, a comparison between the Godel and the van

Stockum solutions shall be done. Firstly, the Godel solution gives a homogeneous universe

(with dust), whereas the van Stockum solution produces an infinite cylinder consisting of

dust particles. Secondly, Godel and the van Stockum universes are not asymptotically

flat (the curvature of spacetime does not tend to become zero as radius increases); due

to the infinitely large rotating universe and the infinitely extending cylinder respectively.

Therefore on the basis of the above two properties, our universe (which is homogeneous

and asymptotically flat) is more alike the Godel universe rather than the van Stockum

one. In the Godel universe the CTCs exist throughout the whole universe, however in

the van Stockum case, the CTCs exist only through certain points (for example if the

1/2 < aR < 1 CTCs will not exist inside the cylinder, however they will exist outside).

Though closed timelike curves exist in both the solutions, closed timelike geodesics only

exist in the van Stockum solution.

In spite of this, both the solutions seem to be unphysical, in the case of Godel universe;

it is because it is rotating and van Stockum’s case because it only applies to a universe

with an infinite cylinder.

23

Chapter 5

The Problem of Chronology

Protection

“There is also strong experimental evidence in favor of the conjecture from the

fact that we have not been invaded by hoardes of tourist from the future.”—

Stephen Hawking

5.1 Chronology Protection Conjecture

In 1992, Stephen Hawking published a paper titled “Chronology Protection Conjecture”

[7] which states that the laws of physics do not allow the appearance of closed timelike

curves. Here the basic argument from Hawking’s paper is introduced and then Matt

Veisser’s reiteration of Hawking’s argument is given. After that, the anti-Chronology

Protection Conjecture by Li Xin Li et.al. is given.

5.1.1 Hawking’s Chronology Protection Conjecture

Hawking’s original paper uses the notation of Newman-Penrose formalism and quantities

in his mathematical reasoning. However, due to the technical constraint of this report,

only the gist of Hawking’s argument is summarized below.

Cauchy Horizons are the separation between closed space-like geodesics and closed

time-like geodesics, it is composed of null geodesics segments. The more mathematical

meaning of Cauchy Horizon is given is the appendix.

Weak energy condition requires that an observer moving forward in time will observe

the mass-energy content of the Universe to be non-negative. It is just a statement that

mass and energy are usually positive.

24

Using the Newman-Penrose quantities and arguments using Cauchy Horizons, and the

condition that weak energy condition must be satisfied, Hawking showed that if no closed

timelike curves are present initially, one cannot create them by warping the metric in a local

region with finite loops of cosmic strings. If the weak energy condition is satisfied, closed

timelike curves require either singularities (as in Kerr’s Black Hole) or counterintuitive

behaviour at infinity (as in Godel , van Stockum, and Gott’s spacetime).

Casimir effect is the fact that quantum phenomena of the creation and annihilation

of virtual particles can lead to forces arising out of vacuum. In order words, it allows

effective negative mass energy density to exist; the weak energy condition is violated by

this effect.

Since the weak energy condition can be violated, Hawking’s paper does not rely on it to

support the Chronology Protection Principle. Instead, it relies on the fact that the Stress-

Energy Tensor diverges near the Cauchy horizon due to quantum effects which is better

summarized by Matt Visser in his “The Quantum Physics of Chronology Protection”.

5.1.2 Matt Visser’s “The Quantum Physics of Chronology Protection”

The paper written by Visser [19] summaries Hawking’s reasoning.

A point x is part of the region of chronology violation if there is a closed timelike

curve passing through it. At the edge, it’s a closed null curve that passes through x.

Therefore as the definition of the Cauchy horizon shows, the boundary between chronology

violation region and chronology protection region, which is called Chronology Horizon, is

a special type of Cauchy horizon. The Chronology Protection Conjecture will show that

it is impossible for any object to venture from the chronology protected region to the

chronology violated region.

If an object, like a photon is placed on a compactly generated Cauchy horizon and

follows a closed null geodesic, there will be a boost of its energy, E by a factor of eh

each time the photon completes a round. Where according to Hawking, h is rather like

the surface gravity of a black hole, it measures the rate of the tip over of the null cones

near the closed null geodesics on the Cauchy horizon. As in the black hole case (Hawking

radiation), it gives rise to quantum effects. It is shown that h ≥ 0 must be true, and

in practice, h > 0 then the factor of increasing energy is more than 1, and the energy is

increasing. Since the curve is null, the time experienced by the photon is zero. Therefore

the energy of the photon increases geometrically in effectively zero time.

25

The Stress-Energy Tensor produces a curvature of spacetime that contains a Chronol-

ogy horizon, which in turn increases the gravitational mass-energy of the photon. The

mass of the photon then contributes to the Stress-Energy Tensor, distorting the space-

time around it so that the Chronology horizon disappears, therefore destroying any hope

of crossing the Chronology horizon and travelling back in time. This is what Hawking

calls the divergence of the Stress-Energy Tensor near the Cauchy horizon. The increasing

mass-energy destroys the closed timelike curve.

5.2 Li Xin Li’s Anti-Chronology Protection Conjecture

Hawking’s Chronlogy Protection Conjecture proposes that when quantum mechanical ef-

fects are taken into consideration, some conditions (such as existence of CTCs) allowed by

classical general relativity do not hold. It emphasizes that the laws of physics do not allow

the existence of CTCs, by arguing that the vacuum polarization Stress-Energy Tensor

always diverges on a compactly generated Cauchy horizon, due to which the space-time

geometry at the Cauchy horizon gets thoroughly disturbed. This leads to the destruction

of the Cauchy horizon and the region containing closed timelike curves.

However, Li Xin Li et.al.[10], respond to his claim by suggesting the following insights.

It is not necessary that the existence of closed timelike curves is prevented by the laws

of Physics, on the basis of the Chronlogy Protection Conjecture , because:

1. The Einstein Equation is a local equation and therefore the divergence of the Stress-

Energy Tensor (under the necessary quantum effects) on the Cauchy horizon does

not necessarily imply its divergence in the region containing closed timelike curves.

Thus closed timelike curves would not be destroyed.

2. As Quantum Mechanical effects are being taken into consideration; the background

is no more classical but quantum. Thus it would be appropriate to use a quan-

tum mechanical theory of gravitation, rather than using Quantum Field Theory or

Classical General Relativity.

They suggest that as it we do not have a satisfactory theory of quantum gravitation,

we can resort to Quantum Cosmology, as it the closest possible we can get. Halliwell

and Hartle [6] suggested that the sum over geometries in the path integral approach to

Quantum Cosmology should include complex geometry, so that the wave function of the

universe converges. This suggests that the certain regions of space-time can be defined

26

in Lorentzian geometry and others in complex geometry 1 , when Quantum Cosmological

theory is being used. In the following section we try to exploit this property to generate a

space-time which is Lorentzian in the two regions containing closed spacelike curves and

closed timelike curves and the region connecting them is complex (instead of a Cauchy

horizon). If the Stress-Energy Tensor of this model doesn’t diverge, under quantum effects,

we can claim that the Chronlogy Protection Conjecture proposal is false.

A complex geometry is defined as a real manifold with a complex metric. The complex

metric is a non-degenerate, symmetric, two-rank complex valued tensor field. Let us

consider a complex geometry such that the line element g is defined as:

ds2 = [N(τ)dτ −M(τ) dψ]2 + dψ2 + [dy − y dψ]2 + [dz − z dψ]2, (5.1)

where (τ , y, z) are the Cartesian coordinates and ψ is the azimuth coordinate. More-

over, the functions N(τ) and M(τ) are complex functions of τ , such that N(τ) = dM(τ)dτ

and for convenience we define w ≡M(τ), thus the metric takes the form

ds2 = [dw − w dψ]2 + dψ2 + [dy − y dψ]2 + [dz − z dψ]2. (5.2)

A necessary condition for the constructed geometry, to be allowed by Quantum Cos-

mology, is that the Euclidean Action has to be finite, which is found to true in the given

model (refer to [10] for a rigorous proof). We define the function M and N, so that the

given model allows closed curves (especially CTCs), as:

w(τ) ≡M(τ) = iτ + p(τ);

p(τ) is a real function defined such that

p(τ) = 0; (|τ − τ0| ≥ a), and p(τ) 6= 0; (|τ − τ0| < a).

Also, the τ0 is the point where the null curves exist (ie. τ0 = (1 + y2 + z2)1

2 ) and a is

given by the condition 0 < a < τ0. A closed curve will have length (squared), l2, given by

(keeping w, y, z constant):

l2 ≡ g

[∂

∂ψ,∂

∂ψ

]= 1 + w2 + y2 + z2. (5.3)

l2 =

1 + y2 + z2 − τ2 > 0 for 0 < τ ≤ τ0 − a,

1 + y2 + z2 − τ2 < 0 for τ ≥ τ0 + a,

1 + y2 + z2 + (iτ + p(τ))2 for |τ − τ0| < a.

(5.4)

1complex here does mean ”complicated”, it refers to the complex domain in mathematics

27

From equation (5.4) we can interpret that closed causal curves do not exist in the

region given by 0 < τ ≤ τ0 − a and closed timelike curves exist in the region given by

τ ≥ τ0 and in the region |τ − τ0| the geometry is complex. Our definition of the function

has appropriately given real geometry in the regions containing Closed Spacelike Curves

and the ones containing closed timelike curves the Cauchy horizon have been replaced

by the complex geometry. It can be further shown that the Stress-Energy Tensor for the

above metric takes the form:

Tµν =

1

960π2(gµν) +

1

6π2

∑

n6=0

e2nπ(2e2nπ + 1)

(1 − e2nπ)4l−4Fµν

, (5.5)

Fµν =

−4l−2w2 + 1 3w −4l−2yw −4l−2zw

3w −3l−2 3y 3z

−4l−2yw 3y −4l−2y2 + 1 −4l−2yz

−4l−2zw 3z −4l−2yz −4l−2z2 + 1

. (5.6)

It can be observed that Stress-Energy Tensor converges everywhere and it’s values are

not large (unless ‘a′ is not very small), refer to the [10] for detailed analysis and proof. As

the Stress-Energy Tensor does not diverge at the Cauchy horizon, the region containing

CTCs is not destroyed.

The heart of this solution lies in the fact that the Cauchy horizon, which is the prob-

lematic region, has to be removed (so that the divergence of the Stress-Energy Tensor can

be avoided, thus allowing CTCs to exist). Here, the Cauchy horizon has been replaced

with a complex region, so that the problem of Chronology Protection does not arise. This

work very successfully refutes the claim of the Chronlogy Protection Conjecture by consid-

ering a quantum cosmological theory instead of a quantum field theory ([10]), and shows

that time travel could be possible after all. However, this reply is still not conclusive as it

has not been based on the full quantum theory of gravitation.

28

Chapter 6

Discussion and Conclusion

“Man ... can go up against gravitation in a balloon, and why should he not

hope that ultimately he may be able to stop or accelerate his drift along the

Time-Dimension, or even turn about and travel the other way.”—H.G. Wells

6.1 Discussion

The solutions to the Einstein field equations proposed by Godel, van Stockum, and others

that allow closed timelike curves, along with the Chronology Protection Conjecture and

the Complex geometry solution, summarizes the development of the idea of time travel

throughout the last few decades. Before we move to a conclusion, we would try to discuss

the pros and cons of different moves that have been taken up in the development of the

understanding of the closed timelike curves.

The Godel universe, unlike many other solutions of the Einstein Field Equations, makes

a genuine attempt to describe a world in which it is possible to time travel using the closed

timelike curves. This is so primarily because it is an exact solution of the field equations,

and secondly the Stress-Energy Tensor in this model corresponds to a physically plausible

entity. Many other solutions, such as wormholes (in order to make them transversable),

require exotic matter or energy (negative mass and energy); however these entities do not

resemble any physical substances. Godel universe on the other hand has no need of exotic

matter.

In spite of this, the present experimental data indicate that universe we live in, is not

a Godel universe, because we are observing that the galaxies are receding away from each

other, rather than spinning around each other and that the universe is expanding rather

than being static. Besides that, the solar system with all the planets in their orbits can

be considered as a giant gyroscope and from there we can determine that the galaxies

are not rotating with respect to us. Also, if the universe had some significant amount

29

of rotation, the Cosmic Microwave Background would vary systematically over the sky,

something which we do not observe [5].

However, Godel universe has the unique property of being homogeneous (In fact, the

Godel Rotating solution and the Einstein Static solution are the only spatially homoge-

neous cosmological solutions with non-vanishing density of matter and equidistant world

lines of matter). This property is observed in our universe, and hence they make the Godel

solution more physical as compared to other solutions that allow closed timelike curves.

Similarly, the Van Stockum solution applies to an infinitely long cylinder. And due to

its properties such as non-homogeneity and asymptotical non-flatness which are contrary

to experimental observation of our universe, it is considered unphysical.

The whole scenario of the closed timelike curves changes as soon as the quantum

mechanical effects are taken into consideration. The Chronlogy Protection Conjecture,

proposed by Hawking, makes a bold step to claim that laws of physics do not allow

the CTCs to exist, because the vacuum polarisation Stress-Energy Tensor diverges on

the Cauchy horizon, due to which the region containing the Cauchy horizon and closed

timelike curves will be destroyed.

However, the introduction of quantum mechanical fluctuations into the background

requires that classical general relativity to be replaced by quantum theory of gravitation

which has not been developed satisfactorily yet. Therefore the allegation made by the

Chronlogy Protection Conjecture is premature. Taking advantage of this point, Li and

Xu, propose that quantum theory of cosmology is to be considered; as it is the closest

we can get to an experimentally verifiable understanding of the quantum gravity theory.

They introduce complex geometry in place of the Cauchy horizon, and prove that the

Stress-Energy Tensor will remain well-behaved, in the regions where Chronlogy Protection

Conjecture had predicted a breakdown. Due to this the region containing CTCs is not

destroyed, making time travel possible.

In spite of this successful argument, this solution considering complex geometry is not

perfect, because it deals with quantum cosmology and not quantum gravity. In the regime

of quantum gravity, Li and Xu may have their stand fortified, or contrarily Hawking’s

claim might turn out to be true, no conclusive position can taken because of the lack of a

full theory of quantum gravity and hence we cannot say if time travel is possible or not.

Nevertheless, if we confine ourselves to the realm of classical relativity, the existence

of time travel is allowed; this compels us to take a step back and re-evaluate our under-

standing of time and causality. From Newton’s idea of absolute time, Einstein takes a

30

step forward to present relative time, but it does not stop there. The solutions of the

relativistic field equations, such as Godel’s and van Stockum’s, suggest a very different

moral. Is time really ‘Real’? Godel said “If time travel is possible, then time itself must

be impossible. A past that can be revisited has not really passed”. This view presents the

idea that time is a psychological entity, essentially related to the change of events, rather

than being a physical or a fundamental entity making up the fabric of our universe. Time

is one of those mysteries in the world, which we think we understand but we actually do

not.

Gott and Li, propose a very strange and a unique stand on the time immemorial

question of the first cause of the universe. The presently accepted view is that our universe

began with a big bang, which itself could have arisen from inflation. However, one can

still question as to what the cause of inflation was, and what the cause of this cause of

inflation was, and so on, ad infinitum. Contrastingly, if closed timelike curves existed in

our universe, during the inflationary epoch there would be no question of the first-cause

at all, thus implying that our Universe could be its own mother.

The existence of closed timelike curves implies that time travel is possible; however,

there are two paradoxes that arises due to that. The first one is Grandfather Paradox,

wherein the past can be changed, modifying the conditions that leads to the cessation of

the existence of the entity that changes it. The only way to solve it is to abandon the

notion of free will. The second paradox is the Bootstrap paradox, where the effect is its

own cause, challenging the notion that cause comes before effect.

The physics community has developed four different reactions towards these paradoxes

[19].

1. The “radical re-write” conjecture is just to change our worldview of the universe to

accommodate time travel. These include parallel universes, a present with multiple

past and future, or even multiple versions of a present.

2. “You cannot change recorded history.” That is, even with closed timelike curves

the time traveller cannot change recorded history. This implies that either free will

does not exist or a mysterious force will always make sure history stays the same no

matter what the time traveller do.

3. Hope that quantum gravity can cut in and produce a law that prevents time travel-

ling, e.g. Hawking’s Chronology Protection Principle.

4. “The Boring Physics Conjecture”, that is agree not to think about it until there is

31

experimental evidence. Assume that our Universe is not the ones that can allow

closed timelike curves to exist and be done with it.

Work has been done on modelling simple physical events in time travelling through

wormholes, which is the billiard ball experiment. It seems that there is no work done on

modelling simple time travel scenario on Godel universe to shed light on the nature of

time travel in that universe. The difference of continuous spacetime may shed a new light

in the research on closed timelike curves and to see what form the paradox takes in Godel

Universe.

6.2 Conclusion

Closed timelike curves are allowed in the context of General Relativity. In particular,

the solution to Einstein’s equation, that is the Godel universe has been studied and ex-

plored extensively. There are other solutions that can allow closed timelike curves and

van Stockum solution in particular has been given here. There has been effort to find a

physical law to prevent time travel by Stephen Hawking and there are some rebuttal too,

all pending further understanding in quantum gravity. However numerous further works

can be work on other methods of finding a physical law that can prevent closed timelike

curves or failing that, these works can shed light on how time travel really works.

32

Appendix A

Introductory General Relativity

Remarks on notation The notation used in General Relativity follows the Einstein

notation where the basis can be written in short notations of xµ where µ ranges from 1 to

4 in a four dimensional spacetime. Thus,

x0 = t, x1 = r, x2 = φ, x3 = z, (A.1)

in cylindrical coordinates. Moreover, λµλµ denotes a summation on µ for all the values of

µ. Or more simply,

λµλµ = λ1λ1 + λ2λ2 + λ3λ3 + λ4λ4, (A.2)

if λ is takes values from 1 to 4. Similarly, partial derivatives can be written as

∂xα

∂xβ= ∂βx

α = xα,β . (A.3)

A.1 Tensors

The Special Theory of Relativity describes physical laws based on inertial reference frames.

And it successfully merged the consistency of the speed of light found by Maxwell with

Newton’s law of motion; however Newton’s law of gravitation still conflicts with the con-

sistency of the speed of light. This problem is solved by the General Theory of Relativity.

The theory is based on the equivalence principle that is the observation that the gravita-

tional mass of an object is the same as the inertia mass of the object. Thus there is no

distinction between locally gravitational frames with accelerated frames. The Theory of

General Relativity states that gravity is nothing but curvature of four dimensional space-

time and the distribution of mass-energy bends spacetime, thus producing the familiar

gravity.

33

The General Theory of Relativity has been formulated in such a way that the laws of

physics remain invariant in the in all frames; fundamentally, the theory must be coordinate

independent. Moreover, the theory should also be able to describe how changes in the

observer’s frame affect his perception a physical phenomena around him. Mathematically,

the transformation of coordinates must be defined. Tensor analysis is a good candidate to

formulate the theory. The properties of tensor are described below.

Tensor is an array of numbers in many dimensions relative to a choice of basis. Basis

refers to the coordinate basis. A tensor has ranks; a tensor of rank 0 is a scalar, or a

single number. A tensor of rank 1 is a vector an array of numbers. A tensor of rank

2 is a matrix, or a 2 dimensional array of numbers. The definition extends this to any

number of dimensions. A tensor is usually denoted with subscripts and superscripts. The

sum of subscripts and superscripts is the rank of a tensor. For example, Tµ is a notation

one tensor and T is notation zero tensor. A tensor with rank n + s can be denoted as

Tµ1µ2...µnν1ν2...νs .

A tensor is defined by the coordinate transformation

Tµ′

1...µ′

n

ν′

1...ν′

s= X

µ′

1

µ1....Xµ′

nµnXν1

ν′

1

...Xνs

ν′

sTµ1...µn

ν1...νs, (A.4)

where

Xν′ν =

∂xν′

∂xν. (A.5)

Therefore the mathematics for these requirements exists in the form of tensors, so the

General Theory of Relativity can be expressed in the tensorial form.

A.2 Einstein’s Equation

Rµν − 1

2Rgµν + Λgµν = 8πTµν , (A.6)

The Einstein Equation above is written in tensorial form and it just equates the curvature

of spacetime on the left side with the mass-energy that causes it on the right side. The

terms are:

• Rµν is called the Ricci Tensor, it tells how much the spacetime differs from Euclidean

space, the traditional notion of space before Special Relativity.

34

• R is the Ricci Scalar, it denotes a single real value to every point of spacetime. If

the value is positive, the volume of an object is smaller at that point in spacetime

compared to the volume at Euclidean space and vice-versa. When the value is 0, the

spacetime is flat, there is no change in volume.

• gµν , is called the metric tensor, it captures the geometric and causal structure of

spacetime.

• Tµν , the Stress Energy Tensor which describes completely the distribution of mass,

energy and stress or pressure,

• Λ, an extra term introduced to represent negative gravitation or more positive grav-

itation. Historically Einstein introduced it as a repellent force to balance against

gravity to maintain a static Universe. Currently it is believed that this value for our

Universe is small but non-zero, the force that is responsible for the acceleration of

expansion of the Universe. It is called Dark Energy too, due to our ignorance of its

nature.

[3]

A.2.1 The Metric Tensor, gµν

Euclidean Space is just three dimensional space. Consider a rod with a length l,

the length can be found in terms of a coordinate by Pythagorean theorem, that is l2 =

x2 + y2 + z2, and the infinitesimal form is dl2 = dx2 + dy2 + dz2. In another coordinate

basis, the length dl2 remains the same that is, dl2 = dx2 + dy2 + dz2 = dx′2 + dy′2 + dz′2.

Minkowski Spacetime is the flat spacetime of Special Relativity where time is treated

as another dimension and therefore the line element ds2 is modified to include the time

axis.

ds2 = −dt2 + dx2 + dy2 + dz2 = −dt′2 + dx′2 + dy′2 + dz′2. (A.7)

ds2 is an invariant in spacetime, it does not change with the change of basis. The time

axis is negative to show that it is a special dimension.

The Metric Tensor is a rank 2 symmetric tensor that describes the geometry of space-

time. It can act on two vectors to a scalar number. The notation can be written as

g[xµ, xν ] ≡ gµνxµxν = c0, (A.8)

35

where c0 is some scalar and g is a function on the vectors defined above. In fact, the

metric tensor maps the infinitesimal change in the elements of the basis, dxµ, to produce

a scalar which is called the line element, ds2.

ds2 = gµνdxµdxν . (A.9)

The metric tensor for Minkowski spacetime denoted ηµν is then

gµν = ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (A.10)

Thus it can be seen that the metric tensor represents the form of the line element which

in turns tells the properties of the spacetime. Metric signature used here is (−,+,+,+)

and is just a convention for writing a metric. It comes from the signs used in Minkowski

metric. The opposite signature (+,−,−,−) can be used as well; however, care must be

taken to use only one type of signature, throughout the work. The metric tensor may

change its form when different geometrical structure is considered. For example in non-

flat spacetime the constant 1 is replaced by the functions of the basis, this would produced

curved spacetime.

A.2.2 Causal Structure

Riemannian manifold (manifold) is a mathematical space that resembles Euclidean

space at a small enough scale. In General Relativity, a manifold,M is then a curved

spacetime that can be approximated to Minkowski spacetime locally. Here, locally is a

term for a small enough scale so that the approximation holds and globally is used for big

scales where the approximation fails. A curve in a manifold, M can be said to be timelike,

spacelike, or null depending on the tangent vectors on every point of the curve. Using the

(−,+,+,+) signature, a tangent vector, X is

timelike if g[X,X] < 0, (A.11)

null if g[X,X] = 0, (A.12)

spacelike if g[X,X] > 0. (A.13)

the curve then is timelike, null or spacelike if the tangent vector along it is timelike, null

and spacelike respectively.

36

A.2.3 Geodesics

Christoffel symbols, Γλµν is a correction term for covariant differentiation. This is

required because normal differentiation acting on a tensor will not produce another tensor.

Physically it is because differentiation requires comparison between two points infinitely

close together, however, in curved spacetime, the basis differs from one point to another,

and two points of completely different basis are not comparable. Therefore Christoffel

symbol is required as a correction term for covariant differentiation and the formula for

Christoffel symbol in terms of the metric tensor is:

Γλµν =

1

2gλγ(∂µgνγ + ∂νgµγ − ∂γgµν),

A Geodesic is the shortest path between two points in a manifold. It describes the motion

of a free particle under the influence of the curvature of spacetime, without any influence

of external forces. The equations for geodesics are

d2xλ

dt2= Γλ

µν

dxµ

dt

dxν

dt. (A.14)

This is analogous to the Newtonian F = ma = 0 with d2xλ

dt2≡ a and Γλ

µνdxµ

dtdxν

dt is the

correction term for the curved spacetime.

Ricci Tensor Rµν and Ricci Scalar R The equations for the two components are

given below:

Rµν = ∂ρΓρµν − ∂µΓρ

ρν + ΓαµνΓ

ραρ − Γα

ρνΓρµα, (A.15)

R = Rµνgµν , (A.16)

where ρ and α are dummy indexes summing from 1 to 4.

A.2.4 Stress-Energy Tensor, Tµν

The next part of the Einstein Field Equation is the stress energy tensor, it corresponds

to the source of the gravitational field in general relativity. It is analogous to mass in

Newtonian gravity. It is a 4 × 4, rank 2 tensor describing the density and flux of energy

and momentum in spacetime.

Tµν =

T00 T01 T02 T03

T10 T11 T12 T13

T20 T21 T22 T23

T30 T31 T32 T33

(A.17)

37

T00 ≡ energy density,

Ti0 ≡ momentum density,

T0i ≡ energy flux,

Tij ≡ momentum flux.

Tµν is a function of position and time, therefore, the Stress-Energy Tensor can model any

distribution of mass-energy. To simplify the term, the universe is usually represented by

dusts or fluids where a dust particle can be understood as a galaxy. The Stress-Energy

Tensor for a perfect fluid can then be written as

Tµν = (ρ+ p)VµVν + pηµν (A.18)

For further understanding of General Relativity, for basic understanding, [15] and [3]

must be reffered to.

38

Appendix B

Solving the Line Element

In the following equations, the elements required to solve the Einstein’s equation, namely

the Christoffel symbols, the Ricci tensors, Ricci scalar and the Stress-Energy Tensor are

defined and listed. The metric that is being considered is represented as the following in

the (t, x, y, z) coordinates:

ds2 = −dt2 + dx2 + dz2 + u(x)dy2 + 2v(x)dtdy, (B.1)

The metric tensor and its inverse are:

gµν =

−1 0 v 0

0 1 0 0

v 0 2u 0

0 0 0 1

, gµν =

− uv2+u

0 vv2+u

0

0 1 0 0

vv2+u

0 1v2+u

0

0 0 0 1

.

where u = u(x) and v = v(x). Using the definition

Γαµν =

1

2gαγ(∂µgνγ + ∂νgµγ − ∂γgµν),

where ∂µ denotes partial differentiation with respect to µ. All the non-zero Christoffel

symbols are

Γ001 =

vv

2(v2 + u), Γ2

01 =v

2(v2 + u), Γ1

02 = − v2,

Γ012 =

vu− uv

2(v2 + u), Γ2

12 =vv + u

2(v2 + u), Γ1

22 = − u2,

where the dot represent differentiation with respect to the basis x.

Now to find the Ricci Tensor, the following relation is used:

Rµρ = ∂νΓνµρ − ∂µΓν

νρ + ΓαµρΓ

ναν − Γα

νρΓνµα.

39

After some involved subsitutions and simplifications, the non-zero Ricci tensors are

R00 =v2

2(v2 + u), R11 =

2(v2 − u)v2 − 4(v2 + u)vv + 4vuv − 2(v2 + u)u+ u2

4(v2 + u)2,

R22 = − u2

+u2 − 2uv2 + 2vvu

4(v2 + u), R02 = − v

2+

uv

4(v2 + u).

The Ricci Scalar is

R = Rµρgµρ =

2u2 + 8vuv − 4(v2 + u)u− 8vv(v2 + u) + 2v2(v2 − 3u)

4(v2 + u)2.

The stress-energy tensor is in the dust form; such that Tµν = ρVµVν but since the

choice of Vµ, the basis system is arbitary and does not affect the physical meaning, the

simple choice of

V µ =

1

0

0

0

, Vν = gµνV

µ =

−1

0

v

0

,

means that the coordinate choosen is co-moving with the system. The physical meaning of

co-moving is that the observer sees the universe as isotropic, or the same in all directions.

However, since the Godel’s universe is not isotropic, it can be seen as isotropic with respect

to the plane perpendicular to the axis of rotation.

Using the convention above, the form of Tµν is

Tµν =

ρ 0 −ρv 0

0 0 0 0

−ρv 0 ρv2 0

0 0 0 0

.

Looking at the Einstein’s equation, there’s one equation that can simplify the calculations,

G33 = R33 −1

2Rg33 = 8πρV3V3 − Λg33,

R = 2Λ.

Applying the results above in Einstein’s equation it is found that,Rµν = 8πρVµV ν, finally

giving us:

R00 = 8πρ,R11 = 0, R02 = −8πρv, R22 = 8πρv2.

Therefore we have four equations containing two unknowns of the function of x and their

derivative with respect to x up to the second order.

40

The rest of the derivation is done just using some mathematical tricks, which will be

outlined here for a mathematically inclined reader. Using the subsitution of

h = v2 + u, (B.2)

h = 2vv + u, h2 = 4v2v2 + 4vvu+ u2,

h = u+ 2(v2 + vv),

in all the Ricci Tensor equations

R11 =2(v2 − u)v2 − 4(v2 + u)vv + 4vuv − 2(v2 + u)u+ u2

4(v2 + u)2= 0, (B.3)

and then we get,

h− h2

2h= v2,

R00 =v2

2(v2 + u)= 8πρ,

v2

2h= 8πρ,

v2 = 16hπρ,

h− h2

2h= 16hπρ. (B.4)

Now to solve equation B.4, the following equations are noted and substituted, to simplify

notations, let a = 16πρ,

d(h−1

2 )

dx= −1

2h−

3

2 h,

− h2

2h= h

1

2 hd(h−

1

2 )

dx,

h+ h1

2 hd(h−

1

2 )

dx= ah,

d(h−1

2 h)

dx= ah

1

2 .

Let

q = h1

2 ,

Thus, (B.5)

q =aq

2.

41

The general solution for the last equation is

q = be√

a2x + ce−

√a2x,

h = (be√

a2x + ce−

√a2x)2,

v = ±√ah,

∫v dx = ±

∫ √a(be

√a2x + ce−

√a2x) dx,

v = ±√

2(be√

a2x − ce−

√a2x) + d. (B.6)

Equation B.6 is the solution pool for the Godel’s type metric. Let f =√

a2 , and b, c, d be

intergration constants. To obtain the exact form that Godel presented in his paper, the

following conditions are fixed.

v ≤ 0, b =1√2, c = d = 0,

v = −efx,

h =1

2e2fx,

u = h− v2,

= −1

2e2fx.

The line element is finally takes the form,

ds2 = −dt2 + dx2 + dz2 − 1

2e2fxdy2 − 2efxdydt.

42

Appendix C

Geodesics

Equation (3.17) can be integrated simply:

u3 = C and x3 = Cs+ c3 (C.1)

where C and c3 are constants. This implies that x3, which is the z axis, can be either

constant (i.e. when C = 0, x3 = c3) or linearly dependent on s. Equation (3.12) can be

rearranged into

1 = (u0 + ex1u2)2 − (u1)

2 − 1

2e2x1(u2)

2 − (u3)2 (C.2)

The sum of u1 × Eqn(3.15) and e2x1u2/2 × Eqn(3.16) is

u1du1

ds+

1

2e2x1

dx1

ds(u2)

2 +1

2e2x1u2

du2

ds(C.3)

Integrating this with respect to s,

(u1)2 +

1

2e2x1(u2)

2 = B2 = constant (C.4)

Defining a new constant D, equation (C.1), with equation (C.2) and (C.4) can be rewritten

in the form

(u0 + ex1u2)2 = 1 +B2 + C2 =

1

2D2 (C.5)

From equations (C.4) and (C.5), we derive

u2 =√

2e−x1

√B2 − (u1)2 (C.6)

and u0 =D√2− ex1u2 =

1√2

{D − 2

√B2 − (u1)2

}(C.7)

To resolve the ambuigities in the signs of the constants from the square roots of the

equations, we require D to be always positive, and B to be positive or negative. From

(C.5), since C2 must be positive,then 12D

2 ≥ B2 + 1. This implies that

D >√

2|B| (C.8)

43

With equation (C.6) and (C.7), equation (3.15) becomes

du1

ds= B2 − (u1)

2 −D√B2 − (u1)2 (C.9)

Since from (C.4), (u1)2 < B2, we can write the following relation:

u1 = B sin θ. (C.10)

Eqn (C.9) becomes

dθ

ds= (D −B cos θ), (C.11)

(C.12)

Integrating Eqn(C.11) and considering D > |B|

s− s0 = − 2√D2 −B2

tan−1

{√D +B

D −Btan

1

2θ

}, (C.13)

,where s0 is a constant. We define new symbols σ and α as

s− s0 = − 2√D2 −B2

σ and α =D −B

D +B(C.14)

⇒ tan1

2θ = −

√α tanσ (C.15)

Using basic trigonometric identities, we can see that

cos θ =1 − α tan2 σ

1 + α tan2 σand sin θ = −2

√α

tanσ

1 + α tan2 σ(C.16)

Using Eqn(C.15),(C.16), Eqn (C.10) becomes

dx1

dσ= u1

ds

dσ= − 4B

D +B

tanσ

1 + α tan2 σ(C.17)

which can be integrated to give

x1 = − 2B

(α− 1)(D +B)log

(1 + α tan2 σ

sec2 σ

)+ c1 (C.18)

The choice of α causes the coefficient of the logarithmic term to become 1, so the term

become

x1 = log(cos2 σ + α sin2 σ

)+ c1 (C.19)

or

ex1 = ec(cos2 σ + α sin2 σ) (C.20)

44

Equation (C.5), with the aid of (C.10) and (C.16) can be rewritten as

dx0

dσ=

√2

D2 −B2(D − 2B cos θ) (C.21)

= −√

2D2

D2 −B2+ 2

√2α

sec2 σ

1 + α tan2 σ(C.22)

Integrating the previous equation yields

x0 = −√

2D2

D2 −B2σ + 2

√2 tan−1(

√α tanσ) + c0 (C.23)

Rewriting the equation (C.6), using (C.16) and (C.19),

dx2

dσ= 2

√2B2

D2 −B2e−x1 cos θ (C.24)

= 2e−c

√2B2

D2 −B2

(1 − α tan2 σ) sec2 σ

(1 + α tan2 σ)2(C.25)

, which can be integrated to yield

x2 = 2e−c1

√2B2

D2 −B2

tanσ

1 + α tan2 σ+ c2 (C.26)

45

C.1 Closed Timelike Geodesics - Chandrasekhar’s method

From the previous section on geodesics, we know that

D2 = 2(B2 + C2 + 1), (C.27)

which results in

α =D −B

D +B=

B2 + 2C2 + 2(√

2(B2 + C2 + 1) +B)2 . (C.28)

With careful use of limits, the range of values of α can be shown to be therefore

(√

2 + 1)2 ≥ α ≥ (√

2 − 1)2 = 0.1716. (C.29)

The metric proposed shows rotational symmetry. A series of coordinates transformation

was introduced by Godel to transform into a system of cylindrical coordinates, resulting

in the alternate form of the metric as mentioned above. Using the transformations, we

obtain

2ex1 cosh 2r = 1 + e2x1 +1

2x2

2e2x1 . (C.30)

From previously found results in the section on geodesic solutions, we find that

x2ex1

√2

=B√

D2 −B2sin 2σ =

1 − α

2√α

sin 2σ. (C.31)

And from the two equations we just derived, we can derive

cosh 2r =1 + e2c1(cos2σ + α sin2 σ)2 + [(α− 1)2/4α] sin2 2σ

2ec1(cos2 σ + α sin2 σ). (C.32)

Assuming r is independent of σ and comparing values, we get e2c1 = α−1, which reduces

the previous equation to

cosh 2r =1

2

(1√α

+√α

), (C.33)

which indeed shows r’s independence of σ. Using the range of values of α in Eqn (C.29),

we get

1 ≤ cosh 2r ≤√

2. (C.34)

Consider the line element mentioned in the previous sections, keeping r and z constant

(i.e. dr2 = 0 and dz2 = 0),

dΣ2 = dt2 + (sinh4 r − sinh2 r)dφ2 + 2√

2 sinh2 rdφdt, (C.35)

which can be reduced to

dΣ2 = (1 − 1

2cosh2 2r)dσ2. (C.36)

At the maximum value of cosh 2r as denoted by the range above, dΣ2 = 0.

46

Appendix D

Definitions for the Problem of

Chronology Protection

These definitions have been adapted from Townsend’s paper [18].

D.1 Cauchy Horizon

Partial Cauchy Surface denoted S, is a surface where no causal curve intersects more

than once. Intuitively, a Cauchy surface is a plane in space-time which is like an instant

of time; its significance is that giving the initial conditions on this plane determines the

future (and the past) uniquely.

Past-inextendable Causual Curve is a timelike worldline that has no past end-point

in a spacetime M. In other words, it’s a timelike world line that has no beginning.

Future Cauchy Development denoted D+(S) is the set of points p ∈ M where the

past-inextendable causual curve through p intersects S. Or more simply, it’s the future

causual influence region of S.

Past Cauchy Domain of Dependance denoted D−(S) is the set of points q ∈ M

where the past inextendable casual curve through q intersects S in its future. So it is the

past causual influence region of S.

Cauchy Surface for the spacetime M is the surface S when the condition D+(S) ∪D−(S) = M is statisfied. The spacetime M is said to be golbally hyperbolic. In such a

spacetime, there is no Cauchy Horizon.

47

Cauchy Horizon is then the boundary ofD+(S) orD−(S) in M where M is not globally

hyperbolic. Motions of particles beyond the Cauchy Horizons cannot be predicted. Future

Cauchy Horizons are denoted H+(S) and Past Cauchy Horizons are denoted H−(S).

D.2 Quantum Effects

This section will attempt to describe some of the quantum effects assumed in the Chronol-

ogy Protection Principle.

Heisenberg uncertainty principle states that a particle cannot have a definite mo-

mentum, p and position, q at the same time. This principle can be extended to energy, E

and time, t.

∆p∆q ≥ 1

2~, ∆E∆t ≥ 1

2~, (D.1)

where ~ = 1.05457148 × 10−34kgm2s−1. Physically it means that a particle can have a

energy, ∆E for a time of ∆t, the shorter the particle exist, the more uncertain its energy

is. It also allows a pair of particle-antiparticle to come out of the vacuum for a short time

before annihilating themselves again. Therefore, the vacuum has a sea of virtual particles

arising and falling constantly.

Casimir Effect is usually described using an example. When two hypothetical metal

plates are placed side by side in a vacuum, with a spacing, d between them, the types of

vitual particles that can arise in between the plates is less than the number that can exist

outside it. Therefore there exist a force that pushes the two metal plates together due to

the difference in vacuum pressure. However, since the pressure around the metal plates

is zero, the presure in between the plates has to be less than zero. Thus it could be said

that negative mass-energy is allowed by Casimir effect.

Black holes are singularities that curves spacetime so much that the escape velocity

beyond a certain surface can be greater than the speed of light, thus trapping everything

that falls into it. The surface of a black hole is called the event horizon.

Hawking Radiation is the radiation that seemingly comes out of a black hole. When

a pair of virtual particle-antiparticle is formed too close to the event horizon of the black

hole, thus before they have time to annihilate, one of the pair is sucked into the event

horizon and the other is shooted out.

48

Appendix E

Acknowledgements

Firstly, we would like to thank Special Program in Science (SPS) for giving us an op-

portunity to study and research, due to which we were able to enlarge our horizons of

knowledge. We would like to thank and express our gratitude to our mentors, Lim Yen

Kheng and Tran Chieu Minh, for the time and effort they have put to guide us in this long

and perilous path. Our special thanks to Yang Tzyh Haur for his guidance and support.

E.1 Acknowledgement for Diagrams

• Fig.(2.1)- Modified from http://www.fourmilab.ch/documents/gtpp/

• Fig.(3.1) - Reference[13]

• Fig.(3.2) - Reference[14]

• Fig.(3.4) - Reference[13]

• Fig.(4) - Reference[11]

49

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