A Brief History of Time-Travel Paradoxes (1.0.9)

Abstract A theory of the origin of time-travel paradoxes is outlined.

 

The Grandfather and Shakespeare Time-Travel Paradoxes

In the well known time-travel paradox “The Grandfather”, a traveler goes back in time and kills his own grandfather, thus removing the grounds of his own existence (it was first proposed by the science fiction writer René Barjavel in 1944 (1)):

Noël Essaillon (physicist-chemist), drawing on the work and collaboration of a young mathematician (Pierre Saint-Menoux), invents a substance (Christmas 3) to travel in time. First developed as capsules to be ingested, it then smeared a suit much better studied for travel time. Saint-Menoux explores first of all the near future then, emboldening himself, a very distant future where he discovers a humanity having evolved towards the exclusive specialization of the tasks. But travel in time is not without danger, and Saint-Menoux will have to learn the hard way that any action has consequences.

The paradox can be generalized by considering that certain past states are conditions of present states, and that altering a past state potentially creates a form of The Grandfather paradox.

There is another time-travel paradox that can be called “The Shakespeare”. In this paradox a traveler goes back in time to give Shakespeare the manuscripts of his plays, thus saving him the trouble of having to write them for himself. The paradox is that the plays have no author: Shakespeare didn’t write them because he was given them by the time-traveler; but the time-traveler didn’t write them either because he was given them by Shakespeare. This paradox is well depicted by Escher’s Drawing Hands:

Imagine going back or forward in time and meeting up with a younger or older self. These are clear-cut cases of what we can call self-intersecting time-lines. The self-intersection can be interpreted as contradictory (The Grandfather) or circular (The Shakespeare) depending on whether or not it is supposed to involve self-interference: if it is supposed to involve self-interference (as it obviously would if one killed one’s ancestor) then the intersection is contradictory; if the self-intersection is supposed not to involve self-interference then it is circular. Corresponding to these two types of self-intersection there are two types of self-intersecting time-lines, one that involves self-interference and one that does not. Corresponding to these two types of self-intersection are two types of particles…

The Riemann Hypothesis, Physics, and Time-Travel

Revisit Euler’s classic argument (10) that the product continued to infinity of this fraction

\frac{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\text{...}}{2\ 4\ 6\ 10\ 12\ 16\ 18\text{...}}

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\text{...}

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

x=1+1/2+1/3+1/4+1/5+1/6...

Then

\frac{x}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}\text{...}

This leaves

\frac{x}{2}=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\text{...}

To eliminate the denominators that are divisible by 3, we divide both sides to get

\frac{x}{2\ 3}=\frac{1}{3}+\frac{1}{9}+\frac{1}{15}+\frac{1}{21}\text{...}

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

\frac{2 x}{2\ 3}=1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\text{...}

Applying this eliminatory process to all of the prime numbers leaves

\left(\frac{2\ 4\ 5\ 10\ 12\ 16\ 18}{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19}\text{...}\right) x=1

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity. It is not as is often maintained all of mass in the compressed down to a point, a prioritization of space over light that creates all that is ugly in physics, including black holes as infinite at their centers (3), the incongruity of relativity and quantum mechanics (4), the infinities of quantum field theory (5), and the flat rotation curves of distant galaxies that seem to call for dark matter (6, 7)…

Rather, all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light.

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of de-centralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place in terms of something known as the Riemann Hypothesis (8, 9), in terms of which the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

\pi (x)

\sum _{n=2}^x \frac{1}{n \log }-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)

\sum _{n=2}^x \frac{1}{H_n}-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)

The Generalized Riemann Hypothesis (10, 11) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominant over any gains in prime-density, the direction of time and of all the arrows of time is given by the GRH. There is a sense in which every change in density is either a contradiction (loss of density) or a loop (gain in density), but if and only if a change in density is outside the bounds imposed by the GRH, do these contradictions or loops take the form of a Grandfather or Shakespeare Paradox and become impossible. Collapsing the difference between the logarithmic/harmonic prime-counting functions and the primes collapses the difference between the conscious mind and the world it is conscious of, and so we obtain the superposition 2 \Re\left(\sum _{n=1}^{\infty } \text{Ei}\left(\rho _n \log (x)\right)\right) by taking difference between the number of primes not greater than x and an approximating formula such as

\left(\frac{\sum _{n=2}^x \frac{a_{1\ 1}}{H_n}+\sum _{n=2}^x \frac{a_{2\ 1}}{n \log }+\text{...}}{n}\right)

and the following wave-form:

Or from another perspective in the same phenomenon we can obtain the superposition 2 \left(\text{Re} \sum _{n=1}^{\infty } \frac{\left(p_x\right){}^{\rho _n}}{\rho _n}\right) by taking the difference between the primes themselves and an approximating formula such as

\frac{a_1 x H_x+a_2 x \log (x)+\text{...}}{n x}

Corresponding to every crossing of the x-axis -when the difference between the subject and the object of consciousness approaches collapse- there must be either a contradiction (a self-intersecting and self-interfering event when the direction is towards a loss of prime-density) or a loop (a self-intersecting and non-self-interfering event when the direction is towards a gain in prime-density) and these self-intersections do not involve time paradoxes if and only if

\rho _n=\frac{1}{2}

The connection there is between number theory and quantum field theory can be simply illustrated (12  by associating the creation operators \left(b_n\right){}^{\dagger } and \left(f_n\right){}^{\dagger } to the prime numbers p_n… Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index (13) -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

\mu n={1 = n has an even number of distinct factors,
-1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator (-1)^F that distinguishes bosonic from fermionic states, with \mu n = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as \sum _{n=1}^{p_x} \mu (n), we see that sums of these states give us essentially the same composite spiral-wave as before.

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the intersections of the x-axis of which that can be identified with self-intersecting time-lines and with time-travel paradoxes of one form or the other. These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a and observe that the self-intersections do not involve time-travel paradoxes if and only if Subscript[\[Rho], n]=1/2. This value 1/2 -associated as it is with a very particular balance of prime-density and sparsity- signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. Infinities -other than the single infinity at root of the universe without which there is nothing- are now as it were nipped in the bud, with no need for renormalization.

Time-Travel Paradoxes on Film

Every film that involves going back in time and changing something with a view to preventing the occurrence of some future events involves the Grandfather paradox, and these films are ubiquitous – a list of books and films on the theme of time travel is a list of narratives involving the Grandfather paradox. An excellent example is The Terminator (1984):

Wikipedia summation of the plot:

In 1984 Los Angeles, a cyborg assassin known as a Terminator arrives from 2029 and steals guns and clothes. Shortly afterward, Kyle Reese, a human soldier from 2029, arrives. He steals clothes and evades the police. The Terminator begins systematically killing women named Sarah Connor, whose addresses he finds in the telephone directory. He tracks the last Sarah Connor to a nightclub, but Kyle rescues her. The pair steal a car and escape with the Terminator pursuing them in a police car.

As they hide in a parking lot, Kyle explains to Sarah that an artificial intelligence defense network, known as Skynet, will become self-aware in the near future and initiate a nuclear holocaust. Sarah’s future son John will rally the survivors and lead a resistance movement against Skynet and its army of machines. With the Resistance on the verge of victory, Skynet sent a Terminator back in time to kill Sarah before John is born, to prevent the formation of the Resistance. The Terminator is an efficient killing machine with a powerful metal endoskeleton and an external layer of living tissue that makes it appear human.

Kyle and Sarah are apprehended by police after another encounter with the Terminator. Criminal psychologist Dr. Silberman concludes that Kyle is paranoid and delusional. The Terminator repairs his body and attacks the police station, killing many police officers in his attempt to locate Sarah. Kyle and Sarah escape, steal another car and take refuge in a motel, where they assemble pipe bombs and plan their next move. Kyle admits that he has been in love with Sarah since John gave him a photograph of her, and they have sex.

The Terminator kills Sarah’s mother and impersonates her when Sarah, unaware of the Terminator’s ability to mimic victims, attempts to contact her via telephone. When they realize he has reacquired them, they escape in a pickup truck. In the ensuing chase, Kyle is wounded by gunfire while throwing pipe bombs at the Terminator. Enraged‚ Sarah knocks the Terminator off his motorcycle but loses control of the truck, which flips over. The Terminator hijacks a tank truck and attempts to run down Sarah, but Kyle slides a pipe bomb onto the tanker, causing an explosion that burns the flesh from the Terminator’s endoskeleton. It pursues them to a factory, where Kyle activates machinery to confuse the Terminator. He jams his final pipe bomb into the Terminator’s abdomen, blowing the Terminator apart, injuring Sarah, and killing Kyle. The Terminator’s still functional torso reactivates and grabs Sarah. She breaks free and lures it into a hydraulic press, crushing it.

Months later, a pregnant Sarah is traveling through Mexico, recording audio tapes to pass on to her unborn son, John. She debates whether to tell him that Kyle is his father. At a gas station, a boy takes a Polaroid photograph of her which she purchases – the same photograph that John will eventually give to Kyle.

Technically the Shakespeare paradox should always involve a repeating time loop -if someone goes back in time to give Shakespeare his plays then there is a repeating time-loop associated with the journey of the plays through time- and there are a number of films involving such time-loops including the Groundhog Day (1993):

Weatherman Phil Connors reassures Pittsburgh viewers that an approaching blizzard will miss western Pennsylvania. He goes with news producer Rita Hanson and cameraman Larry to Punxsutawney, Pennsylvania, to cover the Groundhog Day festivities. Phil makes no secret of his contempt for the assignment, the small town, and the “hicks” who live there.

The next day, Phil awakens at his Punxsutawney bed and breakfast to Sonny & Cher’s “I Got You Babe” on the clock radio. He tapes a half-hearted report on Punxsutawney Phil and the town’s festivities. Rita wants to stay and cover other events, but Phil wants to return to Pittsburgh. The blizzard blankets the region in snow, stranding them in Punxsutawney. Phil shuns the celebrations and retires to bed early.

Phil wakes to “I Got You Babe” and the same announcement from the radio, and discovers the day’s events repeating exactly. Phil relives the day and returns to bed, assuming it was a dream, but it is still Groundhog Day when he wakes: he is trapped in a time loop that no one else is aware of. Realizing there are no consequences for his actions, he spends the first several loops indulging in binge drinking, one-night stands, and reckless driving. He becomes depressed and commits suicide several times, but does not escape the loop.

Phil tries to explain his situation to Rita, for whom he has feelings, by accurately predicting the day’s events. Rita takes sympathy and they spend the entirety of one loop together, but Phil wakes up alone as usual. He decides to use his knowledge of the day’s events to better himself and the lives of others; he learns how to play the piano, sculpt ice, and speak French, but is unable to prevent the death of a homeless man.

During one loop, Phil enthusiastically reports the Groundhog Day festivities, amazing Rita. They spend the rest of the day together, with Phil impressing her with his apparent overnight transformation and charitable deeds. She successfully bids for Phil at a charity bachelor auction. Phil makes a ice sculpture of Rita’s face, and tells her that no matter what happens, even if he is doomed to continue awakening alone each morning forever, he wants her to know that he is finally happy, because he loves her. They retire together to Phil’s lodgings. Phil wakes to “I Got You Babe” again, but finds Rita is still in bed with him; he has escaped the time loop.

Based on a short story called All You Zombies by Robert A. Heinlein, a film that explores the Shakespeare paradox is the convoluted but intriguing Predestination (2014):

The film begins in medias res as a time travelling agent is trying to disarm a bomb that explodes and burns his face. Someone approaches and helps him to grasp his time travelling device, then brings him to a hospital in the future. While the agent is recovering from facial reconstruction, we learn that he has been trying to prevent the “Fizzle Bomber”‘s attack on New York, in 1975. After his recovery he receives his last assignment.

The agent moves to 1970 New York. As a bartender, he starts a conversation with one of the customers. The customer, John, writes true confession articles under the pen name “The Unmarried Mother”. This pseudonym is explained by his own life story, which he tells the bartender. The customer grew up as “Jane” in an orphanage. She excelled in her studies but had difficulty fitting in. Jane decided any children she had would be raised in a proper family, and thus avoided relationships. As an adult she applied for a program called “Space Corp”, which promised women the chance to go to space while providing astronauts R&R, but she was later disqualified because of a medical condition which she was not informed of, which a man named Robertson was interested in. Jane later met a man who said he was waiting for someone. The two fell in love with each other, but later the man disappeared. Robertson approached Jane, revealing that Space Corp worked for the Temporal Agency, and this agency now wanted to recruit her. They broke off contact when it was discovered that Jane was pregnant with her ex-lover’s baby. While performing a Caesarean section, doctors discovered she was intersex, with internalized male sex organs as well as female sex organs. Complications during the birth forced them to remove her female sex organs, and she had to undergo a gender reassignment and begin living as a man. Furthermore, the baby was stolen by a mysterious man. Since then, John has been living a bitter life as “The Unmarried Mother”.

The agent offers to take John back to the day that Jane met the lover who left her, so John can take revenge and kill him for ruining her life. In return, John will take over the agent’s job for whatever duration he wishes. The agent reveals his time travel device and the two jump to that day in 1963. John prepares to kill his past lover before he can meet Jane. While waiting, he encounters Jane, and when they begin talking, John realizes that Jane’s lover was him. The baby born from this “self-fertilization” is stolen by the agent and brought to the orphanage 18 years earlier, in 1945. Therefore, Jane, John, and their baby are the same person, revealing a predestination paradox.

The agent goes to 1975 New York, where he helps the burned man from the beginning of the film. The agent returns to 1963, a few months after he dropped John off. John has to leave Jane behind and is brought to the Temporal Agency. He now takes over the job so the agent can retire in 1975 New York, close to the day of the Fizzle Bomber’s attack. The agent’s time-travel device does not decommission itself as planned and can still be used. He has been ordered to check a launderette at the moment the Fizzle Bomber will be there. The Fizzle Bomber turns out to be the agent’s own future self, now suffering from psychosis as a result of excessive time travel. The Fizzle Bomber insists that his actions have saved and will save more lives than the lives lost, and that they ultimately lead to the reinforcement of the Temporal Agency. He tries to convince the agent that the only way to end the cycle is to spare his life, unlike the Fizzle Bomber did in his past. The agent denies he will ever become the Fizzle Bomber and kills his future self.

The film finally reveals that in 1975, John is the man who travelled to New York and was burned while disarming a bomb. His subsequent facial reconstruction significantly changes his appearance, and it is now clear that Jane, John, the agent, and the Fizzle Bomber are the same person. This agent’s creation was orchestrated by Robertson to create an agent who has no ties to time. This “perfect” temporal agent was responsible for both his own conception and death; he has driven the predestination paradox to its limit.

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REFERENCES

(1) Barjavel, R (1944), Future Times Three

(2) Euler, L (1737), Various observations concerning infinite series

(3) Wald, R (1997), Gravitational Collapse and Cosmic Censorship

(4) Wald, R (1984), General Relativity

(5) Feynman, Richard (1948), A Relativistic Cut-Off for Quantum Electrodynamics

(6) Rubin, V et al (1980), Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122kpc)

(7) de Swart, J. et al (2017), How dark matter came to matter

(8) Riemann, G (1859), On the Number of Primes Less Than a Given Magnitude

(9) Derbyshire, J. (2004), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

(10) Dirichlet, P (1837), Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers

(11) Davenport, H (2000), Multiplicative number theory

(12) Spector, D (1990), Supersymmetry and the M\[Delta]bius Inversion Function

(13) Witten, E (1982), Constraints on supersymmetry breaking