諸知博士の覚え書き
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以下のタイムトラベル問題を想像してください:

あなたは未来から宝くじの当選番号を知らされました。その番号を使った場合、宝くじに当選する確率はいくらでしょうか?

さて、確率を単純に100%と考える人もいるかもしれません。しかし実際の確率は、ランダムに選んだ場合よりもわずかに高くなるだけなのです。これは実験で実証可能です。

タイムトラベル時に深刻な問題を避け、成功を最大限にするためには、このような発生しうるパラドックスについて明確な理解を得ていることが重要です。さあ、基本から始めましょう

因果関係

タイムトラベルについて議論する際は、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” in a 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.

Inachis_io_Lill-Jansskogen.JPG Cyclone_Catarina_from_the_ISS_on_March_26_2004.JPG

In the­ory, the air cur­rents from a sin­gle but­ter­fly wing beat could be­come the de­cid­ing fac­tor in a hur­ri­cane's for­ma­tion.

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 as a 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 re­quired 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 Re­nais­sance 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 “rewrit­ing his­to­ry” can be con­sid­ered an “ac­tion”. This idea will be ex­plained 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 ca­sual 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 mod­ify the past to change an un­de­sir­able event $E$ and en­sure that de­sir­able 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 rep­re­sented with the hor­i­zon­tal line, which goes un­til $E$ oc­curs. The pair of dashed lines ex­tend­ing from it cor­re­spond to the time dis­place­ments 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 bi­fur­cates into the sec­ond one in which $E'$ oc­curs in­stead.

No Ex­cer­cises


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 dis­place­ment, 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 dis­place­ment is equal to the mass times the in­ter­val cubed:

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

For ex­am­ple, if I had an ap­pa­ra­tus ca­pa­ble of 1 µXn, it could dis­place 1 mil­ligram of mat­ter 1 sec­ond, 1 µg of mat­ter 10 sec­onds, etc.

Ex­cer­cises

  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 fu­ture 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 ob­ject through time, an equal and op­po­site re­ac­tion dis­place­ment is also gen­er­ated that moves some other ob­ject in the op­po­site di­rec­tion.

STS-134_solid_rocket_booster_segment_stacking.jpg

A 225 ton gran­ite bal­last mass used in the Chronome­ter Up­scale Nega­tion Test in Mel­borne, 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 dis­place­ment 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 re­main un­af­fected by the changes caused by the prin­ci­pal dis­place­ment, 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.

Ex­cer­cises

  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 trav­el­ling at 0.5c is dis­placed 1 year into the fu­ture, and the re­ac­tion dis­places a sec­ond pro­ton at rest, how far into the past does the sec­ond pro­ton end up?

現在の限界

The fun­da­men­tal en­ergy of dis­place­ment de­scribes the the­o­ret­i­cal limit on the amount of en­ergy re­quired to achieve a given dis­place­ment, and is ap­prox­i­mately 4.95e-21 J/​Xn. the How­ever, mod­ern tech­niques re­quire or­ders of mag­ni­tude more en­ergy: The cur­rent best, the Tachy­onic Ion 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 en­ergy, 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 dis­plac­ing a tar­get into the past in a way that keeps the tar­get in­tact - even a very small mis­match in the cal­i­bra­tion on cur­rent cur­rent tech­niques will con­vert the tar­get into an as-yet-un­known form of mat­ter on dis­place­ment. 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 par­ti­cles and sen­si­tive de­tec­tors. Ap­pa­ra­tus ca­pa­ble of re­ceiv­ing such streams was first de­vel­oped in 1991, plac­ing a hard cap on the ear­li­est date that one can re­li­ably send in­for­ma­tion to. Chap­ter 5 cov­ers de­tails of retro­ca­sual trans­mis­sion schemes used for this pur­pose.

An­other im­por­tant ap­pli­ca­tion of time travel is in com­put­ing. Many newer mi­cro­proces­sors take ad­van­tage of retro­ca­sual con­nec­tions as part of their branch pre­dic­tion and cache prefetch hard­ware, en­abling 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 cov­ered in chap­ter 2, and chap­ter 6 goes into de­tail about how retro­ca­sual 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.

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