Sustaining Consistency in Causal Cycles of Time

7/29/2019 Sustaining Consistency in Causal Cycles of Time

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SPECIAL PROGRAM IN

SCIENCE

Faculty of Science

Sustaining Consistency inCausal Cycles of Time

AUTHORS:

Han Weiding

Jani Hariom Kirit

Ng Xin Zhao

STUDENT MENTORS:

Tran Chieu Minh

Lim Yen Kheng

Thong May Han

STAFF MENTOR:

A/Prof. Edward Teo

November 2, 2010


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Abstract

One of the startling features of the theory of General Relativity is that it permits the

existence of Closed Timelike Curves (CTCs) in special scenarios; these curves allow

time travel. Many paradoxical situations can arise due to causality violation when

time travel is allowed. Consequently, it becomes important to question, whether or

not these curves can exist logically (according to I. D. Novikovs definition), as it

may help us in verifying the possible existence of time travel. The central theme

of our project is to analyze Novikovs hypothesis in the van Stockum Spacetime.

Firstly, the trajectories of the CTCs were determined using numerical integration of

the equations of motion, following which the test of self interaction was performed

(by considering the collision of a billiard ball with its past self). It was found that thelocal causality is coherent with global causality, after collisions involving a particle

traversing a Closed Timelike Geodesic (which is a specific type of a CTC); indicating

that the closed timelike geodesics are self consistent. Hence we conclude that closed

timelike geodesics can exist logically, consequently allowing room for time travel to

be possible in curved space-time.


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To,

Those who believe in the existence of Time Travel...

1


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Contents

1 Introduction 4

2 An Overview 6

2.1 Self Consistency and the Billiard Ball Problem . . . . . . . . . . . . . 6

2.2 The van Stockum Universe . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Closed Timelike Geodesics 11

3.1 The Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Tra jectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 The Simple Circles and the Helices . . . . . . . . . . . . . . . 13

3.2.2 The Fancy Circles . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Runge Kutta Method - Timelike and Geodesic . . . . . . . . . 16

3.3.2 Graph Analysis - Closure . . . . . . . . . . . . . . . . . . . . . 16

3.4 Estimation and Error Analysis . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Theoretical Error . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.2 Program Error . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Self Consistency 24

4.1 Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Non-Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Conclusion 31

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3

6 Future Work 32

7 Acknowledgements 33

A The Four Momentum and the Relationship between E and 34

B C codes 36

B.1 The main C code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

B.2 The graphical method C code . . . . . . . . . . . . . . . . . . . . . . 45

B.3 The code to find the Non-smooth Fancy Circles . . . . . . . . . . . . 48

B.4 Codes to find Simple Circle . . . . . . . . . . . . . . . . . . . . . . . 53

C Other codes 55

C.1 Matlab Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C.1.1 Curve Generator . . . . . . . . . . . . . . . . . . . . . . . . . 55

C.1.2 Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . 55

C.2 Mathematica codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56


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Chapter 1

Introduction

If time travel were possible, wed be inundated with tourists from the

future.

Stephen Hawking

Time travel, a counter intuitive concept in itself, is forbidden for particles with

mass in the realm of Newtonian and Special Relativistic physics, as the former

assumes an absolute view of time and the latter demands that an objects travel underthe speed of light. However, on the invocation of the theory of General Relativity, it

is found that time travel becomes admissible. The transition from Special to General

Relativity replaces the need for superluminal travel with a need for special curvature

in the spacetime geometry. The latter unlike the former is theoretically achievable,

as curvature of spacetime can be produced by having a decently huge gravitating

body. The physical realisation of time travel happens when an object travels on

a path known as a closed timelike curve (hence forth referred as CTC). As the

name suggests, these curves satisfy the property of being firstly timelike, indicating

that they are traversable by objects with mass, and secondly closed in space and

time. This closure results in backward time travel. Furthermore, it must be noted

that existence of time travel in General Relativity (GR) does not violate Special

Relativity (SR) in any sense, because at the local scale GR reduces to SR. The most

famous curved spacetimes in which time travel is allowed, are external Kerr Black

Holes, Wormholes, Cosmic Strings, Godel [2] and the van Stockum universe [7].

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CHAPTER 1. INTRODUCTION 5

In general, many physicists claim that the existence of CTCs is pathological, as

the introduction of time travel brings about many paradoxes and it disturbs our

understanding of causality. Furthermore, it is sometimes alleged that due to the

existence of time travel in GR, GR shouldnt be trusted very much. It thereforebecomes crucial to question whether or not CTCs can exist; and one of the ways to

do this is to test if CTCs can exist logically. This test will based upon whether

there exist any consistent solutions for a particular interaction, involving an object

which is time travelling.

In our work we, aiming to resolve the paradoxes revolving around time travel,

try to check the logical existence of CTCs. This is done by studying the collision

of a ball with its past self1

. If at least one case, for a CTC, is found such thatthe solution of a collision is self consistent, it can be said that there is no obvious

contradiction in the logical existence of that CTC2. The central idea of our work,

derives its inspiration from the famous Billiard Ball Problem [1]. In the billiard

ball problem, newtonian self collisions of a billiard ball are considered; where the

authors use Wormholes as the time machines. In contrast in our work we shall

consider relativistic collisions, using the van Stockum spacetime as the background

universe. This spacetime is of particular interest, as it contains closed timelike

geodesics, unlike the Godel universe.

In Chapter 2, the principle of self-consistency and van Stockum universe are

explored. The trajectories of the closed timelike geodesics are needed to study the

self consistency problem, thus Chapter 3 discusses the equations of motion and

presents some methods by which the closed timelike geodesics can be determined.

In Chapter 4, the test of self consistency is performed and a discussion on the logical

existence of time travel is given. Finally the conclusion and possible future works

are presented.

1Which is similar to a person going back in time and killing oneself.2This argument, in general, can be applied to all CTCs as they differ only in their parameters.


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Chapter 2

An Overview

The notion that one version of time travel is more accurate than

another is ridiculous - except to a physicist.

Dave Goldberg

This section is aimed at shedding light on the self consistency principle and some

of the previous work done in the van Stockum spacetime.

2.1 Self Consistency and the Billiard Ball Prob-

lem

Time travel is all very nice and smooth, until causality violation issues are brought

about. Famous time travel paradoxes from the science fiction stories are the prob-

lems that threaten the logicality of time travel. One story goes that a time traveller

goes back in time and kills his younger self. Consequently, the younger self would

not be able grow up and time travel; but if he did not time travel, then there is no

reason why the younger self could not have grown up and done time travelling. If he

could have time travelled, then he could have gone killed his younger self, and so on.

So, was the younger self killed? Or more importantly, did the person ever travel

back in time? These questions cannot be answered as there is a paradoxical loop

in the story. Many physicists and philosophers exploit this argument and declare

that this paradoxical behaviour of time travel shows that it is absurd and therefore,

impossible.

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CHAPTER 2. AN OVERVIEW 7

In the above problem, although causality holds in the localchain of events, it does

not globally. Therefore the problem fundamentally arises due to this inconsistency

between the local and the global picture of the universe. With an aim to resolve

this problem, six theoretical physicists [1] in 1990 came up with the self consistencyprinciple. The principle claims that the only solutions to the laws of physics that

can occur locally in the Universe, are those which are globally self consistent. One

of the authors - Novikov [4], argues that if there is a set of conditions (on a CTC)

resulting in an inconsistency, it is always possible to find some conditions that give

a self consistent solution. By assuming the self consistency principle, he adds that

nature will abhor the conditions that produce inconsistencies. Hence, events on a

CTC are guaranteed to be self consistent, therefore making it possible to claim thatCTCs can exist logically. Here logical existence is defined in the following way:

If at least one self consistent outcome for an interaction occurring on

a curve can be found, it can be said that there is no obvious

contradiction against its logical existence. Such a curve is defined to

be logically allowable.

Taking inspiration from the billiard problem we shall try to study the self-

collision of a billiard ball in the van Stockum universe. Moreover, we shall try

to see whether Novikovs claim holds in the van Stockum spacetime. It should be

stressed that the content of our work is original because it replaces the ironical

non-relativistic collisions in Wormholes (in the Billiard Ball paper, [1]), with more

realistic collisions in van Stockum universe with a relativistic treatment. In order

to work out the collision problem, the equations of motion (EOM) for a particle on

CTCs have to be found. Thus, the first part of this paper shall deal with a study

of the equations and properties of CTCs in the van Stockum spacetime. Eventually

when the curves have been obtained, we shall turn to the self collision problem and

see what results ensue.

2.2 The van Stockum Universe

The van Stockum spacetime [7], named after its discoverer Willem Jacob van Stockum

, was rediscovered in 1937, independent of an earlier discovery by Cornelius Lanc-


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zos in 1924. It is the first solution to the Einstein Field Equations that permits

CTCs and therefore time travel. The universe consists of an infinitely long, massive

cylinder (made of dust particles) which is rotating about its longitudinal axis. Due

to the rotation of the cylinder, the spacetime curves in a way such that the lightcones start tilting. As one approaches a critical distance the light cones tip into

the domain of negative time (Refer to figure 2.1.). One can possibly follow a path

which goes into the past (at non-luminal speeds), which eventually comes back to

the present1. Such curves are called closed timelike curves, more specifically, if the

above mentioned curves do not require the traveller to accelerate, they are called

closed timelike geodesics (CTGs). As shown by B. R. Steadman [5], CTGs actually

do exist in van Stockum spacetime (our work shall principally focus on CTGs). Thefollowing description of the van Stockum spacetime follows from Tiplers work [6].

The description of the geometry in the van Stockum spacetime is generally done

using Weyl-Papapetrou form (which involves cylindrical polar coordinates along with

time). The metric of the spacetime, in a frame of reference which is rotating at the

same angular velocity as that of the cylinder, using natural units G = c = 1 is then

given by:

ds2 = F(r) dt2 + H(r) (dr2 + dz2) + L(r) d2 + 2M(r) d dt, (2.1)

where t is the time, r the radius, the angle, and z the distance along the axial

coordinate ( < t < , 0 r < , 0 2, < z < ) withG = c = 1.

Ifa is the angular velocity and R the radius of the cylinder, the whole universe can

be divided into two domains, one inside the cylinder (r < R) and the other outside

it (r > R). The exterior solution can further be divided into three categories, when

0 < aR < 1/2, aR = 1/2 and 1/2 < aR < 1 (the upper limit is 1 because it is equal

to the speed of light in our earlier definition). It was seen that the CTCs exist only

in the third case. The functions H, L, M, F in this region assume the following

form:

1The reason why past and present have been put in apostrophes is because such clasifications

hold very little meaning globally when time travel is permitted.


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F(r) =r sin( log(r/R)tan )

R sin , L(r) =

rR sin(3 + log(r/R)tan )

sin + sin 3, (2.2)

H(r) = exp(a2R2)(r/R)2a2R2, M(r) = r sin( + log(r/R)tan )sin2

, (2.3)

where in

tan = (4a2R2 1)1/2, 1/2 < aR < 1.

The coordinate condition LF + M2 = r2 is present. A justification for the

above form of the metric can be given here. The functions H, L, M, F play a role

in determining the curvature of the universe. The elements of the metric tensor

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

is modified only with change in r; secondly, there shall be constants of motion

along the other three coordinates (due to their homogeneity). The time t and the

angular direction are coupled with each other, this generates a rotational effect

as the angle would change with time. Moreover this combination results in frame

dragging effects which are suitable enough for generating the CTCs.

In our analysis the z coordinate shall be suppressed for reasons that shall become

evident later. The energy momentum tensor is given by pressure free fluid, which is

nothing but dust particles. The van Stockum spacetime is shown in figure 2.1.

The characterization of the curves is done according to the following manner, if

the value of the metric is positive, zero or negative, then the curve corresponding

to those set of points by definition is spacelike, null or timelike2 respectively. In

order to visualize a simple examples of a CTC, let us consider a circular path in

the spacetime; this curve will satisfy dr = 0, dt = 0 and dz = 0 with d = 0,making the metric (2.1) depend only on the function L. From its definition L is a

sinusoidal, and can thus take both positive and negative values. Those circles that

have a radius for which L becomes negative, are CTCs3.

In order to find the equations of the CTGs, the equations of motion have to be

solved. But unfortunately as the functions of the metric are very complicated, it

2Spacelike trajectories can be traversed by the particles travelling at superluminal velocities,

null by photons and timelike by particles with mass at velocities less than that of light.3Some of the examples mentioned by B. R. Steadman in [5] are 7 .0 103, 2.0 109 units etc.


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Figure 2.1: Van Stockum spacetime showing the existence of closed timelike curve

due to the tipping of the cones

is difficult to solve these equations analytically, therefore a numerical approach is

taken. The fourth order Runge-Kutta method is used to integrate the equations of

motions and the a graphical approach is used to find curves that are closed. In the

following chapter, the above processes shall be discussed in detail.


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Chapter 3

Closed Timelike Geodesics

Time hasnt stopped for any troubles, heartaches, or any other

malfunctions of this world, so please dont tell me it will stop for you.

C.S. Lewis

To test the self consistency principle the CTGs must be found. In order to

find the CTGs, the equations of motion (EOM) have to be solved. In the first

portion of this chapter, the EOM are obtained by casting the Lagrangian of the van

Stockum metric into the Euler-Lagrange equations. Then the EOM are numerically

integrated (Using the fourth order Runge Kutta algorithm) to find the trajectories of

the timelike geodesics. Moreover, the closure condition is imposed using a graphical

analysis method. Once the appropriate parameters satisfying the closure condition

are found, a CTG can be uniquely determined.

3.1 The Equations of motion

In the case of curved geometry, the Lagrangian L Lagrangian can be naively definedas the difference between the kinetic and the potential energies of a system. is

covariantly stated as:

L =4

, =1

gxx, (3.1)

where x represents the total differential of the x coordinate, with respect to

an affine parameter , which is the proper time in our case. Using the definition of

the metric (2.1), the Lagrangian becomes:

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CHAPTER 3. CLOSED TIMELIKE GEODESICS 12

2L = 2

ds

d

2= Ft2 + H(r2 + z2) + L2 + 2Mt. (3.2)

The EOM are then obtained by substituting the L into the Euler-Lagrange equa-tions:

d

d

Lx

=Lx

.

Thus the above equation will give us four equations, one for each of the co-

ordinate. However, as the , z and t coordinates are not explicitly present in the

Lagrangian, three constants of motions are obtained, one along a coordinate (except

along r) each as shown below.

Lr

= Pr = Hr = constant, (3.3)L

= = L + Mt = constant, (3.4)

Lz

= Pz = Hz = constant, (3.5)

Lt = E = M Ft = constant. (3.6)

This means that the momentum along the () and the z (Pz) coordinate, and

the energy (E) will be conserved. In our work we shall suppress the z coordinate by

demanding that the change in z coordinate be zero (as mentioned earlier in section

2.2). This must be done because any change in z will not result in CTGs. Since

Pz is conserved and the increase in will not change the sign of H, the sign of z

cannot change as well, thus dooming z to always increase or decrease indefinitely.

Hence closure can never occur in z. One may think that the above argument can

be applied to r too, and argue that closure cannot occur in the radial direction.

Therefore dismissing any possibility of closure. However as Pr is not conserved, the

above argument does not hold for r. Hence it could be possible to find closure.

The E and 1 can be solved to get the EOM along the and the t axes. These

1The E and are to treated as if they are independent for the rest of the paper. However

strictly speaking they are related by the 4-momentum invariance refer to appendix


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are given by:

=ME+ F

r2, (3.7)

t =M LE

r2, (3.8)

z = 0. (3.9)

As there is no homogeneity along r, momentum is not conserved along that

direction. Therefore the EOM along r can be only obtained from the Euler-Lagrange

equation and it takes the following form:

r4 (2Hr + H

r2) 2F(F L + r2) + 2MM(LE2 F2) + LE2(F L + r2) =2ME(2M2 + r2) + LF(F + 2ME) + FLE(2M LE) 4F2r + 4LE2r 8MEr,

where the prime represents the differentiation with respect to r. The equation

for r is highly nontrivial and very involved. Fortunately this equation can be avoided

by invoking the constraint that the curve be timelike. This constraint will result

in a first order differential, which can be found by demanding that (ds/d)2 from

equation (3.2) assume a negative value, (which is 1 for timelike curves). Using therelations (3.2), (3.7), (3.8), (3.9), the equation for r, becomes:

r2Hr2 = LE2 F2 2ME 2r2. (3.10)

The above four equations (3.7), (3.8), (3.9) and (3.10) completely describe the

motion of a particle in the van Stockum spacetime.

3.2 Trajectories

3.2.1 The Simple Circles and the Helices

Before we begin to analyze the general EOM, we shall first consider a specific case

of a circle (similar to the qualitative analysis in section 2.2). The acceleration along

r, for a Closed Timelike Circle (which is characterized by t = r = z = 0) is given by

r = r4

2

L

L2. (3.11)


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A particle can remain on this Closed Timelike Circular geodesic only if there exist

solutions for r = 0. It is found that there are infinitely many nontrivial solutions

which occur for L = 0. When solved for r, using the definition of L from (2.2), the

above condition results in:

rk = R exp [2(k 2)cot ], such that k N. (3.12)

Figure 3.1: The Simple Circle, E = 393.548, = 100, a =

5/4, R = 1, k = 1

This example, as mentioned earlier, is a circular geodesic which exists in a plane

of constant time. Such curves seem very bizarre and physically unrealistic. Therefore

they can be called a Simple Circles. These circles are highly unstable; if the particle

travelling on them slightly deviates from the ideal curve they fall into a helix (it is a

timelike geodesic which is not closed). The above statement is equivalent to saying

that for non integral values of k (or k + N) in equation (3.12), the r willcorrespond to helical trajectories. A particle travelling on them can move only

unidirectionally in time, either continuously upwards or downwards. Objects move

forward or backward in time according to the following conditions:

t > 0; rk=even < r < rk=odd, (3.13)

t < 0; rk=odd < r < rk=even. (3.14)


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Figure 3.2: The Helix, E = 393.548, = 100, a =

5/4, R = 1, k = 0.99

A detailed discussion of the above two cases has been done in [5] by B. R.

Steadman. The diagrams for the two cases have been illustrated in the following

figures 3.1 and 3.2.

3.2.2 The Fancy Circles

In this section the geodesics that are not as geometrically simple as the Simple

Circles shall be discussed. They are named the Fancy Circles, in anticipation that

they could be circular. The Simple Circles hold very little physical meaning as they

exist in a plane of constant time. If more meaningful trajectories are to be found

then we have to let loose some of the strict conditions that were assumed earlier.

The values of, t and r are generally nonzero. This will give some degree of freedom

to change the radial, angular and time position for a particle. This time, the three

EOM (3.7), (3.8) and (3.10) must be solved together. Unfortunately, as the functions

H, L, M, F in the EOM are highly non-linear, therefore analytical integration of

the EOM is impossible. Therefore we shall resort to numerical methods.


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3.3 Algorithm

3.3.1 Runge Kutta Method - Timelike and Geodesic

The EOM are first order linear differential equations. The Runge-Kutta methodis one of the standard iteration based procedures, used to numerically integrate

a first order ordinary differential equation with initial values. Hence, the Runge-

Kutta method 2 can be used for numerical integration of the EOM with respect to

the proper time, . Moreover, the C programming language will be used to write

our program. The code can be viewed in appendix B. The values of a and R

are kept constant throughout program (a, R are intrinsic properties of the universe;

hence the trajectories found for a particular set of a, R can also be correspondingly

found for other sets of values). The control variables of the program are E and

. Furthermore, the initial position of the particle r03, 0 and t0 must also be

specified. For ease of calculation, the initial conditions t0 and 0 will be kept as 0.

In conclusion, to run the program the necessary inputs are E, , r0 and the step

size h.

The program generates a file containing a list of the three coordinates ( r, and

t) of the curve along with the proper time and the number of iterations. These

points are plotted using MatLab (the codes can be viewed in appendix C)to generate

the trajectories. By selecting smaller step size, the error can be reduced. In order

to authenticate the results produced from the program, the value of the Lagrangian

was evaluated at every point of the trajectory. It turns out that it is a constant, 1,indicating that the curves are timelike, thus validifying that the program is running

accurately.

3.3.2 Graph Analysis - Closure

Having carried out the above procedure, the trajectory of the timelike geodesic of

a particle (for a given E and ) can be completely given. However, to find CTGs,

the closure condition must also be imposed. Fundamentally, to impose closure we

need the parameters E and as well as a suitable r0.

2

In our case the Fourth order Runge Kutta method (RK4) shall be used.3The r0 should be inserted such that the value of r is real.


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Figure 3.3: The Graph of r, t, versus r

In the case of the Simple Circle, this relation was already imposed by demanding

that the curve be a circle ( t = r = 0). For more general cases, closure can be found

by demanding that after some evolution of the particle, if the value of r returns to

r0, the value oft should return to t0 and to 2m (where m is an integer). However,

strictly speaking there are two types of the Fancy Circles. One is when the value

of r has the same sign as the initial radial velocity r0 and the other is when it is

opposite. The first case is called the Smooth Fancy Circle, and the second one the

Non-smooth Fancy Circle4.

Let us consider the following definitions (with reference to the figure 3.3):

tfull the value of time, when r = r0 and r = +r0

full the value of angle, when r = r0 and r = +r0

tpartial

the value of time, when r = r0 and r =

r0

partial the value of angle, when r = r0 and r = r0

Smooth Fancy Circles

The specific conditions for Smooth Fancy Circles are

1. Time condition: tfull = t0

4The former curve is differentiable everywhere, whereas the latter is non-differentiable at the

self intersection or non-smooth point.


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Figure 3.4: The graph oftfull versus Eand . The points of intersection between the

curved structure and t = 0 are the possible candidates which could satisfy closure.

Here a = 0.9, R = 1, h = 214

2. Angle condition: n full = 2m + 0 (with m and n as integers)

When a Smooth Fancy Circle is generated 3.5 it turns out that the trajectoryis flower-like and not circular (however, the name fancy circle shall be retained

to avoid confusion). The number of petals, n (named so due to the flower pattern

generated) is the multiple offull needed to ensure closure in space, m is the number

of rounds around the cylinder. It can be seen that the first condition can be easily

fulfilled. However, the second condition requires that full be a rational multiple,

mn

of 2. Mathematically, this can always be done because a rational number can

always be approximated from an irrational number up to any arbitary degree ofaccuracy. Therefore, if the right E and are found then a Smooth Fancy Circles

can be found.

A graphical plotting method is used to determine the phase regions of E and

that can statisfy the above conditions. Firstly, the value of is varied, keeping E

constant. Then E is varied and is reset to scan through the phase region to plot

out the value of tfull. Once the graph is generated, one can find the appropriate

E and that satisfy tfull = 0, to obtain closure. In the following portion of this

subsection, the method to obtain a Smooth Fancy Circle is elaborated upon.


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In order to save computing time, the suitable phase regions of E and are found

such that the generic curves in the figure 3.3 can be found. The t curve cuts inside

the egg-shaped r curves 5 and the curve remains constantly above or below 06.

This can then help reduce the phase region to plot the graph in figure 3.4.

Figure 3.5: The Top View of a Smooth Fancy Circle, E = 0.1, = 31.46, a = 0.9,

R = 1

To obtain a Smooth Fancy Circle, firstly the E and must satisfy that tfull = 0.

Figure 3.4 shows how the tfull changes with respect changes in E and . The ragged

surface indicates the error and uncertainty in tfull which does not affect the results

too much. The points on this manifold which intersect with the plane of t = 0,

satisfy the time condition. Amongst these points, those which satisfy the second

condition give us the appropriate E and . For example E = 0.1 and = 31.46

satisfy both the conditions, with m = 9 and n = 40 hence we can obtain a Smooth

Fancy Circle. The trajectory of the above Smooth Fancy Circle with 40 petals and

gone through 9 rounds is given in the figure 3.5 and figure 3.6.

5This is to ensure that in average, the increase in t (t is positive) is equal to the decrease in t(t

is negative) so that tfull = 0 can be fulfilled.6This is to ensure that keeps on increasing or decreasing with time.


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Figure 3.6: A Side View of a Smooth Fancy Circle, E = 0.1, = 31.46, a = 0.9,

R = 1

Non-smooth Fancy Circles

On the other hand, the specific conditions for a Non Smooth Fancy Circles are:

1. Time condition: tpartial = n tfull

2. Angle condition: n full + partial = 2m (with m and n as integers)

The E and are found by making sure that there exists one r0 that satisfy

the above two conditions at once. From the above statement, it might seem as if

a Non-smooth Fancy Circle is characterized by a specific starting point r0. This

is not true as any starting point in the Non-smooth Fancy Circle will generate the

whole curve.

Similarly the construction of the Non-smooth Fancy Circle is done as shown be-

low in the program in appendix B.3. In the program, the ranges of E and are

varied and then the number of petals, n and rounds, m are specified to find a particu-

lar curve. The program repeats the calculation of the values tpartial, tfull, partial, full

for different starting values, r0 from the minimum possible r to the maximum value

of r (refer to figure 3.3) at a particular E and .


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Figure 3.7: The Top View of a Non-Smooth Fancy Circle, E = 0.01, = 4.63295,

a = 0.9, R = 1

Figure 3.8: Another View of a Non-Smooth Fancy Circle, E = 0.01, = 4.63295,

a = 0.9, R = 1

The value of r0 such that tpartial = n tfull is satisfied is named r0t. The valueof r0 such that n full + partial = 2m is satisfied is named r0.

The program stops when r0t = r0, indicating that the two conditions above are

satisfied. The values of E and such that this happens is then used to generate

the Non-smooth Fancy Circle. One of the example of a Non-smooth Fancy Circle is


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Figure 3.9: The scale of the error in r evaluated at the nth iterative step for a = 0.9,

R = 1, E = 0 and = 39.15, where h = 216

given in the figure 3.7 and figure 3.8.

3.4 Estimation and Error Analysis

3.4.1 Theoretical Error

The truncation error in Fourth order Runge Kutta (RK4) method is proportional

to the fifth order of the step size (h5) for each iteration. Let this error be denoted

by O(h5). To approximate this we use the following method.

Let u be the approximate solution to r() at 0 + n h through n iterations of theRK4 method.

r() = u + n O(h5)

Let v be the approximate solution to r() at 0 + n h through 2n iterations withhalf the step size.

r() = v + 2n O( h2

5

)

Solving for O(h5) we get

n O(h5) =

u v(1 24)

(3.15)

The above error is called the local error in r, which is evaluated for the nth

step.The scale of the error is given in the figure 3.9.


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As it can be observed this error is not observable until large n. If one desires to

improve the accuracy, make the step size smaller.

3.4.2 Program Error

Theres one estimation that is necessary for the program to work, called the sign

switch effect. To understand it, the following form of the equation for r must be

seen clearer (refer to figure 3.3).

r =

LE2 F2 2ME 2r2r2H

. (3.16)

The sign switch effect comes in when the value of r reaches 0, the value of r

then will not change from the maximal value, rmax or minimal value, rmin once it

got there. It means that at some point of r or equivalently, r2 close to 0, the sign

for r must change to make r periodic and moving. To accommodate to different E,

and the step size h, the quantity dr2

drr, (refer to appendix B.1) the estimation

of the change of r2 in one iteration, is used to determine the lower boundary for

which r2 becomes close to 0 and for r to change sign. The criteria for changing signs

introduces error in the determination of the turning points. Let r be the error in

r incurred. Then

r = |rmax rmax|

where rmax is the maximum radius of the orbit, where r is zero.

rmax is the value of r when the sign switch mechanism activates.

r cannot be analytically derived, but can be calculated numerically. Under theconditions E = 0, = 39.15, a = 0.9, R = 1, rmax = 15.63275029 (10 s.f.), and

from the program rmax = 15.545398065, yielding r = 0.087352225.

The error in value at the turning points, denoted by , is given by = r

r.

For the conditions mentioned above, = 0.0000233023. The error in t similarlyis given by t = t

rr. For the above conditions, t = 0.00115663. Hence if two

sections of a curve come within the above mentioned range of error, the curve can

be considered closed.


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Chapter 4

Self Consistency

If the universe of discourse permits the possibility of time travel and of

changing the past, then no time machine will be invented in that

universe.

Larry Niven

The consistency problem in the van Stockum spacetime is studied in this portion

of the paper, by making use of the CTGs that have just been developed. We shall alsotry to see whether Novikovs hypothesis still holds. Based on the initial conditions,

there are two broad classes of collisions that can be constructed. One which involves

a ball to travel in a Smooth Fancy Circle, and the other in a Non-smooth one. The

collisions shall be discussed in a General Relativistic framework. In the end, a

summary of both the cases, along with the pros and cons of the self consistency

principle, is presented.

4.1 Smooth Curves

Let us consider a Smooth Fancy Circle, which is uniquely determined by parameters

E1 and 1. It is always possible to pick one particular point on the curve and

construct a timelike geodesic, which has the form of a helix, passing through it. Let

the parameters of the Helix be E2 and 2. Furthermore, having obtained these two

curves, let us assume that there are two identical balls B1 and B2 (both point-like

and both possessing the same mass), such that B1 is travelling on the Fancy Circle

24


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CHAPTER 4. SELF CONSISTENCY 25

Figure 4.1: The collision between the balls B1 and B2, travelling on the Fancy Circle

and the Helix respectively (the fancy circle and the helix has not been used, as the

diagram is too complicated to see the collision clearly).

and B2 on the Helix. Moreover, we also demand that the initial conditions of B1

and B2 are arranged such that they reach the intersection point of the Fancy Circle

and the Helix at the same time. Having set up the boundary conditions, let us now

study the collision between these two balls.

Given the above initial conditions, the conservation of energy E and angular

momentum can be invoked, to determine the evolution of the state of the balls

after collision. According to our definition, the energy and the angular momentum

ofB1 is given by E1 and 1 respectively, and that ofB2 is given by E2 and 2. The

conservation equations are:

E1 + E2 = E

1+ E2, (4.1)

1 + 2 =

1+

2, (4.2)

where the primed values represent the final energies and angular momenta. The

above two equations only impose that the sum of the energies and the angular

momenta be constants. Considering the E values in a special case, if the magnitude

ofE1 is 0.01 and E2 is 0.04, energy conservation only demands that the sum E

1+ E2

be equal to 0.05. Over and above this, if the solution is expected to be physically

acceptable it is a necessary that the resultant conditions after a collision be consistent

with the initial conditions of the balls (here the self consistency principle is being


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assumed). In the above specified case, there can exist infinitely many results (for

example: E1

= 0.02, E2

= 0.03 or E1

= 0.0001, E2

= 0.0499 etc.) that are

mathematically acceptable; but not all are physically allowable.

To find out the physically permissible results, let us consider the case whenE1 is 0.02, E

2 is 0.03. It means that the balls B1 and B2 will, after collision,

travel on trajectories that are characterized by the energies given by 0.02 and 0.03

respectively. Consequently, none of the balls travel on the Smooth Fancy Circle

after the collision (given by energy 0.01 in this case) leaving no ball to explain the

origin of B1. This just means that when B2 comes to the intersection point of

the Helix and the Smooth Fancy Circle, it will not undergo any collision as there

is no ball there. So by contradiction, the only values for E

1 and E

2 is the trivialcase E1 = 0.01 and E

2 = 0.04 (when the balls retain their initial conditions) or

E1 = 0.04 and E

2 = 0.01 (when they switch their energies). So that the origin ofB1

can be explained by having either of the balls, B1 or B2, replace the ball B1 after

the collision1. A similar argument can be made for 2.

Thus there is only one non trivial consistent solution3, which requires B1 to travel

on the helix and B2 on the smooth fancy circle after the collision. The two particles

have exchanged their trajectories. After some thought, one will realise that they are

the same particle! It is possible to construct a self consistent solution for a collision

problem on a Smooth Fancy Circle (given by some specific parameters). Due to

this, by definition, there can logically exist a consistent Smooth Fancy Circle. This

result can be generalised for all Smooth Fancy Circles, as they only differ in their

parameters.

4.2 Non-Smooth Curves

Now let us replace the Smooth Fancy Circle with a Non-smooth one and redevelop

the collision problem. The parameters of the Non-smooth curve are defined as E1

1This is an important fact which applies not only for this case but also for the ones that shall

be discussed later.2Due to the 4-momentum invariance, the case for angular momentum is taken cared of, refer to

appendix3such that E1 = E2, E

2 = E1,

1 = 2 and

2 + 1


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Figure 4.2: The collision between the balls B1 and B2 in Case A

and 1 and all other definitions made earlier remain the same. In order to work out

the collision, just as before, another geodesic which intersects the Non-smooth curve

has to be considered. It turns out that there are various different ways of doing this.

Firstly, as the curve is Non-smooth, it is already self intersecting, therefore it would

be natural to construct the self collision of the ball moving on the Non-smooth circle

itself (Case A). Moreover, another possible geodesic is a Helix passing through the

Fancy Circle at the Non-smooth point (Case B)4. The approach to the problem shall

remain same as in the previous section4.1.

4.2.1 Case A

As Non-smooth Fancy Circle is being considered, the problem of collision becomes

drastically simpler. This is because we already have one curve that intersects itself.

To find out the trajectory of the whole curve, firstly the C program can be made

to run from the point just before the collision point until just after the collision

point as shown in figure 4.2. Let the Non-smooth portion be called trajectory 1

and the other two curves that are sticking out from the the collision point, together,

as trajectory 2. The balls B1 and B2 are initially travelling on trajectory 1 and

4This is analogous to the collision that was constructed in the Smooth Fancy Circle case.


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trajectory 2 respectively. Now the conservation equations (4.1) and (4.2) can be

considered. The constants E1, E2 and 1, 2 are equal to each other as both the

trajectories are in essence the same. Therefore the equations (4.1) and (4.2) now

read as:

2E1 = E

1+ E

2, (4.3)

21 =

1+

2. (4.4)

Just as before, the above two equations form the necessary mathematical condi-

tions, but in order to be physically acceptable either of the resultant energies and

momenta must replace be equal to E1 and 1 respectively. The only possible case

when this can happen, is when E1 = E2 = E1 and 1 = 2 = 1. As before, there

are two cases. The collision results in B1 travelling out of the Non-smooth circle and

B2 travelling into the circle, so that eventually B2 discovers itself be B1. Or B1 is

knocked back into the circle, and B2 keeps on trajectory 2, never entering trajectory

15.

4.2.2 Case B

In this case, the collision is constructed in a way similar to that of Smooth curves

such that a helical geodesic passes specifically through the non-smooth point. The

main difference is that this problem is a three and not a two body collision. The

non-smooth circle is parametrized by E1 and 1 and the Helix by E2 and 2. Now

ball B1 is travelling on the Non-smooth curve and ball B2 on the Helix. At the

non-smooth point the ball B1 not only meets B2, but also its own past self called B3

(with the same parameters as B1, that is E1 and 1). Therefore, when the problemis being evaluated, all the three balls must be considered. The equations are then:

2E1 + E2 = E

1+ E

2+ E

3, (4.5)

21 + 2 =

1+

2+

3, (4.6)

5However, this case seems to violate the Second Law of Thermodynamics, that B1 gets looped

infinitely many times and still remains the same (having the same entropy). The Second Law does

not directly forbid this case, but it makes this case a weak one in macroscopic situations


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Figure 4.3: The collision between the balls B1 and B2, travelling on the Non-smooth

Fancy Circle and the Helix respectively along with the third ball B3

However, one of the 3 balls will have to enter the Closed Timelike Geodesic and

become B3 (to ensure consistency and time travel). So either B1 becomes B3 (E

1=

E1 and

1= 1) or B2 becomes B3 (E

2= E1 and

2= 1)

6. The other 2

parameters of the remaining 2 balls are not constrained in anyway, so it can be

anything that branches out of the collision point.

Using the results from the analysis of the case A (subsection 4.2.1) and case B

(subsection 4.2.2), one can conclude that Non-smooth Fancy Circles can also exist

logically.

6The last possibility ofB3 becoming B3 is not considered here as it seems to violate the Second

Law of Thermodynamics as mentioned before


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4.3 Summary

In this chapter two broad cases of billiard ball collision were studied. The central

aim of the whole procedure was to determine which CTGs can exist logically. It was

found that all CTGs can exist logically, provided that the interaction between parti-

cles travelling on them adhere to the mathematical and the physical constraints (in

the form of the energy-momentum conservation and self consistency respectively).

These results testify the validity of the claim made by Novikov and his colleagues

in the van Stockum universe. It is indeed true that all the CTGs (at least for the

cases that have been considered till now) can exist logically. It would be interesting

to consider collisions of billiard balls travelling along other possible trajectories.

Although the self consistency principle severely restrains the evolution of par-

ticles after an interaction, it has the amazing ability of sifting out the logically -

sensible, consistent solutions from the inconsistent ones. It must be noted however

that the principle of self consistency does not in any way show that time travel is

possible or not. It is just a tool to resolve the possible paradoxes that may arise

due to time travel.


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Chapter 5

Conclusion

Whatever begins, also ends.

Seneca

In order to address the issues of time travel paradoxes in the van Stockum space-

time, firstly the equations of motion were obtained. Then the trajectories of particles

travelling on a Closed Timelike Geodesic were obtained by making use of the Fourth

order Runge Kutta and the graphical analysis methods. It was found that in generalthe CTGs can be divided into two categories, one the Smooth Fancy Circles and the

other the Non-smooth Fancy Circles ones.

Finally, assuming the self consistency principle, the test of particle collision was

performed. It was found that it is possible to find at least one self consistent solution,

for any general collisions involving the Fancy Circles (Smooth and Non-smooth). As

all the Fancy Circles just differ by their parameters, the above conclusion can be

extended for all CTGs. Therefore, it can be claimed that the CTGs are logicallypossible (according to the definition given in section 2.1), at least as far as point-like

particles are dealt with. Although our work does not manage to resolve the time

travel paradoxes, it shows that at least one consistent case can be obtained for any

general collisions even when curved spacetime geometry is considered.

This project has been written for the Special Program in Science under the theme

of Cycles and Sustainability. Our work is related to the theme in the following way:

in spite of time travel (which can be achieved by using CTGs that are cycles in

time), it is possible to preserve the consistency in the evolution.

31


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Chapter 6

Future Work

There is never enough time to do everything, but there is always

enough time to do the most important thing.

Brian Tracy

There are various different possible paths that can be studied as a follow up for

this project. As far as the CTGs are concerned, the method used for finding the non

smooth fancy circle can be generalized. Moreover, a classification of all the CTGscan be done by using the number of petals and turns as the parameters (if there

exists a unique mapping). Along the line of the self consistency problem, it would be

interesting to consider collisions of billiard balls travelling along other trajectories.

One of the possible examples being the collision between two particles travelling on

two smooth or non smooth fancy circles. The same argument concerning self con-

sistency can be considered under the light of the Chronology Protection Conjecture

proposed by Stephen Hawking [3]. This study might provide more insight of the

time travel behaviour in general.

32


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Chapter 7

Acknowledgements

Time is a great teacher, but unfortunately it kills all its pupils.

Hector Berlioz

Firstly, we would like to express our gratitude to our staff mentor Associate

Professor Edward Teo for devoting his time to guide us through this project. We

would like to convey our thanks to our mentor, Tran Chieu Minh, for the time

and effort he has put in, to guide us in this long and perilous path. Moreover,we would like to thank Lim Yen Kheng, from whom we have gained our present

understanding of General Relativity. Our special thanks to Thong May Han for her

presence and support. Last but not the least, we would like to express our gratitude

to the Special Program in Science (SPS) for giving us an opportunity to study and

perform research, allowing us enlarge the horizons of our knowledge.

33


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Appendix A

The Four Momentum and the

Relationship between E and

In order to solve the collision problem the following two equations can be invoked:

E1 + E2 = E

1 + E

2, (A.1)

1 + 2 =

1+

2, (A.2)

Moreover, just like the classical cases, a relation connecting the different four

momenta can be framed. This relation is the invariance of the length of the four

momentum. According to the definition of our affine parameter proper time, four

momentum P and inverse metric tensor g the relation can be written as:

P2 = gPP = 2,

1r2H

E Pr 0

LH 0 MH 00 r2 0 0

MH 0 F H 0

0 0 0 r2

E

Pr

0

= 2. (A.3)

The above equation can be simplified into the equation of r (A.4). It can be

written as

r2Hr2 = LE2 F2 2M E 2r2. (A.4)

34


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APPENDIX A. THE FOUR MOMENTUM AND THE RELATIONSHIP BETWEENEAND35

The above equation forms a relation between E and for a particular r and r.

For the collision problem there are four unknowns E1, E

2,

1and

2, and three

equations (A.2), (A.2) and (A.4). This indicates that there is one free condition

that can be imposed. If it is imposed that the solution be self consistent, we wouldat least require that either E1 = E

2 or 1 =

21. Let us just say that E1 = E

2.

According to equation (A.2), one can immediately conclude that E2 = E

1. Then

using the equation (A.4), the relation between 1 and

2 can be written as

(r2Hr2 + 2r2)1 (r2Hr2 + 2r2)2 = F(21 22 ) 2M E1(1 2). (A.5)

At the intersecting point r1 and r

2are both equal; moreover, the values of (r2)1

and (r2)

2 must also be equal. Hence the left hand side of the equation becomes zero,

leaving:

(1 2)[F(1 + 2) + 2ME1] = 0. (A.6)

For any general value of the radius, the above equation can only hold if 1 =

2.

Hence, it was shown that when E2 = E

1, then 1 =

2. This result when applied

to the equation (A.2), to give 2 =

1. These results show that even if the extra

condition (invariance of the length of Four Momentum) is used, the consistent case

is mathematically permitted. Therefore the parameters E and , while considering

the consistent CTGs, can be trated independently of each other.

1To show that the two balls, B1 and B2, have switched identities after the collision, it must be

shown that both the corresponding parameters are equal.


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Appendix B

C codes

B.1 The main C code

The fourth order Runge-Kutta method. To be compiled by Miracle C program.

#include

#include

/*Define constant*/

#define pi 3.1415926535

/*put a file in*/

FILE *p1;

/*functions prototypes*/

/*main functions*/

long double Lg(long double v, long double w, long double x, long double y,

long double a, long double R, long double ep, long double E, long double P);

long double fl(long double v, long double w, long double x, long double y,


long double fk(long double v, long double w, long double x, long double y,

long double a, long double R, long double ep, long double E, long double P,

long double g);

long double ft(long double v, long double w, long double x, long double y,


long double fg(long double v, long double w, long double x, long double y,


long double fh(long double v, long double w, long double x, long double y,

36


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APPENDIX B. C CODES 37


/*HLMF*/

long double H(long double v, long double a, long double R, long double ep);

long double L(long double v, long double a, long double R, long double ep);

long double M(long double v, long double a, long double R, long double ep);

long double F(long double v, long double a, long double R, long double ep);

long double Hp(long double v, long double a, long double R, long double ep);

long double Lp(long double v, long double a, long double R, long double ep);

long double Mp(long double v, long double a, long double R, long double ep);

long double Fp(long double v, long double a, long double R, long double ep);

void main()

{

/*Declare the variables used*/

long int n, n1, e, m, m1, m2, e1, pr; /*iteration counters*/

long double tau, r, phi, t, r_dotsquare, phi_dot, t_dot; /*variables*/

long double k1, k2, k3, k4, q1, q2, q3,q4, w1,

w2, w3, w4; /*Runge-Kutta dummies*/

long double l1, l2, l3, r1, r2, r3, t1 ,t2, t3, u1,

u2, u3, o1, o2, o3; /*Runge-Kutta dummies*/

long double cond, input, rcheck, revived, g, sp1, sp2,

sp3, sp4, Wcheck, condW; /*checkers*/

long double a, R, ep, E, P; /*Parameters*/

long double tP, r_0, tPW, Pn, PW, Dr_dot2, rn, Dr_dot3,

rn_dotsquare, prmax; /*values of variables*/

long double h, W; /*small increments*/

long double step, stepW; /*small increments inputs*/

long double Lgn,Lgnn; /*the Lagrangian*/

/*Open a file for writing in the data*/

p1 = fopen("p5.dat", "w");

/*Input constants of motions*/

revived=1;

input=1;

while (input==1)


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{printf("Please ensure that the value aR is between 1/2 and 1\n");

printf("Put in the angular velocity of the cylinder, a\n");

scanf("%f", &a);

printf("Put in the Radius of the cylinder, R\n");

scanf("%f", &R);

input=0;

if (a*R>=1 | a*R


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t_dot=fh(r, phi, t, tau, a, R, ep, E, P);

Lgn=Lg(r,phi,t,tau,a,R,ep,E,P);

printf(

"r_dot^2=%Lg phi_dot=%Lg t_dot=%Lg Lagrangian=%Lg\n\n",

r_dotsquare, phi_dot, t_dot, Lgn);

if (r_dotsquare >= 0)

{rcheck=1; }

else { rcheck=0;

printf("r is not suitable, choose a different r\n");

}}

r_0=r;

revived=0;

printf(

"Please enter a value for step, where h=2^(-step),\n step=");

scanf("%f", &step);

/*set h*/

h=pow(2,-step);

printf(

"Please determine the number of rounds to simulate, prmax=\n");

scanf("%f", &prmax);

/*print initial value*/

printf(

"r=%Lg phi=%Lg t=%Lg tau=%Lg h=%Lg E=%Lg P=%Lg a=%Lg R=%Lg ep=%Lg\n",

r, phi, t, tau, h, E, P, a, R, ep);

/*start the loop*/

n=0;

n1=0;

while (cond == 0)

{rn_dotsquare=ft(r, phi, t, tau, a, R, ep, E, P);

k1 = h*fk(r, phi, t, tau, a, R, ep, E, P, g);

q1 = h*fg(r, phi, t, tau, a, R, ep, E, P);

w1 = h*fh(r, phi, t, tau, a, R, ep, E, P);

l1 = (r+k1/2);

t1 = (phi+q1/2);


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u1 = (t+w1/2);

o1 = (tau+h/2);

k2 = h*fk(l1, t1, u1, o1, a, R, ep, E, P, g);

q2 = h*fg(l1, t1, u1, o1, a, R, ep, E, P);

w2 = h*fh(l1, t1, u1, o1, a, R, ep, E, P);

l2 = (r+k2/2);

t2 = (phi+q2/2);

u2 = (t+w2/2);

o2 = (tau+h/2);


q3 = h*fg(l2, t2, u2, o2, a, R, ep, E, P);

w3 = h*fh(l2, t2, u2, o2, a, R, ep, E, P);

l3 = (r+k3);

t3 = (phi+q3);

u3 = (t+w3);

o3 = (tau+h);


q4 = h*fg(l3, t3, u3, o3, a, R, ep, E, P);

w4 = h*fh(l3, t3, u3, o3, a, R, ep, E, P);

rn=r;

r = r+(k1+2*(k2+k3)+k4)/6;

phi = phi+(q1+2*(q2+q3)+q4)/6;

t = t+(w1+2*(w2+w3)+w4)/6;

tau = tau+h;

n = n+1;

e = n1*10000+n;

Lgn=Lg(r,phi,t,tau,a,R,ep,E,P);

r_dotsquare=ft(r, phi, t, tau, a, R, ep, E, P);

/*Edge boundary values*/

if((r-rn)*fl(rn,phi,t,tau,a,R,ep,E,P)>= 0)

{Dr_dot3=(r-rn)*fl(rn,phi,t,tau,a,R,ep,E,P);}

else

if((r-rn)*fl(rn,phi,t,tau,a,R,ep,E,P)< 0)

{Dr_dot3=-(r-rn)*fl(rn,phi,t,tau,a,R,ep,E,P);}


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if(h*sqrt(ft(r, phi, t, tau, a, R, ep, E, P))*

fl(rn,phi,t,tau,a,R,ep,E,P)>= 0)

{Dr_dot2=h*sqrt(ft(r, phi, t, tau, a, R, ep, E, P))*

fl(rn,phi,t,tau,a,R,ep,E,P);}

else

if(h*sqrt(ft(r, phi, t, tau, a, R, ep, E, P))*

fl(rn,phi,t,tau,a,R,ep,E,P)< 0)

{Dr_dot2=-h*sqrt(ft(r, phi, t, tau, a, R, ep, E, P))*

fl(rn,phi,t,tau,a,R,ep,E,P);}

if (r>r_0 & r_dotsquare


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cond=1;

revived=1;}

/*for cond*/}

/*for revived*/}

fclose(p1);

/*for main*/}

/*Define functions*/

long double Lg(long double v, long double w, long double x, long double y,

long double a, long double R, long double ep, long double E, long double P)

{long double c;

c=H(v, a, R, ep)*ft(v,w,x,y,a,R,ep,E,P)+L(v, a, R, ep)*

pow(fg(v,w,x,y,a,R,ep,E,P),2)+2*M(v, a, R, ep)*fg(v,w,x,y,a,R,ep,E,P)*

fh(v,w,x,y,a,R,ep,E,P)-F(v, a, R, ep)*pow(fh(v,w,x,y,a,R,ep,E,P),2);

return(0.5*c);

}

long double fl(long double v, long double w, long double x, long double y,


{long double c;

c=(((pow(E,2)*Lp(v,a,R,ep)-pow(P,2)*Fp(v,a,R,ep)-2*E*P*Mp(v,a,R,ep))/

v-(-2*Hp(v, a, R, ep)*v+(2*H(v,a,R,ep)+v*Hp(v,a,R,ep))*

(pow(E,2)*L(v,a,R,ep)-pow(P,2)*F(v,a,R,ep)-2*E*P*M(v,a,R,ep)))/

H(v, a, R, ep))/(v*H(v, a, R, ep)));

return(c);

}

long double ft(long double v, long double w, long double x, long double y,


{long double c;

c=(-2+(pow(E,2)*L(v,a,R,ep)-pow(P,2)*F(v,a,R,ep)-2*E*P*M(v,a,R,ep))/

pow(v,2))/(H(v, a, R, ep));

return(c);

}



long double g)


45/60


{

long double c, d, m, p;

c=(-2+(pow(E,2)*L(v,a,R,ep)-pow(P,2)*F(v,a,R,ep)-2*E*P*M(v,a,R,ep))/

pow(v,2))/(H(v, a, R, ep));

m=100;

d=sqrt(c);

p=-sqrt(c);

if (g==1){

return(d);

}

else if (g==-1)

return(p);

/*for fk*/}

long double fg(long double v, long double w, long double x, long double y,


{

long double d;

d=(E*M(v,a,R,ep)+P*F(v,a,R,ep))/pow(v,2);

return(d);

}

long double fh(long double v, long double w, long double x, long double y,


{

long double b;

b=(P*M(v,a,R,ep)-E*L(v,a,R,ep))/pow(v,2);

return(b);

}

long double H(long double v, long double a, long double R, long double ep)

{

long double c;

c=(exp(-pow(a,2)*pow(R,2)))*pow((v/R),(-2*pow(a,2)*pow(R,2)));

return(c);

}

long double L(long double v, long double a, long double R, long double ep)


46/60


{

long double c;

c=(v*R*(sin(3*ep+log(v/R)*tan(ep))))/(sin(ep)+sin(3*ep));

return(c);

}

long double M(long double v, long double a, long double R, long double ep)

{

long double c;

c=(v*(sin(ep+log(v/R)*tan(ep))))/(sin(2*ep));

return(c);

}

long double F(long double v, long double a, long double R, long double ep)

{

long double c;

c=(v*(sin(ep-log(v/R)*tan(ep))))/(R*sin(ep));

return(c);

}

long double Hp(long double v, long double a, long double R, long double ep)

{

long double c;

c=(-2*pow(a,2)*pow(R,2))*(exp(-pow(a,2)*pow(R,2)))*pow((1/R),(-2*pow(a,2)*

pow(R,2)))*pow((v),(-2*pow(a,2)*pow(R,2)-1));

return(c);

}

long double Lp(long double v, long double a, long double R, long double ep)

{

long double c;

c=(R/(sin(ep)+sin(3*ep)))*(sin(3*ep+log(v/R)*tan(ep))+tan(ep)*

cos(3*ep+log(v/R)*tan(ep)));

return(c);

}

long double Mp(long double v, long double a, long double R, long double ep)

{

long double c;


47/60


c=(1/sin(2*ep))*(sin(ep+log(v/R)*tan(ep))+tan(ep)*cos(ep+log(v/R)*tan(ep)));

return(c);

}

long double Fp(long double v, long double a, long double R, long double ep)

{

long double c;

c=(1/R*sin(ep))*(sin(ep-log(v/R)*tan(ep))-tan(ep)*cos(ep-log(v/R)*tan(ep)));

return(c);

}

B.2 The graphical method C code

These codes generate the graph figure 3.4. To save space, the codes that are available

above are not repeated but are represented by the dots.


.

.

.

printf("determine the accuracy of r to be printed\n");

scanf("%f", &u);

printf("Please enter a value for the change in P, \n Pr=");

scanf("%f", &Pr);

printf("Please enter a value for the change in E, \n Er=");

scanf("%f", &Er);

printf("Please enter a value for when to stop for P, \n Pend=");

scanf("%f", &Pend);

printf("Please enter a value for when to stop for E, \n Eend=");

scanf("%f", &Eend);


printf(


r, phi, t, tau, h, E, P, a, R, ep);/*start the loop*/


48/60


Evar=0;

P_0=P;

while(Evar==0)

{P=P_0;

Pvar=0;

while(Pvar==0)

{r=r_0;

phi=0;

t=0;

tau=0;

g=2;

n=0;

n1=0;

cond=0;

sp1=0;

sp2=0;

sp3=0;

sp4=0;

/*printf("%lu %Lg %Lg %.12Lg %.12Lg %.12Lg %.12Lg\n",

e , E , P, r, phi, t, tau);*/

while (cond == 0)

.

.

if (g==-1 & sp1==0) {

/*fprintf(p1, "%lu %Lg %Lg %.12Lg %.12Lg %.12Lg %.12Lg\n",

e , E , P, r, phi, t, tau);*/

sp1=1;

r_max=r;

}

if (g==-1 & sp2==0 & r


49/60


t_1p=t;

}

if (g==1 & sp3==0) {


e, E , P, r, phi, t, tau);*/

sp3=1;

r_min=r;

}

if (g==1 & sp4==0 & r>=r_0) {

/*fprintf(p1,

"%lu %Lg %Lg %.12Lg %.12Lg %.12Lg %.12Lg %.12Lg %.12Lg\n",

e, E , P, r, phi, t, tau, u*r_diff, Lgn);*/

sp4=1;

phi_1=phi;

cond=1;

printf("r=%Lg phi_1p=%Lg phi_1=%Lg t_1p=%Lg t_1=%Lg Lgn=%Lg \n",

r , phi_1p, phi_1, t_1p, t_1,Lgn);

}

.

.

/*for cond*/}

P=P+Pr;

printf("P=%Lg\n", P);

if(P>=Pend)

{Pvar=1;

printf(

"Next E=%Lg, congrats, The SPS Time Traveller group rocks\n", E+Er);}

/*for Pvar*/ }

E=E+Er;

if(E>=Eend)

{Evar=1;

printf("Program ends Finally!\n");}

/*for Evar*/ }

/*for revived*/}


50/60


fclose(p1);

/*for main*/}

.

.

.



long double g)

{

long double c, d, m, p;

c=(-2+(pow(E,2)*L(v,a,R,ep)-pow(P,2)*F(v,a,R,ep)-2*E*P*M(v,a,R,ep))/pow(v,2))/

(H(v, a, R, ep));

m=100;

d=sqrt(c);

p=-sqrt(c);

if (g==1 | g==2){

return(d);

}

else if (g==-1)

return(p);

/*for fk*/}

B.3 The code to find the Non-smooth Fancy Cir-

cles

long double r_min, phi_1,phi_1p,t_1p, r_max,r_0t,sp5,r_0p,sp6,rounds,petals,

r_0var, r_step, r_0real, t_1;/*the new ones for p11*/

/*Open a file for writing in the data*/


.

.

.

printf("determine the number of rounds to be considered\n");


51/60


scanf("%f", &rounds);

printf("determine the number of petals to be considered\n");

scanf("%f", &petals);

printf("Please enter a value for the change in P, \n Pr=");

scanf("%f", &Pr);

printf("Please enter a value for the change in E, \n Er=");

scanf("%f", &Er);

printf("Please enter a value for the change in r_0, \n r_step=");

scanf("%f", &r_step);

printf("Please enter a value for when to stop for P, \n Pend=");

scanf("%f", &Pend);

printf("Please enter a value for when to stop for E, \n Eend=");

scanf("%f", &Eend);


printf(


r, phi, t, tau, h, E, P, a, R, ep);

/*start the loop*/

Evar=0;

P_0=P;

while(Evar==0)

{P=P_0;

Pvar=0;

while(Pvar==0)

{r=r_0;

phi=0;

t=0;

tau=0;

r_0t=100;

r_0p=10;

g=2;

n=0;

n1=0;

cond=0;


52/60


sp1=0;

sp2=0;

sp3=0;

sp4=0;

sp5=0;

sp6=0;


e , E , P, r, phi, t, tau);*/

while (cond == 0)

{ .

.

.

if (g==-1 & sp1==0) {


e , E , P, r, phi, t, tau);*/

sp1=1;

r_max=r;}

if (g==-1 & sp2==0 & r=r_0) {

/*fprintf(p1, "%lu %Lg %Lg %.12Lg %.12Lg %.12Lg %.12Lg %.12Lg %.12Lg\n",

e, E , P, r, phi, t, tau, u*r_diff, Lgn);*/

sp4=1;

phi_1=phi;

cond=1;


53/60


printf("r=%Lg phi_1p=%Lg phi_1=%Lg t_1p=%Lg t_1=%Lg Lgn=%Lg \n",

r , phi_1p, phi_1, t_1p, t_1,Lgn);}

.

.

.

/*for cond*/}

r_0var=0;

sp5=0;

sp6=0;

while(r_0var==0)

{r=r_0;

phi=0;

t=0;

tau=0;

g=2;

n=0;

n1=0;

cond=0;

sp1=0;

sp2=0;

sp3=0;

sp4=0;


e , E , P, r, phi, t, tau);*/

while (cond == 0)

{

.

.

.

/*for cond*/}

if (t_1p>=-petals*t_1& sp5==0)

{r_0t=r_0;

sp5=1;

printf("r_0t found~! r_0t=%Lg\n\n", r_0t);}


54/60


if (phi_1p>= -2*rounds*pi-petals*phi_1 & sp6==0)

{r_0p=r_0;

sp6=1;

printf("r_0p found~! r_0p=%Lg\n\n", r_0p);}

if((r_0t-r_0p)>= 0)

{r_0real=(r_0t-r_0p);}

else

if((r_0t-r_0p)< 0)

{r_0real=-(r_0t-r_0p);}

if (r_0real=r_max-2*r_step)

{r_0var=1;

printf(

"No non-smooth Closed Timelike Curve in these conditions,r_min=%Lg,r_max=%Lg\n",

r_min, r_max);

}

r_0=r_0+r_step;

/*for r_0var*/}

P=P+Pr;

printf("P=%Lg\n", P);

if(P>=Pend)

{

Pvar=1;

printf("Next E=%Lg, congrats, The SPS Time Traveller group rocks\n", E+Er);

}

/*for Pvar*/ }

E=E+Er;

if(E>=Eend)

{

Evar=1;

printf("Program ends Finally!\n");


55/60


}

/*for Evar*/ }

/*for revived*/}

fclose(p1);

/*for main*/}

/*Define functions*/

.

.

.



long double g)

{long double c, d, m, p;

c=(-2+(pow(E,2)*L(v,a,R,ep)-pow(P,2)*F(v,a,R,ep)-2*E*P*M(v,a,R,ep))/pow(v,2))/

(H(v, a, R, ep));

m=100;

d=sqrt(c);

p=-sqrt(c);

if (g==1 | g==2){

return(d);}

else if (g==-1)

return(p);

/*for fk*/}

B.4 Codes to find Simple Circle

.

.

.

r=R*exp(2*(k*pi-2*ep)/tan (ep));

E=M(r,a,R,ep)*P/L(r,a,R,ep);

.

.


56/60


.


57/60

Appendix C

Other codes

C.1 Matlab Codes

C.1.1 Curve Generator

This code generate the curves from the data of the C code.

load p5.dat; % read data into the my_xy matrix

r2 = p5(:,2); % copy first column of my_xy into x

t2 = p5(:,4); % and second column into y

phi2 = p5(:,3);

tau2 = p5(:,5);

[X,Y,T]=pol2cart(phi2,r2,t2);

plot3 (X, Y, T, DisplayName, X, Y, T,Color,green); figure(gcf)

xlabel(x)

ylabel(y)

zlabel(t)

grid on

C.1.2 Graphical Method

This code generates the graph from the data generated by the program the graphical

method C code, the figure 3.4 is generated by this code.

load P9PositiveP.dat; % read data into the my_xy matrix

r =P9PositiveP(:,4); % copy first column of my_xy into x

55


58/60

APPENDIX C. OTHER CODES 56

t = P9PositiveP(:,6); % and second column into y

phi = P9PositiveP(:,5);

tau = P9PositiveP(:,7);

E = P9PositiveP(:,2);

P = P9PositiveP(:,3);

[X,Y,T]=pol2cart(phi,r,t);

j=reshape(t,30,299);

e=[0.001:0.001:0.299];

p=[20:1:49];

rp=reshape(r,30,299);

yth=j-j;

surf (e, p, j, rp, DisplayName, e, p, j, rp); figure(gcf)

hold on

surf (e, p, yth, rp, DisplayName, e, p, j, rp); figure(gcf)

C.2 Mathematica codes

The code below is used to generate the graph in figure 3.3.

Manipulate[

Plot[{Sqrt[\[ExponentialE]^(a^2 R^2) (r/R)^(

2 a^2 R^2) (-2 -

P^2 ( Cos[(-1 + 4 a^2 R^2)^0.5 Log[r/R]] -

Sin[(-1 + 4 a^2 R^2)^0.5 Log[r/R]]/(-1 + 4 a^2 R^2)^0.5)/(

r R)

- (2

G P Csc[2 ArcTan[(-1 + 4 a^2 R^2)^0.5]] Sin[

ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 + 4 a^2 R^2)^0.5 Log[

r/R]])/r

+

1/(4 r (-1 + 4 a^2 R^2)^0.5)

G^2 R (1 + (-1 + 4 a^2 R^2)^1.)^(3/2)

Sin[3 ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 +

4 a^2 R^2)^0.5 Log[r/R]])], -Sqrt[\[ExponentialE]^(a^2 R^2) (r/R)^(


59/60

APPENDIX C. OTHER CODES 57

2 a^2 R^2) (-2 -

P^2 ( Cos[(-1 + 4 a^2 R^2)^0.5 Log[r/R]] -

Sin[(-1 + 4 a^2 R^2)^0.5 Log[r/R]]/(-1 +

4 a^2 R^2)^0.5)/(r R)

- (2

G P Csc[2 ArcTan[(-1 + 4 a^2 R^2)^0.5]] Sin[

ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 + 4 a^2 R^2)^0.5 Log[

r/R]])/r

+

1/(4 r (-1 + 4 a^2 R^2)^0.5)

G^2 R (1 + (-1 + 4 a^2 R^2)^1.)^(3/2)

Sin[3 ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 +

4 a^2 R^2)^0.5 Log[r/R]])],

Csc[2 ArcTan[(-1 + 4 a^2 R^2)^0.5]] Sin[

ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 + 4 a^2 R^2)^0.5 Log[r/R]]

P /r - 1/(4 r (-1 + 4 a^2 R^2)^0.5)

G R (1 + (-1 + 4 a^2 R^2)^1.)^(3/2)

Sin[3 ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 + 4 a^2 R^2)^0.5 Log[

r/R]], (

P ( Cos[(-1 + 4 a^2 R^2)^0.5 Log[r/R]] -

Sin[(-1 + 4 a^2 R^2)^0.5 Log[r/R]]/(-1 + 4 a^2 R^2)^0.5))/(

r R) + 1/r

G Csc[2 ArcTan[(-1 + 4 a^2 R^2)^0.5]] Sin[

ArcTan[(-1 + 4 a^2 R^2)^0.5] + (-1 + 4 a^2 R^2)^0.5 Log[r/

R]]}, {r, 0, 20}, PlotRange -> {-100, 100},

AxesLabel -> {r, r_dot}], {{a, 0.9, "Angular Velocity"}, 0.001,

100}, {{R, 1, "Radius"}, 0.006, 999}, {{G, 0.1, "Energy"}, 0,

1000}, {{P, 31.46, "Angular Momentum"}, 0, 50}]


60/60

Bibliography

[1] Novikov I.D. Echeverria F. Klinkhammer G. Thorne K.S. Friedman J., Morris M.S.

and Yurtsever U. Cauchy problem in spacetimes with closed timelike curves. Physical

Review D, 42:1915, 1990.

[2] K. Godel. An example of a new type of comsmological solutions to the einsteins field

equations of gravitation. Reviews Modern Physcial, 21:447, 1949.

[3] S. W. Hawking. Chronology protection conjecture. Physical Review D, 46:603, 1992.

[4] Novikov I.D. Time machine and self consistent evolution in problems with self inter-

action. Physical Review D, 45:1989, 1992.

[5] B. R. Steadman. Causality violation on van stockum geodesics. General Relativity and

Gravitation, 35:1721, 2003.

[6] Frank J. Tipler. Rotating cylinders and the possibility of global causality violation.

Physical Review D, 9:9, 1973.

[7] van Stockum W. J. The gravitational field of a distribution of particles rotating about

an axis of symmetry. Proc. Roy. Soc. Edinburgh, 57:135, 1936.

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Sustaining Consistency in Causal Cycles of Time

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