## 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 intrinsically 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 reference 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.

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## 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 encountered 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 sensitive.

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.

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## Algebraic Properties of Thread Distance

Thread distance, notated $d(A,B)$ for any given threads A, B, allows 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 practical 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:

- $d(A,B)\in\Bbb R$ — Thread distance is a real number.
- $d(A,B)\ge0$ — Thread distance is non-negative.
- $d(A,A)=0$ — Thread distance from a thread to itself is zero.
- $d(A,B)=d(B,A)$ — Thread distance is reflexive; it's the same measured in either direction.
- $d(A,B)\le d(A,C) + d(B,C)$ — Thread distance obeys the triangle 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

- 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 $d(K, T)$?
*Advanced*Let $f(\vec x) = d(E+\vec x, E)$. Prove that $\nabla\times\nabla f(\vec x)=0$.

## Thread Convergence and Time Loops

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 limits of loop iteration.

Given a time loop with a thread sequence $A^{(1)}, B^{(1)}, A^{(2)}, B^{(2)} ...$, 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 $A^{(1)}, A^{(2)} ...$ and $B^{(1)}, B^{(2)} ...$ are both Cauchy sequences, 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 sequence 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.

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## 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)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.

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

(3)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)After modifying the past so that $E'$ occurs instead, we wish to instead revert the change to $E$. Unfortunately, when we measure the thread potential:

(5)Computing the relative probabilities:

(6)Since $E$ is only 1/10th as likely as $E'$, it will be much more difficult to return to $E$ than it was originally to get to $E'$.

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**Chapter 2: Time Loops | Chapter 4: Aperiodic Loops and Chaos (Coming Soon!)**