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

In order to understand how time loops work, it is necessary to introduce 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 primordial 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 decides 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 second-order loop:

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

In this case, the main world lines $B$ and $B'$ each transmit mutually exclusive messages $M$ and $M'$, each prompting the other to send its message. In this case, the situation could have been bootstrapped 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

- Draw a complete timeline diagram for a basic fourth-order periodic time loop.
*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 between $B_1$ 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 receiving 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 $M_3$. 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)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

- Generalize the example problem to five messages. Draw out the complete timeline diagram and compute the probability of receiving each message.
*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 sensitive 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. However, 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 random 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 *fail* 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

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