Swansea University

Department of Physics

MPhys Thesis

Time Machines in General

Relativity

Author:Luke Clement

Supervisors:Prof. Graham Shore

Prof. Tim Hollowood

May 20, 2015

Time Machines in General Relativity

Luke Clement

May 20, 2015

Abstract

In this project the violation of causality is discussed, involving themanipulation of space-time geometries to probe for the existence ofclosed time-like curves (CTCs). Two such constructs are studied, thefirst involving the conical space-time generated by cosmic strings in rel-ative motion, and the second consisting of the collision of gravitationalshock waves and the curious space-times they generate which cause asignal to experience a discontinuous jump back in time. The conclu-sions drawn show that from a classical viewpoint, these constructs donot form CTCs, however they do serve the purpose of furthering ourunderstanding of the conditions necessary to achieve causality viola-tion.

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Contents

1 Introduction 3

2 Superluminal Motion in Special Relativity 6

3 General relativity 8

4 Cosmic Strings 104.1 Gott Time Machine . . . . . . . . . . . . . . . . . . . . . . . 12

5 Gravitational Shock Waves 175.1 Smoothed-out Shock Wave Analysis . . . . . . . . . . . . . . 215.2 Non-interacting Shock Waves . . . . . . . . . . . . . . . . . . 235.3 Interacting Shock Waves . . . . . . . . . . . . . . . . . . . . . 26

6 Conclusions 31

7 Appendix 337.1 Proof of Geodesic Equation . . . . . . . . . . . . . . . . . . . 33

8 Acknowledgements 36

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1 Introduction

The intent of this project is not to establish a working time machine, but

rather to look at certain constructs one can build, using the principles of

Einsteins theory of general relativity. We can then examine the physical

consequences of the application of these principles in extreme conditions.

Considering the behaviour of the laws of physics under extreme theoretical

conditions is an important tool for physicists in order to push the boundaries

of our knowledge, and to raise further questions of our understanding of the

universes inner workings.

The idea of time travel is a fascinating and subsequently popular concept

in physics, perhaps more so for the non-scientific community, but much

theoretical work has been done in the field since Einstein first introduced

his theory [1]. The work consists of manipulating space-time into various

constructs which would allow closed time-like curves (CTCs). These are

signal trajectories that travel through distorted space-time and end up in

the past light-cone of the emitter, the implication being that causality is

violated as a signal will arrive before its own emission. This would allow

photons of light and massive particles that travel along this time-machine

trajectory to effectively experience motion backwards in time.

Physicists and philosophers alike have thought about the problems con-

cerning the violation of causality. In a paper written by Igor Novikov along

with several others in 1990, they addressed the idea of paradoxes arising

from CTCs stating in a principle, now known as the Novikov Self-consistency

Principle [2], that any events which would result in a time paradox are im-

possible. Stephen Hawking also proposed a prominent conjecture in his pa-

per published in 1992, titled Chronology Protection Conjecture [3] which

states that the laws of physics prevent the existence of CTCs. A third less

accepted view on the resolving of temporal paradoxes is the Many-worlds

Interpretation. This applies the concept in quantum physics that every

quantum event creates multiple time-lines that branch off from one another.

It considers the procedure of entering the past to be a quantum event which

would imply the avoidance of paradoxes as the signal sent back in time would

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simply access a different time-line, and therefore not have any affect on the

original time-line. Although these viewpoints are not entirely compatible

they do all share the view that time paradoxes are very unlikely to arise,

and until a working theory of quantum gravity is formulated, ascertaining

this is rather unlikely.

These conjectures and principles (excluding the Many-worlds Interpre-

tation) are the results of theoretical work attempting to solve Einsteins

field equations under various conditions, with the purpose of finding gen-

eral theorems. This project however, will look in more detail at previous

research which involves creating relatively simplistic models consisting of a

space-time configuration allowing the existence of CTCs, and summarising

and comparing these attempts. This is an equally fruitful approach to work

on CTCs as it allows us to increase our understanding on the assumptions

and conditions that we impose when trying to find constructs allowing the

existence of CTCs. Various research has been done on CTCs, many of which

utilise such space-times as those generated by black holes, wormholes, the

Godel universe, cosmic strings and gravitational shock waves, the latter two

are looked at in detail in this project. One of the earliest attempts of finding

a CTC can be found in Willem van Stockums 1937 paper [4] in which he

derives a solution to the Einstein equation where the gravitational field is

generated by an infinitely long cylinder of dust, rapidly rotating creating an

eternal circular CTC around it. Another similar example that I mentioned

above is Godels solution of the Einstein equation [5] which involves the uni-

verse with a non-zero cosmological constant, filled with rotating dust. This

works in much the same way as van Stockums dust cylinder but both are

considered to be unphysical as van Stockums solution requires an infinitely

long source, and Godels requires a non-zero cosmological constant as well

as a universe unlike our own. These are examples of the considerations

physicists have had to take when discussing how physically realisable these

various constructs are and we look to apply these considerations throughout

this project.

Firstly we look at the foundations of all the research done, with special

relativity and the concept of superluminal motion. The violation of causality

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is not prohibited in special relativity. A signal travelling faster than the

speed of light is enough to violate causality between two space-like events.

We discuss this effect in detail in order to reinforce the concept of backwards

in time motion which, of course, plays an important role in the various

attempts discussed in this project. In the following section we then look

at general relativity and highlight the differences between it and special

relativity, with regards to superluminal motion, and the requirements for

backwards in time motion. Once these concepts are discussed sufficiently,

we move on to looking at examples that make use of them in an attempt to

find CTCs, first of all with cosmic strings.

Cosmic strings are supposed topological defects in the universe that un-

der the right conditions distort the space-time surrounding them to have the

geometry of a cone. The first construct we look at, the famous time machine

theorised by J. Richard Gott in 1991, uses the space-times created by two

cosmic strings moving in opposite directions with relative speed in order

to find a trajectory through these distorted space-times which describes a

CTC [6]. We then discuss whether this is a physically viable time machine

and consider work done by Deser, Jackiw, and t Hooft shortly after Gotts

paper was published which scrutinises the model.

After the conclusion of the cosmic strings section, we turn our attentions

to gravitational shock waves as our new source for distorting space-time

Gravitational shock waves have the rather interesting property that, under

certain assumptions, any particles travelling towards them experience a dis-

continuous jump backwards in time. This effect is clearly self-explanatory

in the reasons one would look to explore to see if this can be exploited for

the purposes of finding a CTC. We look at a similar construct discussed by

Graham Shore in his paper on constructing time machines [7], which takes

inspiration from the Gott model and considers the case of two shock waves

moving towards one another, and consider a trajectory which describes a

CTC.

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2 Superluminal Motion in Special Relativity

We first look at superluminal motion in special relativity in order to reinforce

the important concepts of the theory relating to causality, and to highlight

the differences between special and general relativity. Consider a frame S

moving with velocity , relative to S for particles travelling faster than light,

see Figure 1. Clearly tq < 0, therefore motion from O to Q is backwards in

Figure 1: Sketch of a path OQ in a frame S and another relative frame S

moving with velocity

time.

We need to consider two important properties of special relativity. Firstly

that all initial observers (with relative speeds v < c) agree that all time-like

paths involve motion forwards in time. Secondly, we also note that for

space-like paths (i.e. path of a superluminal particle), certain observers say

motion is forwards in time, others say it is backwards in time.

The condition for S to have OQ as backwards in time is that the angle in the x-axis is greater than that of the path OQ. Let OQ have warp

6

factor , this implies a speed = c. Since tan = c , the condition is

tan =1

= > c

(1)

In terms of the Lorentz transformation, (c=1)

tQ = (tQ xQ) (2)

As tQ < 0 i.e. tQ xQ < 0 this implies

>tQxQ

=1

(3)

In special relativity, path OP must be possible since the first postulate

Figure 2: Sketch of path OP in frame S which shows P in the past light-coneof O

requires the equivalence of all (global) inertial frames. Since OQ implies

backwards in time motion in frame S, backwards in time motion such asQP must also be possible in frame S. For violation of causality in special

7

relativity, we require both superluminal motion, and that all global inertial

frames are equivalent. This second condition is encoded by the fact that the

space-time in special relativity is flat Minkowski space-time.

3 General relativity

In the theory of general relativity, the space-time is Riemannian rather than

Minkowski as in special relativity. Riemannian space-time is defined by a

metric of homogeneous quadratic form

ds2 = gdxdx (4)

This space-time is locally flat, and the metric describes mathematically the

weak equivalence principle (WEP). This principle states that at each point in

space-time, there exists a locally inertial frame. This is not enough by itself

to ensure that faster-than-light motion implies the violation of causality. To

do this we also make the assumption that in all locally inertial frames, the

laws of physics are identical at different points in space-time and these laws

reduce to their special relativity form at the origin of each locally inertial

frame. This is the Strong Equivalence Principle (SEP), and it takes the place

of globally inertial frames in special relativity in terms of linking faster-than-

light motion and the violation of causality. Particle trajectories in curved

space-time are described by geodesics, i.e. paths that extremise the space-

time interval along them. The general equation to describe a geodesic in

any space-time is given by

d2x

d2+

dx

d

dx

d= 0 (5)

Where are the Christoffel symbols defined as

=1

2g(

d

dxg +

d

dxg d

dxg) (6)

8

These describe the curvature of the coordinates in a given coordinate sys-

tem, specifically the coordinates of space-time general relativity. See the

Appendix for a comprehensive proof of the geodesic equation. Having dis-

cussed the principles and differences between the two theories of relativity,

we can now apply these when looking at the constructs in the following

sections.

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4 Cosmic Strings

Cosmic strings are hypothetical defects in the universe which are supposed

to have formed due to a symmetry-breaking phase transition in the early

universe. In 1976 Tom Kibble first contemplated the idea of topological de-

fects in the early universe in a paper in which he discussed the consequences

of symmetry-breaking phase transitions [8]. Cosmic strings are formed when

an axial or cylindrical symmetry is broken, and are incredibly thin, typically

around 1029cm, similar to the Compton length of a particle, hence in manytheories they are idealised to being infinitely thin [9]. To derive the metric

associated with a cosmic string we use the Gauss-Bonnet theorem which

connects the geometry of a surface to its topology. The Gauss-Bonnet the-

orem states, supposing is a two-dimensional Riemannian manifold, with

boundary K.dS = 2pi(M)

kg.dl (7)

Where K is the Gaussian curvature, the element dS is the surface area and

dl is the line element along the boundary , dl = rd. kg is the geodesic

curvature, which for a circle is given by kg = 1/r. (M) is the Euler

Characteristic, which for a two-dimensional disk is (M) = 1. In our case

the Gaussian curvature K is given by the Ricci Tensor Rxx. Therefore,RxxdS = 2pi

1

rrd (8)

This leads after substitutions to the Einstein Equation,

R = 8piT +1

2gR = 8pi

(T 1

2gT

)(9)

where T is the trace of the energy-momentum tensor. We need only know

the x-components of the Riemannian Tensor. The energy-momentum tensor

is aligned along the z-axis, so the x-components do not contribute. This

10

leads to the Gauss-Bonnet theorem becoming8pi(x)(y)dxdy = 8pi = 2pi 2piA (10)

Then A = 1 4, so the metric associated with a cosmic string is given by

ds2 = dt2 + dr2 + (1 4)2r2d2 + dz2 (11)

From this we can see by redefining the angle and considering the range of

this new angle that there is a wedge of = 8pi missing from flat Minkowski

space-time. This metric solves the Einstein Equation (with = 0) if the

energy-momentum tensor is

T00 (1 4)2(~r) (12)

Where ~r is the transverse position vector (r, )

We can alter this set up slightly by considering the case of a spinning cos-

mic string with angular momentum. For a cosmic string spinning about the

z-axis the metric becomes

ds2 = (dt2 + Jd)2 + dr2 + (1 4)2r2d2 + dz2 (13)

This solves the Einstein equations with

T00 (1 4)2(~r) (14)

T0i Jijj2(~r) (15)

After redefining the angular and time components, = (1 4) and t =t+ J respectively, the metric looks flat

ds2 = dt2 + dr2 + r2d2 + dz2 (16)

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We have 0 < < (1 4)2pi and we must identify points t and t + 2piJ ,so the t coordinate is helical. Thus the space-time differs from Minkowski

space-time in its global properties, but locally it is the same. If we consider

the path with constant t, r, and z, we have

ds2 = d2(J2 (1 4)2r2) (17)

This is negative for sufficiently small r, i.e.

r < rcrit =J

1 4 (18)

These geodesics are closed time-like curves.

4.1 Gott Time Machine

John Richard Gott proposed a well-known model for the possibility of CTCs

using two cosmic strings moving with relative velocity. [10] The existence of

these CTCs are due to the fact that the conical space-time around a cosmic

string can be exploited to make trajectories that appear to be superluminal.

By patching together the space-times of two cosmic strings, both with mass

M , for simplicity, it appears to allow CTCs that traject through the conical

space-times. We assume in this case that the space-times of each string does

not have an effect on the other. We derive the Gott condition for CTCs as

follows:

Figure 3: Cross-sectional sketch of the cosmic string space-time in the (x, y)plane. Comparisons between two trajectories are drawn, the straight pathfrom A to B, and path ACCB which crosses the wedge. The points C andC are identified, and the paths intersect the wedge at right angles.

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y = x sin d cos (19)

x2 l2 = y2 d2 (20)

The path ACCB begins at

EA = (t = lv,x) (21)

and ends at

EB = (t =l

v, x) (22)

This corresponds to a space-time interval EAEB of

4l2

v2+ 4x2 (23)

This is space-like ifl2

v2< x2 (24)

Using equations (19) and (20) we reach the equation

1 l2

v2= sin2 2d

xsin cos+

d2

x2cos2 d

2

x2(25)

From this we arrive at the Gott condition

v > cos (26)

So for v > cos a signal along the path ACCB can arrive at B sooner than

a signal directly from A to B. EA and EB are space-like separated and can

be made simultaneous with a suitable Lorentz transformation, which cor-

responds to the string moving with velocity v. Using this we can find aCTC as shown in Figure 4.

Patching space-times with deficit angles together can only be done if cer-

tain global conditions are satisfied. This was investigated further by Deser,

Jackiw and t Hooft in 1992 and they showed that this Gott system is un-

physical once the interactions of the strings is introduced.[11] First we must

identify the points x and x, where x = Mx and M is the rotation through

13

Figure 4: Sketch showing Gott Model. The path ACCBDDA is a closedtime-like curve in this space-time with two strings moving in opposite direc-tions. The existence of this CTC requires the Gott condition, v > cos tohold

the deficit angle 2 = 2piM allowing also a Lorentz transformation

x = L1v MLvx (27)

For two cosmic strings we have

x = LvML1v L1v MLvx (28)

But this is meant to be viewed from infinity as a single cosmic string, as far

as the global matching is concerned. So we require

x = L1v MLvx (29)

for some v, M For consistency we require

Tr M = TrL2vML

2v M (30)

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To show this, we need to define the matrices M and L2v.

M =

1 0 00 cos 2 sin 20 sin 2 cos 2

(31)This is a rotation through 2.

L2v =

cosh 2 sinh 2 0sinh 2 cosh 2 00 0 1

(32)This is a Lorentz boost.

We need, for some A,

M =

1 0 00 cos 2A sin 2A0 sin 2A cos 2A

(33)We calculate

TrL2vML2v M = 2 cos

2 2 2 cosh 2 sin2 2+ 1 + (1 cos 2)2 sinh2 2(34)

and

Tr M = 2 cos 2A+ 1 (35)

Now equating (34) and (35) we get

cos 2A = cos2 2 sin2 2 cosh 2 + 12

(1 cos 2)2 sinh2 2 (36)

This is equivalent to

cosA = cos2 sin2 cosh 2 sin A2

= sin cosh (37)

15

Recall = piM , A = piM and tanh = v = cosh = 11v2 , then

sinpiM

2= sinpiM

11 v2 sin

piM

2= sinpiM(v) (38)

Now for positive values of M , the L.H.S of (38) is less than 1, and this

condition implies

v < cospiM (39)

if v > cospiM , then M is not a rotation but a boost. This would mean that

the centre-of-mass motion of the cosmic string pair is tachyonic. Provided

we require non-tachyonic centre-of-mass motion we must have v < cospiM =

cos [11][13]. But this is the exact opposite of the Gott condition for the

existence of CTCs derived earlier. However this does not rule out CTCs

completely, in their paper Jackiw, Deser and t Hooft did show that CTCs

can exist in the space-time inherent in the Gott Model at space-like infinity.

Though they regarded it as an unrealistic boundary condition, we cant

however rule it out, given our lack of a working quantum theory of gravity,

as suggested by M.P. Headrick and J.R.Gott in their paper on space-times

containing CTCs [12]. We cannot rule out certain boundaries as unphysical

without first understanding the universal nature of gravity and these space-

times. Further papers investigating the Gott model have looked at the total

energy-momentum associated with the model and reached two important

conclusions. The first co-authored by S.M. Carroll, E. Farhi and A.H. Guth

shows that the Gott time machine cannot exist in an open universe with a

timelike total energy-momentum [13]. The second conclusion reached by t

Hooft in his 1992 paper, is that in a closed universe the space-time suggested

by Gott would collapse to having zero volume before any CTCs are able to

form [14].

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5 Gravitational Shock Waves

In this section we look at the possibility of exploiting the gravitational effect

of shock waves on a signal to hopefully show the existence of CTCs in this

space-time. This section closely follows the work done by Graham Shore

in his 2002 paper on constructing TIme Machines [7]. The metric for a

gravitational shock wave is described by the Aichelburg-Sexl metric [15]

ds2 = dudv + f(r)(u)du2 + dr2 + r2d2 (40)

where

u = t z (41)

v = t+ z (42)

This metric describes a shock wave travelling at the speed of light in the

positive z direction and f(r) is the profile function in the transverse direction

defining the characteristics of the shock wave. Well only consider the case

of a homogeneous beam of constant density (r) as a matter source for the

shock wave. This solves the Einstein equation as its profile function is given

by f(r) = 4pir2 for impact parameters less than the size of the beam [7].We will exclusively look at scenarios with this condition in the following

sections. In this first instance we assume the shock wave lies along t = z,

then we have the geodesic Lagrangian associated with this space-time metric

L = uv + f(r)(u)u2 + r2 + r22 (43)

Now using the Euler-Lagrange equation:

d

d(L

x) =

L

x(44)

We can calculate the geodesic equations

u = 0 (45)

17

Figure 5: Sketch of a Gravitational Shockwave lying along t = z

v f(r)(u)u2 2f (r)ru = 0 (46)

r 12f (r)(u)u2 r2 = 0 (47)

+2

rr = 0 (48)

Now we set u which leads tod

du(u) = 0 = u = 1 (49)

with the now signifying a derivative with respect to u. We also take to be

a constant, and with these alterations we can rewrite the geodesic equations

as follows

u = 0 (50)

v f(r)(u) 2f (r)(u)r = 0 (51)

r 12f (r)(u) = 0 (52)

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We also have the extra equation

L = , where =0, for massless particlesconstant > 0, for massive particles

So we have

L = = v + f(r)(u) + r2 (53)

v = + f(r)(u) + r2 (54)

We have the radial equation,

r =1

2f (r)(u) (55)

We can solve this equation by writing a solution of the form

r = b+ a1u(u) + a2u(u) (56)

Using the fact u(u) = (u) and u(u) = 0 we have

r = a1(u) + a2(u) (57)

and the second derivative is

r = a1(u) a2(u) (58)

=1

2f (r)(u) (59)

=1

2f (b)(u) (60)

Therefore

a1 a2 = tan(2

) + tan(

2) =

1

2f (b) = tan(

2) (61)

We can look at the case where a null geodesic in the negative z direction col-

lides with a shock wave moving in the exact opposite direction with impact

19

parameter b. This is described by the geodesic equations

v = f(b)(u) + tan2

2u(u) (62)

r = b u tan 2

(63)

where the deflection angle satisfies tan 2 = 12f (b) In this case we can

(a) (z, t) plane(b) (z, r) plane

Figure 6: Sketches of null geodesics in this Aichelburg-Sexl metric in the(z, t) and (z, r) planes. A geodesic is shown that colliding with the shockwave at u = v = 0 and jumps discontinuously back in time from (t = 0, z =0) to the point (tP = 0, zP =

12f(b)) These jumps are illustrated by the

dotted lines in the figures. The geodesic is then deflected by an angle where tan 2 = 12f (b) towards the r = 0 axis.

see that a geodesic can be reflected back in the positive z direction, given

a large enough impact parameter leading to > pi2 . The main idea to

take note of here is that the geodesics make a discontinuous jump from the

collision surface at the origin u = v = 0 to a point further back in time at

u = 0, vP = f(b). For homogeneous beams, all impact parameters b lead to

a discontinuous jump backwards in time for any geodesic colliding head-on

with the shock front. We look to manipulate this phenomenon in order to

create a situation that allows CTCs. However this case with an infinitely

thin shock wave is not a realistic outlook and in the next section we explore

the results of allowing a shock wave to have a more realistic physical width.

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5.1 Smoothed-out Shock Wave Analysis

Now we consider the case where the shock wave is no longer infinitely thin

in order to show that there is nothing truly problematic about the delta

function shock wave. We can do this by smoothing out the delta function,

to give the shock wave a physical width

(u) 2a3

pi(u2 + a2)2(64)

where a is the width of the shock wave. When a = 0 we get the delta

function back and this is essentially the case of the infinitely thin shock

wave. Also we can see that the integral of this function with respect to the

affine parameter u, is normalised

2a3

pi(u2 + a2)2du = 1 (65)

The derivative of the function is given by

u((u)) = 8a3u

pi(u2 + a2)3(66)

For a homogeneous shock wave beam, the profile function is given by

f(r) = r2 (67)

where 4pi, and the derivative is

df

dr= 2r (68)

Now solving the radial equation, with being a parameter that gives the

freedom to turn on an angle of incidence,

r 12

(2r)(

2a3

pi(u2 + a2)2

)= 0 (69)

21

The solution to this equation is then

r =

ba2 + u2 sin

[1 + 2api

(pi2 + arctan

[ua

])]a

1 + 2api

(70)

where = pi2 corresponds to zero angle of incidence. We can investigate

what happens to r as u approaches negative and positive infinity, by power

expanding r, again with zero angle of incidence.

At negative infinity:

r = b+ a3b

3piu2+O

[1

u

]3 b as u (71)

At positive infinity:

r =b sin[

pipi + 2a]u

a

1 + 2api

b cos[pipi + 2a] + ... as u (72)

Now using this we can calculate the derivative of v with respect to u as

written in (54). For a null geodesic i.e. = 0

v =

b2(pi(a2 + u2) + (a2pi piu2 + 4a3) cos[

1 + 2api (pi + 2 arctan[ua ])

]+ 2a

piupi + 2a sin

[1 + 2api (pi + 2 arctan[

ua ])

]2a2(a2 + u2)(pi + 2a)

(73)

where and a are the strength and width of the shock wave respectively,

and b is the impact parameter. We can now integrate this equation in order

to calculate v and setting appropriate values for these three variables we can

plot v and r in the (z, t) and (z, r) planes.

As we can see, with this altered delta function giving the shock wave a

physical width, there is no longer a discontinuous jump backwards in time,

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(a) (z, t) plane (b) (z, r) plane

Figure 7: Plots of v and r in the (z, t) and (z, r) planes. A geodesic is shownthat experiences continuous motion backwards in time.

instead we see a geodesic approaching the shock wave, and then experiencing

motion backwards in time, as well as being deflected in its direction. This

analysis of the smoothed-out delta function where weve looked at the more

realistic case of a shock wave with a physical width, has shown the resulting

continuous backwards in time motion, and therefore shown that there is

nothing truly problematic about the delta-function shock wave. We have

not yet however, shown that this delta-function shock wave gives rise to a

CTC, as a single shock wave isnt sufficient for a signal to be sent into the

past light-cone of the emitter. In the next section we take inspiration from

the case of the Gott time machine and try to create a model with two shock

waves that would allow the existence CTCs.

5.2 Non-interacting Shock Waves

We now turn our attenition to a construct where two shock waves travelling

towards one another meet at the origin, as shown in Figure 8. A signal

following a null geodesic is struck by the first shock wave travelling in the

opposite direction at point A, it has impact parameter a. It experiences a

discontinuous jump back in time to point B. For certain impact parameters,

its angle of deflection a can be large enough to alter its direction along the

z-axis, and reflect it back into the path of the second shock wave. When it

23

collides with the second shock wave at point C in an angled collision with

a second impact parameter b and angle of incidence , it is again sent back

through time to D and carries along its path towards A. Given certain pa-

rameter choices, we try to show that point D is within the past light-cone

of A.

Figure 8: Sketch of two gravitational shock waves colliding at the origin.The shock waves are generated by homogeneous beams, the first lying alongu = 0 and the second along v = 0 as they are unaffected by each otherin this case. Each beam has energy densities described by profile functions4pi(r) = r2 and r2 respectively. Trajectory shown consists of a nullgeodesic colliding with shock wave 1 at A, jumping discontinuously back intime to point B and changing direction towards point C where it collideswith shock wave 2 and experiences another discontinuous jump back in timeto point D and carries along a new direction towards point A.

Approaching A, the initial signal follows null geodesic

v = v0 (74)

r = a (75)

Following impact with shock wave 1 and jump to B, BC geodesic is then

24

described by

v = v0 + f1(a) + u(tana2

)2 (76)

r = a u tan a2

(77)

where tan a2 = 12f 1(a). We can see

vB = v0 + f1(a) (78)

rB = a (79)

The signal then heads toward the second shock wave where it collides at an

angle at C. (vC = 0, uC = u0, rC = b), v0 and impact parameter b can be

chosen, and the angle of incidence is = pi a. We find

vC = u0 = (v0 + f1(a))(tan a2

)2 (80)

rC = b = a+ (v0 + f1(a))(tana2

)1 (81)

After the second discontinuous jump back in time, the geodesic carries on

along the trajectory DA which is described by

u = u0 + f2(b) + v

(tan

2

)2(82)

r = b v tan 2

(83)

where

tan

2= tan

b2

+ tan

2= tan

b2

+

(tan

a2

)1(84)

and the point D is located at

uD = u0 + f2(b) (85)

rd = b (86)

As the space-time in the region where v > 0 and u < 0 is flat, the space-time

25

interval for the DA trajectory is then described by

s2DA = (uA uD)(vA VD) + (rA rD)2 (87)

We can eliminate u0 and v0 in favour of the two profile functions,f1(a) and

f2(b)

s2DA = f1(a)f2(b) (a b)(f1(a)

(tan

a2

)2+ f2(b)

(tan

a2

))(88)

As we are only looking at the case where the shock waves are generated

by homogeneous beams with profiles f1(r) = r2 and f2(r) = r2, thisleads to

s2DA = (ab)b2 + (1b

a)a2 (89)

where and are the energy densities of the two beams which we are free

to choose. The initial signal trajectory which can also be chosen, is specified

by the impact parameter a, and the location of the first collision v0. v0 can

be traded for the second impact parameter b which can also be chosen, hence

all parameters in the space-time interval in s2DA can be chosen freely, and

therefore we can do so in such a way which results in a negative space-time

interval

s2DA < 0 (90)

which implies that D is in the past light-cone of A, hence it appears that it

is indeed possible for this construct to allow CTCs.

5.3 Interacting Shock Waves

So far in this section we have been looking at the case of two shock waves

which do not interact with each other. Neglecting the effect of each shock

waves distortion of space-time on each other is not a realistic assumption,

and well now consider the effect it has on this two shock wave case. Firstly

as discussed in [7], we should address a few properties of the interaction

between these gravitational shock waves before reaching any conclusions on

the existence of CTCs arising from this construct. First is the observation

26

originally made by Roger Penrose [16] that for certain impact parameters

there is a possibility that a closed trapped surface could form in the collision

region. This was then discussed more recently and applied to the case of

beam collisions by Kohlprath and Veneziano [17]. This means that the im-

pact parameters which are needed to complete a time machine trajectory

in this construct could well lead to the formation of an event horizon, as any

signals which fall within the requirements to follow the trajectory will not

be able to escape the continual loop. The second effect of the interaction to

consider is that the future region of the collision is not flat, and the metric

describing the space-time is difficult to derive. Lastly and arguably most im-

portantly for us, is the fact that for shock waves generated by homogeneous

beams, the shock wavefronts themselves are distorted into concentric spher-

ical shells, which converge to focal points. This has important consequences

on the ultimate outcome of this construct.

Its possible to describe the shock wavefronts as being made up of a series

of null geodesics and therefore we can use the same geodesic equations as

in the previous subsections to describe the wavefronts themselves. First of

all the beams collide at the origin, u = v = 0, jumping and deflecting, and

focusing to their respective points vF =1 and uF =

1 for shock wave 1

and 2. A reasonable assumption to make at first is that given a very small

distortion from the shock waves, so that the signal trajectory is far away

from the focusing zone, this would give more credence to the non-interacting

case we discussed in the previous subsections. For this to be realised, we

would require that uF uC and vF vA. We can write these in terms ofthe parameters we are free to choose, the impact parameters a, b and also

, to find

uC = u0 =1

(1 b

a

)(91)

vA = v0 = ab (92)

So the conditions to ensure the signal trajectory is far from the focusing

zone are

1 ba 1 (93)

27

ab 1 (94)

with the latter equation being equivalent to tan a2 tanb2 1 These con-

ditions should be very familiar as they are the exact coefficients that we

derived earlier in equation (89) describing the space-time interval between

D and S, and therefore these conditions are constraints on the magnitude of

the backwards-in-time motion. Once again we return to the non-interacting

case where the shock waves are flat discs propagating at the speed of light.

They then collide, causing the generators to jump and deflect as shown in

Figure 9.

Figure 9: Sketch showing the discontinuous jump experienced by the gen-erators of Shock wave 2 as they collide with Shock wave 1 at the point E(z = 0). Each generator jumps to a point forming the surface E+ and isthen focused towards the point zF =

1 , (r = 0). The W are spherical sur-

faces consisting of the focusing of converging, equal-time wavefronts. Theimportance of point B is discussed below

The wavefronts are spherical shells that converge towards the focal point

zF =1 , r = 0 and the geodesic equations describing the path of their

generators after the collision are

v = a2(1 u) (95)

r = a(1 u) (96)

28

it then follows that the wavefronts are the surfaces

r2 = 1v(1 u) (97)

The most important surface however, is the described by the equation

z = 12r2 (98)

This is the surface after the discontinuous jump, E+, and is the limit where u

approaches 0 from above (u 0+) for the shock wavefronts. We also identifythe corresponding surface E as the limit where u approaches 0 from below(u 0) for the shock wavefronts before the collision. The future region ofthe surface E+ is the curved space-time zone. The significance of this is that

a point B behind the collision surface E+ at a later time is in this curved

space-time. Taking a look back at Figure 8, we can see that the point B,

at which the signal collides with Shock wave 2, is of course in this future

curved region. The key observation to make here is that, the collision of the

wave fronts result in the generators being jumped back in time by the same

amount as the signal when it collides with Shock wave 1. Specifically, the

generators of the wavefront of Shock wave 2 with r = a , collide with Shock

wave 1 at the origin u = v = 0, and jump back in time along the u=0 line

by an amount f1(a) corresponding to the profile function of Shock wave 1.

Looking at the collision between the signal and Shock wave 1 at point A, we

derived earlier that the shift back in time for the signal, along the u = 0 line

is also by that same amount f1(a). The logical conclusion is that the point

B must be further forward in time and have a greater z than the wavefront

of Shock wave 2.

Therefore when we take into account the interaction between the shock

waves, the scenario changes from the one we see in Figure 8 as the shock

waves themselves experience jumps back in time and are distorted. This

means that the point C is no longer a point at which the signal collides

with Shock wave 2 as the signal is behind the curved shell of the wave front,

and remains so forever since they are both travelling at the speed of light.

29

Furthermore, as we described earlier, the metric describing the space-time

of the future, curved region is unknown, and given that the signal travels in

this future region, it is not possible to use the same geodesic equations as

before to calculate its path. Therefore we conclude that when the interaction

of the shock waves is accounted for, the formation of CTCs is not possible.

6 Conclusions

Throughout this project, we have looked at a couple of models constructed

in order to allow the existence of closed time-like curves. We have discussed

them in detail and concluded that while at first they seemed to allow CTCs,

ultimately they fail to produce violation of causality.

To begin, we discussed the foundations of the project in the theories of

special and general relativity. We briefly looked at the necessary conditions

for the violation of causality in both theories and discussed the similarities

and more importantly the conflicts between them. Once these had been

discussed sufficiently, we looked to examine two constructs claiming to allow

the formation of CTCs.

Firstly the Gott Time Machine formed by two cosmic strings passing each

other with relative velocity. We used the Gauss-Bonnet theorem to derive

the conical space-time geometry generated by the cosmic strings, which we

then used to find that a trajectory circling around the two strings in the

Gott model was effectively superluminal, subject to the Gott condition that

we derived as v > cos. But, following subsequent work on the model [11],

we came to the conclusion that for this CTC to form under appropriate

boundary conditions, the opposite of this Gott condition was necessary i.e.

v < cos. We briefly discussed the CTC that forms at space-like infinity

and whether or not this was realisable, especially given our lack of a working

quantum theory of gravity.

We then turned our attention to gravitational shock waves. We showed

that a signal colliding with a gravitational shock wave experiences a dis-

continuous jump back in time and is deflected by an angle determined by

its impact parameter and the profile function of the shock wave. Given

30

that this infinitely thin, delta-function shock wave at first seems unphysical

we then introduced a smoothed-out delta function in order to show that a

shock wave with a real physical width still produces the same effect on a

signal colliding with it. This effect was slightly different in that the signal

experiences a continuous motion backwards in time but nevertheless, shows

that there is nothing too problematic about the delta function shock wave.

We then looked at the case of two colliding shock waves described in [7],

and a signal trajectory utilising these discontinuous jumps and deflections,

which resulted in the signal returning to its own past. However this was an

unphysical idealisation as the interaction between the shock waves was not

accounted for. This interaction was then introduced and the consequences

discussed, eventually reaching the conclusion that given the shock waves

themselves are made up of a series of null geodesics, they also experience

the same discontinuous jump back in time and that this is sufficient to ensure

the failure of the construction in allowing CTCs to form.

Though neither of these constructs resulted in the allowance of CTC

formation, there are likely to be many other constructs yet to be considered

which could prove more fruitful. These constructs were rather simplistic

models, in (2+1)-dimensions and much work is being done on more complex

constructs, extending to higher dimensions. Whether the existence of CTCs

is prohibited by the chronology protection conjecture [3] remains to be seen,

but a significant amount of work is now being done on quantum gravity

which could, quite possibly, lead to progress in our understanding of the

conditions necessary for the existence of CTCs.

31

7 Appendix

7.1 Proof of Geodesic Equation

Define notation, F = gxx where x = dx

d

We need to extremise the space-time along the path, i.e.

I =

ds =

dgxx =

dF (99)

In general, if

I =

dtf(x, x) (100)

where x = dxdt then the variation is

I =

dt(

f

xx+

f

xx) (101)

=

dt(

f

xx+

f

x

d

dtx) (102)

=

dtx(

f

x ddt

f

x) (103)

So if I = 0, for any path change x(t) we require

f

x ddt

f

dx= 0 (104)

Now, applying the general formula implies

I = 0 = F12

x dd

(F

12

x) = 0 (105)

= 12

1

F12

xgx

x dd

(1

F12

gx) = 0 (106)

= 1F

12

(1

2

xgx

x x

gxx gx) + 1

2

1

F32

gx dF

d= 0

(107)

32

Now the part within the parentheses of equation (107) is equal to zero if

x +1

2g(

xg +

xg

x)xx = 0 (108)

i.e. if

x + xx = 0 (109)

This is the geodesic equation. Now we show that if this geodesic equation

holds, then also dFd = 0 and so I = 0 We calculate

dF

d=

xgx

xx + 2gxx (110)

=

xgx

xx 2gxxx (111)

Using the geodesic equation

= xxx(

xg

xg

xg +

xg) (112)

from the definition of , and by switching dummy indices, we get

= xxx(

xg

xg) = 0 (113)

So, assuming the geodesic equation holds, then dFd = 0 and therefore

I = 0 (114)

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[13] S. M. Carroll, E. Farhi, and A. H. Guth. Gott Time Machines Cannot

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8 Acknowledgements

Id briefly like to thank both of my supervisors, firstly Professor Graham

Shore, for offering continued guidance at a difficult time and whose previ-

ous work inspired the majority of this project, and secondly Professor Tim

Hollowood, for volunteering to take over the supervisory role when it was

needed.

35