The narrow tunnel entrance was full of magma. Long before the 
fight, Sebastian’s secret weapon had started it flowing. He had only 
needed to distract the creature long enough that it would still be 
underground when lava filled the chamber. Even if the creature did 
not burn, the lava would eventually cool and encase the creature in 
stone. 


Noticing Sebastian’s smile the creature turned its attention to the 
flowing lava. The flow of lava was building speed as the pressure 
from the tunnel forced the magma through more and more quickly. 
Surprisingly, the monster did not react to this new development. It 


simply began to shift again. Its skin became much tougher. Its long 
body began to shrink, becoming thicker. Its appendages became 
squatter, and its claws became short and blunt. As the claws 
retracted, Sebastian fell to the ground, laughing now. 


This, finally, seemed to get the monster. It turned back to Sebastian, 
and in sounds of severing skin and scraping stone, began to speak. 
“FAILURE. THIS WILL NOT HOLD ME LONG. THROUGH MY 
PATIENCE OR PERSISTENCE, | WILL ESCAPE.” 


Sebastian coughed, “Not long. But maybe long enough.” The 
creature turned and met the lava, now pouring through the entrance 
in a torrent. 


Sebastian realized now, what it all meant. His sacrifice would keep 
this monster imprisoned, at least for a little while. It would prevent 
more deaths, and maybe give his brothers time to find a permanent 
solution. 


Finally, he was at peace. 


Elementary Time Travel 


Elementary Time Travel 


And Causal Analysis 


2nd Edition 


A. J. Mansfield 


Table of Contents 


Chapter 1: Introduction to Time Travel 


1.1 Causality 

1.2 Displacements 

¢ 1.3 Reaction Displacements 
¢ 1.4 Current Limitations 


Chapter 2: Time Loops 


¢ 2.1 The Bootstrap Paradox 
¢ 2.2 Estimating Loop Structure 
* 2.3 The Lottery Problem 


Chapter 3: Threads 


¢ 3.1 Thread Space 

* 3.2 The Rzewski Field 

* 3.3 Algebraic Properties of Thread Distance 
¢ 3.4 Thread Convergence and Time Loops 


* 3.5 Thread Potential 
Chapter 4: Aperiodic Loops and Chaos (coming soon!) 


Chapter 5: Retrocausal Data Transmission (coming soon!) 
Chapter 6: Applications to Information Processing (coming soon!) 


Start | Chapter 1: Introduction to Time Travel 


Chapter 1: Introduction to Time Travel 


Imagine the following time travel problem: 


You recieve a winning lottery number from the future. 
What is the probability of winning the lottery using that 
number? 


Now, one might naively assume the chance is 100%. But in fact, the 
actual probability is only slightly better than if you guessed ran- 
domly, and this can be verified experimentally. 


In order to avoid making serious mistakes and maximize success 
when time travelling, it is important to have a clear understanding of 
this and other seeming paradoxes that can arise, and for that we 
must start with the basics. 


Causality 


When discussing time travel, it is often convenient to discuss things 
in terms of “alternate timelines”, referred to as world lines. While it 
is an open question whether other world lines are “actually real” ina 
philosophical sense, the conclusions that can be drawn using this 
model are consistently borne out by experimental results, and it 
serves as a useful mental model for understanding time travel. 


In theory, the air 
currents from a sin- 
gle butterfly wing 
beat could become 
the deciding factor 
in a hurricane's for- 
mation. 


In some cases, the precise sequence of events in a world line can 
be dramatically influenced by a cascade of events, starting with 
some small predicating event. This phenomenon is referred to asa 


bifurcation, and it is useful to think of a world lines as ‘splitting’ into 
two or more separate lines. 


The typical example of this is the butterfly effect, named for the idea 
that due to the chaotic nature of weather systems, a single flap of a 
butterfly's wing could become the deciding factor in the formation or 
strength of a hurricane. 


In practice, predicating events are both a blessing and a curse: while 
very useful for modifying past events, careful precautions are re- 
quired to avoid inadvertent changes. 


Suprisingly, it is also not uncommon for two or more world lines to 
spontaneously converge to nearly identical sequences of events. 
These event sequences common to multiple world lines are referred 
to as attractor fields, and it is useful to think of world lines as 
“merging together” into one. 


A real world example of an attractor field is the one surrounding the 
fall of the Daevites. Regardless of the precise date, be it 500 BC or 
500 AD, their fall appears to lead inevitably to the events in the Re- 
naissance period up to the present. 


Classical thought about chaotic systems would lead one to believe 
that nearly every small event, every nuclear decay, protein fold, or 
cosmic ray, would result in large-scale bifurcations. However, in the 
context of time travel, this does not appear to be the general case: a 
small change may trigger a small-scale temporary bifurcation, but 
the two world lines quickly re-converge. This may be thought of as a 
generalization of the principle of least action, in as much as “rewrit- 
ing history” can be considered an “action”. This idea will be ex- 
plained more formally in chapter 3, but this approximation is good 
enough for now. 


A timeline diagram is a way of graphically representing the different 
types of causal relationships that can occur when time travelling. 
There are many different ways one can draw a timeline diagram; the 
style used in this text is one of the most common styles. 


Here is an example diagram showing time travel being used to mod- 
ify the past to change an undesirable event $E$ and ensure that de- 


wearing a pig mask and an outfit typical of private schooling 

* SCP-747-04 is a male, approximately fourteen (14) years of 
age, wearing a rabbit mask and a winter coat 

* SCP-747-05 is a female, approximately twelve (12) years of 
age, wearing a giraffe mask and a striped sari 

* SCP-747-06 is a female, approximately five (5) years of age, 
wearing a goat mask and a bright pink dress typical of a 
beauty pageant 

¢ SCP-747-07 is a male, approximately nine (9) years of age, 
wearing a cat mask and blue overalls; SCP-747-07 was also 
recovered with a note! which has since been separately 
stored 


Each instance of SCP-747 cannot speak, nor do they seem to be 
aware of humans unless taking particular interest in specific 
individuals. Mainly, SCP-747 shows interest in hand-made dolls and 
each other. It is unknown if SCP-747 has sentience, though they 
appear to be aware of their surroundings and will avoid walking into 
obstacles. SCP-747 is able to move through obstacles no thicker 
than ten (10) centimeters with ease. SCP-747 is also semi- 
corporeal; instances are able to lift and touch objects, but only for 
short periods of time before they tire. 


SCP-747 has the ability to transform humans into dolls over the 
course of approximately twenty-one (21) days. To do this, instances 
of SCP-747 surround the human and hold hands before walking 
around the human for a period of five (5) to seven (7) seconds. 
Afterwards, SCP-747 disperses and continues normal behavior. It is 
unknown which humans they choose to transform or why. SCP-747 
occasionally shows interest in a person before attempting to 
transform them. The person is able to interrupt the process by 
moving away or avoiding SCP-747; however, most people describe 
a feeling of 'thoughtlessness' or 'blankness' when in contact with 
SCP-747, thereby making the interruption difficult. 


After interaction with SCP-747, there are no immediate symptoms. 
There is currently no cure or treatment for the conversion process. 
Within fifteen (15) minutes, the target will experience numbness of 
the tips of extremities, similar to that caused by cold. Afterwards, 
over a period of approximately twenty-one (21) days, the symptoms 


sirable event $E'$ occurs instead. 


In this diagram, the double bar at the left indicates the beginning of a 
world line as it pertains to the chart. The original world line is repre- 
sented with the horizontal line, which goes until $E$ occurs. The pair 
of dashed lines extending from it correspond to the time displace- 
ments intended to correct E. In this case the displacement we care 
about is on top, the bottom one is a reaction displacement discussed 
in the next section. The top displacement triggers a predicating 
event represented by the split, and then the world line eventually bi- 
furcates into the second one in which $E'$ occurs instead. 


No Exercises 


Displacements 


The xyank is named after Dr. Thaddeus Xyank, who discovered 
many of the theoretical foundations of time travel in the 1950s and 
60s. 


In order to quantify time travel, we measure the total temporal dis- 
placement, represented with $\xi$, to describe 'how much' time 
travel a given event is. Temporal displacement is measured in 
xyanks (abbreviated "Xn") equivalent to 1 kg s3. By convention, 
positive values are used to represent displacements into the future, 
and negative values represent displacements into the past. 


The First Law of Time Travel states that, given an object of mass $m 
$ and the displacement interval $t$ the object travels, the total dis- 
placement is equal to the mass times the interval cubed: 


(1) 
\begin{align} \xi = m\; t*3 \end{align} 


For example, if | had an apparatus capable of 1 Xn, it could dis- 
place 1 milligram of matter 1 second, 1 ug of matter 10 seconds, etc. 


Exercises 


1. An 7 kg object is displaced by 4 Xn. Does it end up in the past 
or the future, and how far? 

2. Given a 5 kg test mass, what displacement would be needed 
to send it 5 minutes into the future? 

3. A 62.0 kg human is displaced 46.7 kXn at 5:00 on Monday, 
when does he arrive? 

4. Advanced A certain object starts out weighing 0.450 kg. The 
object is repeatedly displaced 5 Xn into the future, doubling its 
mass between displacements. In the limit, how far into the fu- 
ture will the object ultimately be displaced, not counting 
elapsed time between displacements? 


Reaction Displacements 


The Second Law of Time Travel states that for any displacement, 
there must be an equal and opposite displacement, or, the sum of all 
displacements is Zero. 


(2) 
\begin{align} \sum \xi = 0 \end{align} 


As a result, in order to generate a displacement to move some ob- 
ject through time, an equal and opposite reaction displacement is 
also generated that moves some other object in the opposite direc- 
tion. 


A 225 ton granite bal- 
last mass used in the 
Chronometer Upscale 
Negation Test in Mel- 
borne, Australia. 


In current real-world applications, the absolute displacement values 
achieved are incredibly tiny, usually on the order of a few 
nanoxyanks or less. As a result, commercial applications generally 
use an appropriately-sized ballast mass to limit the total reaction dis- 
placement interval. In some cases the reaction displacement can 
even be dissipated into the equipment or its surroundings without 
needing a ballast mass, however for safety reasons this is generally 


not done except at extremely low displacements. 


However, the reaction displacement can have useful applications in 
observing the results of time travel: an object so displaced will re- 
main unaffected by the changes caused by the principal displace- 
ment, allowing for comparisons across world lines. In the case of a 
person, they would be able to remember the events of their original 
world line. 


Exercises 


1. A researcher displaces an alpha particle (m = 6.646e-27 kg) 1 
day into the past. The reaction displacement is used to retain 
the contents of a hard drive (m = 0.327 kg). How long must 
the researcher wait before examining the hard drive? 

2. An integrated circuit needs to generate a displacement of 
-68.3 fXn per clock as part of its operations. Because of the 
sensitive nature of the circuit, the total reaction displacement 
time needs to be limited to under 15.0 ps per clock. How large 
does the ballast need to be? 

3. Advanced In relativity, particles that are moving close to the 
speed of light gain additional mass according to their speed, 
by a factor of $\gamma = 1 / \sqrt{1-v’2/c*2}$. If a proton trav- 
elling at 0.5c is displaced 1 year into the future, and the reac- 
tion displaces a second proton at rest, how far into the past 
does the second proton end up? 


Current Limitations 


The fundamental energy of displacement describes the theoretical 
limit on the amount of energy required to achieve a given displace- 
ment, and is approximately 4.95e-21 J/Xn. the However, modern 
techniques require orders of magnitude more energy: The current 
best, the Tachyonic lon Manual Emission and Origin Unification 
Transmitter (TIMEOUT) experiment at CERN, requires on the order 
of 1e20 J/Xn! To put that into perspective, one Xn costs more than 
the entire energy consumption of the planet in 2013. 


Techniques that function at ambient conditions require still more en- 


ergy, limiting the types of targets that can be used to just those that 
are stable under vacuum at cryogenic temperatures. 


Finally, no currently known techniques are capable of reliably dis- 
placing a target into the past in a way that keeps the target intact - 
even a very small mismatch in the calibration on current techniques 
will convert the target into an as-yet-unknown form of matter on dis- 
placement. Fortunately, this limitation does not appear to apply to 
forward displacements. 


Due to these limitations, transport of people, objects, or animals into 
the past is largely out of the question. However, it is relatively 
straightforward to transmit digital information using streams of parti- 
cles and sensitive detectors. Apparatus capable of receiving such 
streams was first developed in 1991, placing a hard cap on the earli- 
est date that one can reliably send information to. Chapter 5 covers 
details of retrocausal transmission schemes used for this purpose. 


Another important application of time travel is in computing. Many 
newer microprocessors take advantage of retrocausal connections 
as part of their branch prediction and cache prefetch hardware, en- 
abling much higher performance and clock speeds than before. This 
is not without its limitations - it is very difficult to reliably transmit 
high-entropy information to the past - but significant advances have 
been made with this technology. The reason for this limitation is cov- 
ered in chapter 2, and chapter 6 goes into detail about how retro- 
causal connections can be used for integrated circuits. 


Based on transmissions received from our future, it is believed that 
most, if not all, of these limitations will eventually be overcome, but 
as yet nothing more specific about time travel technology has been 
received. 


No Exercises 


Cover | Chapter 2: Time Loops 


Chapter 2: Time Loops 


In the first chapter, we discussed time travel in mostly informal 
terms, to help introduce the subject and give a feel for time travel. 
This chapter continues this, with additional informal techniques for 
analyzing time loops. In the next chapter we will introduce a more 
formal approach, but for the time being it is more important that the 
reader gain a more practical understanding first. 


The Bootstrap Paradox