(原題: Chapter 3: Threads)

元記事: https://scp-wiki.wikidot.com/time-travel-ch3

評価: 0+x

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.

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 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.

No Ex­er­cises


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

  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

FWHSjH9.png

An ex­am­ple of a thread se­quence pro­jected into 2D con­verg­ing to a 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 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.

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 rep­re­sented 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}

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)
\begin{align} \nabla^2d(E')=0.1 \end{align}

Com­put­ing the rel­a­tive prob­a­bil­i­ties:

(6)
\begin{align} \frac{P(E)}{P(E')} = \frac{\nabla^2d(E')}{\nabla^2d(E)} = \frac{0.1}{1} = 0.1 \end{align}

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 $E'$.

No Ex­er­cises


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


en tale



ページ情報

執筆者: 08sr_
文字数: 10491
リビジョン数: 1
批評コメント: 0

最終更新: 14 Jun 2022 06:14
最終コメント: 批評コメントはありません

ページコンソール

批評ステータス

カテゴリ

SCP-JP

本投稿の際にscpタグを付与するJPでのオリジナル作品の下書きが該当します。

GoIF-JP

本投稿の際にgoi-formatタグを付与するJPでのオリジナル作品の下書きが該当します。

Tale-JP

本投稿の際にtaleタグを付与するJPでのオリジナル作品の下書きが該当します。

翻訳

翻訳作品の下書きが該当します。

その他

他のカテゴリタグのいずれにも当て嵌まらない下書きが該当します。

言語

EnglishРусский한국어中文FrançaisPolskiEspañolภาษาไทยDeutschItalianoУкраїнськаPortuguêsČesky繁體中文Việtその他日→外国語翻訳

日本支部の記事を他言語版サイトに翻訳投稿する場合の下書きが該当します。

コンテンツマーカー

ジョーク

本投稿の際にジョークタグを付与する下書きが該当します。

アダルト

本投稿の際にアダルトタグを付与する下書きが該当します。

既存記事改稿

本投稿済みの下書きが該当します。

イベント

イベント参加予定の下書きが該当します。

フィーチャー

短編

構文を除き数千字以下の短編・掌編の下書きが該当します。

中編

短編にも長編にも満たない中編の下書きが該当します。

長編

構文を除き数万字以上の長編の下書きが該当します。

事前知識不要

特定の事前知識を求めない下書きが該当します。

フォーマットスクリュー

SCPやGoIFなどのフォーマットが一定の記事種でフォーマットを崩している下書きが該当します。


シリーズ-JP所属

JPのカノンや連作に所属しているか、JPの特定記事の続編の下書きが該当します。

シリーズ-Other所属

JPではないカノンや連作に所属しているか、JPではない特定記事の続編の下書きが該当します。

世界観用語-JP登場

JPのGoIやLoIなどの世界観用語が登場する下書きが該当します。

世界観用語-Other登場

JPではないGoIやLoIなどの世界観用語が登場する下書きが該当します。

ジャンル

アクションSFオカルト/都市伝説感動系ギャグ/コミカルシリアスシュールダーク人間ドラマ/恋愛ホラー/サスペンスメタフィクション歴史

任意

任意A任意B任意C

ERROR

The 08sr_'s portal does not exist.


エラー: 08sr_のportalページが存在しません。利用ガイドを参照し、portalページを作成してください。


利用ガイド

  1. portal:6943710 (18 Dec 2020 16:27)
特に明記しない限り、このページのコンテンツは次のライセンスの下にあります: Creative Commons Attribution-ShareAlike 3.0 License