Top: original billiard ball trajectory.Middle:
the ball emerges from the future at a different trajectory from the
original, and collides with its past self, changing its trajectory.Bottom:
the changed trajectory causes the ball to enter and exit the time
machine in exactly the same way that changed its trajectory. The changed
trajectory is its own cause, without an origin.
A causal loop is a theoretical proposition in which, by means of either retrocausality or time travel, a sequence of events (actions, information, objects, people) is among the causes of another event, which is in turn among the causes of the first-mentioned event. Such causally looped events then exist in spacetime, but their origin cannot be determined. A hypothetical example of a causality loop is given of a billiard ball
striking its past self: the billiard ball moves in a path towards a
time machine, and the future self of the billiard ball emerges from the
time machine before its past self enters it, giving its past self
a glancing blow, altering the past ball's path and causing it to enter
the time machine at an angle that would cause its future self to strike
its past self the very glancing blow that altered its path. So, the
question here in this paradox is, how was the ball struck in the first
place?
Terminology in physics, philosophy, and fiction
Backwards
time travel would allow for causal loops involving events, information,
people or objects whose histories form a closed loop, and thus seem to
"come from nowhere." The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical, with several authors referring to a causal loop involving information or objects without origin as a bootstrap paradox, an information paradox, or an ontological paradox. The use of "bootstrap" in this context refers to the expression "pulling yourself up by your bootstraps" and to Robert A. Heinlein's time travel story "By His Bootstraps". The term "time loop" is sometimes used to refer to a causal loop,
but although they appear similar, causal loops are unchanging and
self-originating, whereas time loops are constantly resetting.
An example of a causal loop paradox involving information is
given by Everett: suppose a time traveler copies a mathematical proof
from a textbook, then travels back in time to meet the mathematician who
first published the proof, at a date prior to publication, and allows
the mathematician to simply copy the proof. In this case, the
information in the proof has no origin. A similar example is given in the television series Doctor Who
of a hypothetical time-traveler who copies Beethoven's music from the
future and publishes it in Beethoven's time in Beethoven's name. Everett gives the movie Somewhere in Time
as an example involving an object with no origin: an old woman gives a
watch to a playwright who later travels back in time and meets the same
woman when she was young, and gives her the same watch that she will
later give to him.
Krasnikov writes that these bootstrap paradoxes – information or
an object looping through time – are the same; the primary apparent
paradox is a physical system evolving into a state in a way that is not
governed by its laws.
He does not find this paradoxical, and attributes problems regarding
the validity of time travel to other factors in the interpretation of
general relativity.
A 1992 paper by physicists Andrei Lossev and Igor Novikov labeled such items without origin as Jinn, with the singular term Jinnee. This terminology was inspired by the Jinn of the Quran, which are described as leaving no trace when they disappear.
Lossev and Novikov allowed the term "Jinn" to cover both objects and
information with reflexive origin; they called the former "Jinn of the
first kind", and the latter "Jinn of the second kind".
They point out that an object making circular passage through time must
be identical whenever it is brought back to the past, otherwise it
would create an inconsistency; the second law of thermodynamics
seems to require that the object become more disordered over the course
of its history, and such objects that are identical in repeating points
in their history seem to contradict this, but Lossev and Novikov argued
that since the second law only requires disorder to increase in closed systems, a Jinnee could interact with its environment in such a way as to regain lost order. They emphasize that there is no "strict difference" between Jinn of the first and second kind.
Krasnikov equivocates between "Jinn", "self-sufficient loops", and
"self-existing objects", calling them "lions" or "looping or intruding
objects", and asserts that they are no less physical than conventional
objects, "which, after all, also could appear only from either infinity,
or a singularity."
The term predestination paradox is used in the Star Trek
franchise to mean "a time loop in which a time traveler who has gone
into the past causes an event that ultimately causes the original future
version of the person to go back into the past." This use of the phrase was created for a sequence in a 1996 episode of Star Trek: Deep Space Nine titled "Trials and Tribble-ations", although the phrase had been used previously to refer to belief systems such as Calvinism and some forms of Marxism
that encouraged followers to strive to produce certain outcomes while
at the same time teaching that the outcomes were predetermined.
Smeenk and Morgenstern use the term "predestination paradox" to refer
specifically to situations in which a time traveler goes back in time to
try to prevent some event in the past, but ends up helping to cause
that same event.
Self-fulfilling prophecy
A self-fulfilling prophecy may be a form of causality loop, only when the prophecy can be said to be truly
known to occur, since only then events in the future will be causing
effects in the past. Otherwise, it would be a simple case of events in
the past causing events in the future. Predestination does not necessarily involve a supernatural power, and could be the result of other "infallible foreknowledge" mechanisms. Problems arising from infallibility and influencing the future are explored in Newcomb's paradox. A notable fictional example of a self-fulfilling prophecy occurs in the classical play Oedipus Rex, in which Oedipus becomes the king of Thebes
and in the process unwittingly fulfills a prophecy that he would kill
his father and marry his mother. The prophecy itself serves as the
impetus for his actions, and thus it is self-fulfilling. The movie 12 Monkeys heavily deals with themes of predestination and the Cassandra complex, where the protagonist who travels back in time explains that he can't change the past.
Novikov self-consistency principle
General relativity permits some exact solutions that allow for time travel. Some of these exact solutions describe universes that contain closed timelike curves, or world lines that lead back to the same point in spacetime. Physicist Igor Dmitriyevich Novikov discussed the possibility of closed timelike curves in his books in 1975 and 1983, offering the opinion that only self-consistent trips back in time would be permitted. In a 1990 paper by Novikov and several others, "Cauchy problem in spacetimes with closed timelike curves", the authors suggested the principle of self-consistency, which states that the
only solutions to the laws of physics that can occur locally in the
real Universe are those which are globally self-consistent. The
authors later concluded that time travel need not lead to unresolvable
paradoxes, regardless of what type of object was sent to the past.
Physicist Joseph Polchinski argued that one could avoid questions of free will by considering a potentially paradoxical situation involving a billiard ball sent back in time. In this situation, the ball is fired into a wormhole
at an angle such that, if it continues along its course, it will exit
in the past at just the right angle to hit its earlier self, knocking it
off course, which would stop it from entering the wormhole in the first
place. Thorne referred to this problem as "Polchinski's paradox".
Two students at Caltech, Fernando Echeverria and Gunnar Klinkhammer,
went on to find a solution that avoided any inconsistencies. In the
revised scenario, the ball would emerge from the future at a different
angle than the one that had generated the paradox, and delivers its past
self a glancing blow instead of knocking it completely away from the
wormhole. This blow changes its trajectory by just the right degree,
meaning it will travel back in time with the angle required to deliver
its younger self the necessary glancing blow. Echeverria and Klinkhammer
actually found that there was more than one self-consistent solution,
with slightly different angles for the glancing blow in each case. Later
analysis by Thorne and Robert Forward
showed that for certain initial trajectories of the billiard ball,
there could actually be an infinite number of self-consistent solutions.
Echeverria, Klinkhammer and Thorne published a paper discussing these results in 1991; in addition, they reported that they had tried to see if they could find any
initial conditions for the billiard ball for which there were no
self-consistent extensions, but were unable to do so. Thus it is
plausible that there exist self-consistent extensions for every possible
initial trajectory, although this has not been proven.
The lack of constraints on initial conditions only applies to spacetime
outside of the chronology-violating region of spacetime; the
constraints on the chronology-violating region might prove to be
paradoxical, but this is not yet known.
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as an ad hoc solution, and supposes that there are far more damaging implications of time travel. Krasnikov similarly finds no inherent fault in causal loops, but finds other problems with time travel in general relativity.
Quantum computation with negative delay
Physicist David Deutsch shows in a 1991 paper that quantum computation with a negative delay—backwards time travel—could solve NP problems in polynomial time, and Scott Aaronson later extended this result to show that the model could also be used to solve PSPACE problems in polynomial time.
Deutsch shows that quantum computation with a negative delay produces
only self-consistent solutions, and the chronology-violating region
imposes constraints that are not apparent through classical reasoning. Researchers published in 2014 a simulation validating Deutsch's model with photons.
However, it was shown in an article by Tolksdorf and Verch that
Deutsch's CTC (closed timelike curve, or a causal loop) fixed point
condition can be fulfilled to arbitrary precision in any quantum system
described according to relativistic quantum field theory
on spacetimes where CTCs are excluded, casting doubts on whether
Deutsch's condition is really characteristic of quantum processes
mimicking CTCs in the sense of general relativity.
