Imag­ine the fol­low­ing time travel prob­lem:

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

Now, one might naively as­sume the chance is 100%. But in fact, the ac­tual prob­a­bil­ity is only slightly bet­ter than if you guessed ran­domly, and this can be ver­i­fied ex­per­i­men­tally.
In or­der to avoid mak­ing se­ri­ous mis­takes and max­i­mize suc­cess when time trav­el­ling, it is im­por­tant to have a clear un­der­stand­ing of this and other seem­ing para­doxes that can arise, and for that we must start with the ba­sics.
Ca­u­sal­ity
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.

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 ca­usal 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 Ex­er­cises

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.
Ex­er­cises

An 7 kg ob­ject is dis­placed by 4 Xn. Does it end up in the past or the fu­ture, and how far?
Given a 5 kg test mass, what dis­place­ment would be needed to send it 5 min­utes into the fu­ture?
A 62.0 kg hu­man is dis­placed 46.7 kXn at 5:00 on Mon­day, when does he ar­rive?
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.

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.
Ex­er­cises

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?
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?
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?

Cur­rent Lim­i­ta­tions
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 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 retro­causal trans­mis­sion schemes used for this pur­pose.
An­other im­por­tant ap­pli­ca­tion of time travel is in com­put­ing. Many newer mi­cro­proces­sors take ad­van­tage of retro­ca­usal 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 retro­causal 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.
No Ex­er­cises

Cover | Chapter 2: Time Loops