In order to understand how time loops work, it is necessary to intro- 
duce a notion of the ‘connectedness’ between world lines. While 
presently, every known world line is, one way or another, reachable 
from every other known world line, it is theorized that in the primor- 
dial universe, there may have been additional world lines leading to 
the current set, but without any way back. These extra theoretical 
world lines, while unobservable, are very important for explaining the 
bootstrap paradox, and in predicting what types of time loops are 
likely to have formed. 


As an example, consider this loop: 


The upper line here is the spurious line that bootstraps the time 
loop. Although it does not receive a message, it spontaneously de- 
cides to transmit message $M$§ to the past. The past receives this 
message, and then also transmits the same message $M§ to the 
past, forming a time loop. 


In some cases there may be more than one potential spurious world 
line that could have lead to a given loop. For example, take this sec- 
ond-order loop: 


(Note that reaction displacements have been omitted from this dia- 
gram in order to simplify it.) 


In this case, the main world lines $B$ and $B'$ each transmit mutu- 
ally exclusive messages $M$ and $M'$, each prompting the other to 
send its message. In this case, the situation could have been boot- 
strapped by either $A$ or $A'$. 


This can easily be generalized to an arbitrary number of cases: 


Each world line $B_k$ transmits a distinct message $M_{k+1}$ that 
then prompts $B_{k+1}$ to send its own $M_{k+2}$, up to $B_n§, 
which transmits $M_1$ again, causing the cycle to repeat. In this 
case, the entry point could have been any of these, depending on 
what message $A$ transmitted initially. 


Exercises 


1. Draw a complete timeline diagram for a basic fourth-order pe- 
riodic time loop. 

2. Advanced In some cases, a given world line may be part of a 
time loop more than once. Draw a timeline diagram for a loop 
in which world line $C$ sends and receives messages from 
both $B_1$ and $B_2$, without any direct communication be- 
tween $B_1$ and $B 2$. 


Estimating Loop Structure 


In many cases, it may only be possible to observe some portions of 
a time loop. But even in these cases, it may still be possible to infer 
part or all of the loop's structure from the portion you can observe, 
by treating the potential structure as if it were a Markov chain and 
solving the corresponding stochastic matrix. 


For example, take a case where each world line is sending and re- 
ceiving a message that is either $M_1$, $M_2$, or $M_3$. Before 
receiving their message, they flip a coin. If heads, they will add 1 to 
the message and send it, unless it's $M_3$, in which case they will 
just send $M_36. If it's tails, however, they throw the message away 
and just send $M_1$. What is the probability of receiving each of 
these messages? 


If we diagram out all possible transitions on our timeline, we get the 
diagram above. This can be then written out as a stochastic matrix: 


(1) 

\begin{align} B = \begin{bmatrix}B_1&B_ 2&B_ 3\end{bmatrix} = B\; 
\begin{bmatrix}0.5 &0.5&0\\0.5&0&0.5\\0.5&0 & 
0.5\end{bmatrix} \end{align} 


This could be solved algebraically, but in this case it is easier to just 
simulate it - in this case it converges after only two iterations. The 
resulting asymptotic probabilities are 0.5 for $M_1$, and 0.25 for 
$M_2$ and $M_ 3$. 


Exercises 


1. Generalize the example problem to five messages. Draw out 
the complete timeline diagram and compute the probability of 
receiving each message. 

2. Advanced Solve the time loop in 2.1 exercise 2. Because of 
the way world line $C$ participates in the loop multiple times, 
this should affect its probability differently than in a simple 
loop. 


The Lottery Problem 


We are now equipped to understand the lottery problem presented 
at the beginning of Chapter 1, and understand why, as we asserted, 
you are only slightly more likely to win than by random guessing. 


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


First off, some basic information about lotteries: 

The selection methods used by modern lotteries are extremely sen- 
sitive to even very small changes, and the act of transmitting the 
message to the past will very likely destroy any correlation between 
the number drawn in the transmitting and receiving world lines. How- 


ever, the Intermediate Value Theorem from calculus guarantees that 
at there is at least one message that, if transmitted, would end up 
being correct. The message might need to include some extra ran- 
dom data along with the number, but for the sake of argument it 
works to assume that it's not necessary in our case. 


Because it's impossible to determine beforehand the exact message 
that needs to be sent, the best possible strategy that the spurious 
lines can take is brute force, where each possible message is sent 
in turn until it works. Then once we have the working message, each 
iteration after than can just re-transmit that same message. 


However, each time, there is also a nonzero probability that you 
might fai/ to transmit the next message correctly, be it a transcription 
error, random software glitch, or anything else. Because the number 
of steps you will likely have to take in order to arrive at the correct 
lottery number, this probability of failure builds up and compounds 
on itself, since you only need to fail once to prevent ever reaching 
the correct number. And since the chance of making no mistakes 
over the billions or trillions of iterations required to converge to the 
right number, using the number received from the future only gives 
you a very slight edge over randomly guessing. 


Exercises 


1. Assuming 100 steps, and a 1% chance of making an error that 
resets to step 1, compute the expected value for the length of 
the loop, and find the probability of reaching step 100. 

2. Advanced Derive the general equation describing the proba- 
bility of reaching step $n$ given a uniform failure probability 
$p$ at each step. 


Chapter 1: Introduction to Time Travel | Chapter 3: Threads 


Chapter 3: Threads 


Thread Space 


In earlier chapters, we discussed time travel in terms of transitions 
between world lines, and shown how it can be used to predict the 
behavior of simple time loops. However, in order to understand more 
complex phenomena, it is necessary to generalize this. 


Thread space refers to the space of all possible combinations of 
things that could ever occur, down to the tiniest detail of even intrin- 
sically random events (like nuclear decay or quantum interactions). 
A single instance of such a combination is referred to as a thread. 
Note that in relativistic contexts it's necessary to consider each ref- 
erence frame having its own threads, but that will not be covered in 
this text. 


It's also useful to consider offset threads; that is, given a spacetime 
vector $\vec x§$, then $A + \vec x$ is also a thread. We can usually 
consider all the offset threads of a given thread together with the first 
thread, but it becomes important when discussing thread distance 
and teleportation. 


No Exercises 


The Rzewski Field 


The Rzewski field was named for Dr. Carlos Rzewski, who won the 
Dirac medal of the FTPI for this discovery in 1975. 


In order to properly understand the relationships between individual 
threads, we also need to introduce the Rzewski field, one of the 
fundamental fields in the universe. (Note that in some contexts it 
may also be referred to as the subspace field.) The Rzewski field 
defines a unique value associated with each point in spacetime 


across every thread. It is theorized to be the underlying reason that 
that there are points in spacetime that are distinct from one another, 
as opposed to having a universe containing only a single point. This 
is also what makes different threads distinct from each other, and, 
most importantly for practical purposes, can be measured to directly 
determine how similar two threads are to each other. 


There are a number of different ways this can be measured, but one 
of the most common and useful is thread distance, measured in 
humes. In your other coursework you may have already encoun- 
tered humes, when measuring how “anomalous” something is with a 
Kant counter or similar device. In time travel, we use a different tool, 
the divergence meter. Instead of comparing to a set of fixed pocket 
dimensions, a divergence meter allows measuring thread distance 
directly relative to other threads, and is generally much more sensi- 
tive. 


Note that thread distance does not directly tell us what is different 
between two threads, but it does tell us how different the threads 
are, and it can be used to help find where major changes may have 
occurred. 


No Exercises 


Algebraic Properties of Thread Distance 


Thread distance, notated $d(A,B)$ for any given threads A, B, al- 
lows us to define a metric space and induces a topology that allows 
us to reason about thread space. While the precise details of the 
Rizewski field are very important for theoretical causality, for practi- 
cal purposes we need not concern ourselves with it, except for a few 
basic concepts. 


Since thread distance is a metric, we have the following properties: 


B)\in\Bbb R$ — Thread distance is a real number. 

MA Bese Thread distance is non-negative. 

tee 0$ — Thread distance from a thread to itself is zero. 
d(A,B)=d(B,A)$ — Thread distance is reflexive; it's the same 


A, 


Sees 


3 


will progressively worsen. The process will accelerate under stress 
or panic to a minimum of ten (10) days. The conversion has been 
divided roughly into three (3) stages, beginning from initial influence 
to the transformation into a doll. If at any point a doll made from 
SCP-747's influence is destroyed, SCP-747 will begin to show a 
greater interest in humans until another human is 'targeted' for 
replacement. 


Stages of Transformation in Detail 


Stage 1: Loss of Minor Senses 

The targeted human loses the senses of smell, hearing, 
and taste over a period of approximately seven (7) days. 
Their mental state appears stable. Subject may display 
worry about condition. Any signs of mental instability at 
this stage are signs of an accelerated transformation. 
SCP-747 will ignore the person as if they were normal. 
The target's skin color and eye color will progressively 
change in appearance to that of what they will become 
as a doll. Any personnel that show symptoms must be 
contained according to the containment procedures 
listed until late stage two (2), in which the personnel are 
to be moved to SCP-747's containment chamber. 


Stage 2: Loss of Major Senses 

Subject begins to lose the final two senses over a period 
of approximately thirteen (13) days. Their mental state 
will begin to destabilize due to loss of the major senses, 
first touch and then sight. This degeneration of senses is 
gradual, and victims have been found initiating self-harm 
in attempts to feel something. SCP-747 will begin 
showing interest in the person, from noticing their 
entrance into their containment to actively attempting to 
follow and interact with the subject. Should subject 
attempt suicide or die in this stage, SCP-747 will 
immediately seek a replacement with any human. Once 
the targeted human loses all their senses, SCP-747 will 
immediately show great interest with the body. The skin 
will become ragged, similar to textile in both texture and 
appearance, and the subject's eyes will begin to harden 


measured in either direction. 

¢ $d(A,B)\le d(A,C) + d(B,C)$ — Thread distance obeys the tri- 
angle inequality; the sum of distances to some third thread will 
be at least as large as the direct distance between two 
threads. (i.e. There are no ‘shortcuts’.) 


These properties are important because it allows us to use analytical 
tools to reason about thread space, and in particular it allows us to 
define the concept of thread potential, discussed in section 3.5. 


Exercises 


1. Given that $d(K, Q) = 1.5\,\mathrm{Hm}$ and that $d(T, Q) = 
7.0\,\mathrm{Hm}$, what is the maximum possible value for 
$d(K, T)$? 

2. Advanced Let $f(\vec x) = d(E+\vec x, E)$. Prove that $\nabla 
\times\nabla f(\vec x)=0$. 


Thread Convergence and Time Loops 


An example of a 
thread sequence 
projected into 2D 
converging toa 
world line. 


In chapter 2, we discussed time loops in terms of world lines, as if 
each iteration of the loop was exactly identical to the previous. In 
practice, each iteration of a world line will inevitably have at least 
some small difference, stemming from Bell's theorem and the fact 
that it's impossible to observe anything without changing its state. 
As a result, it makes more sense to talk about world lines as the lim- 
its of loop iteration. 


Given a time loop with a thread sequence $A‘{(1)}, B“{(1)}, A*{(2)}, 
B‘{(2)} ...6, then if we can split this sequence up into only finitely 
many convergent Cauchy sequences, it is possible to define our 
world lines as the limits of those sequences. In our example, if 
$A{(1)}, AX{(2)} ...$ and $B“{(1)}, B“{(2)} ...6 are both Cauchy se- 


quences, then we can refer to $A = \lim_{n\to\infty} A*{(n)}$ and $B 
= \lim_{n\to\infty} B*{(n)}$ as world lines. In other terms, if after an 
arbitrary number of times around the loop, it becomes arbitrarily 
hard to distinguish between each $A{(n)}$ and $A“{(n+1)}$, then it 
still makes sense to consider them as world lines. 


However, in some cases it is not possible to split up a thread se- 
quence in this way, and any such sequence will instead converge to 
a set of closed curves or higher-order manifolds in thread space. 
These world manifolds can sometimes still be considered in a similar 
way to world lines, but systems containing world manifolds are not in 
general solvable using algebraic techniques. Some methods for 
solving these more difficult systems are presented in chapter 4. 


No Exercises 


Thread Potential 


One other important property of thread distance is the way it varies 
over time and space. In particular, it is continuously differentiable, 
and ‘at infinity’ it is identically zero. That is: 


(1) 
\begin{align} \lim_{|\vec x|\to\infty} d(A+\vec x,B+\vec x)=0 
\end{align} 


The thread potential 
for the event repre- 
sented by the upper 
(blue) curve is 5 
times the lower 
(red) one, meaning 
that it is only 1/5th 
as likely. 


Measuring thread distances between separate threads is useful for 
determining how similar they are, and reasoning about convergence. 
However, and in some ways even more importantly, we can also 


measure thread distance between points that are only separated by 
space and time. Doing this makes it possible to define a potential 
field based on thread distance ‘to infinity’, called thread potential 
and notated $\nabla‘*2 d(E)$, with some extremely useful properties. 


(2) 
\begin{align} \nabla*2 d(E) = \nabla \cdot \nabla d(E+\vec x, \infty) 
\end{align} 


This quantity turns out to be enormously important in later chapters, 
because it allows us to directly relate the probabilities of different 
events to each other: 


(3) 
\begin{align} \nabla*2 d(E_1)\, P(E_1) = \nabla‘2 d(E_2)\, P(E_2) 
\end{align} 


The ratio of the probabilities of two events, is also one of the main 
determining factors when estimating how easy or difficult it would be 
to change those events via time travel. It also enables us to locate 
and map out nearby events that will be susceptible to modification, 
by following the gradient of the thread potential to its peak. 


Example 1 


We measure the thread potential of some event $E$ to be: 


(4) 
\begin{align} \nabla*2d(E)=1 \end{align}