Bootstrap Paradox

A young engineer finds an envelope on her desk. Inside are complete plans for a time machine, in her own handwriting, with a note: build this. She does. Decades later she uses the machine to travel back, and leaves the plans on her younger self's desk.

Nothing in this story contradicts itself. Every event has a cause. She built the machine because she had the plans. She had the plans because her older self delivered them. Her older self delivered them because she had built the machine. The history is complete, consistent, and closed.

And yet one question has no answer: who designed the time machine? Not the engineer, who copied the plans. Not her older self, who only delivered what she had copied. The design exists, but nothing in the story ever produced it.

This is the bootstrap paradox, named after Robert Heinlein's 1941 story "By His Bootstraps," in which a time traveler's actions depend on instructions he himself supplied. It is the quieter companion of the grandfather paradox. The grandfather paradox asks what happens when a time loop contradicts itself. The bootstrap paradox asks what happens when a time loop is perfectly consistent, but contains something that came from nowhere.

How a Loop Closes

A time loop of this kind is called a causal loop: a chain of causes that eventually becomes its own cause. It needs a path through spacetime that returns to its own past, which in General Relativity means a closed timelike curve. Given such a path, two kinds of loop are possible. ^[1]

Information loops. Information travels around the loop and becomes its own source. The plans in the envelope are one example. A standard one in the philosophy literature has a time traveler teach her younger self the theory of time travel, which she knows only because her older self taught it to her. ^[1]

Object loops. A physical thing travels around the loop and has no beginning. Imagine a pocket watch handed to a young man by an old stranger, carried for fifty years, and then handed by the now-old man to his younger self. The watch is never manufactured. Its entire existence is a closed circuit.

The object version is harder to accept, and, as discussed below, physics treats it more harshly. A watch carried for fifty years wears down. To be handed over again, it must be in exactly the state it was in when first received.

Why It Is Not a Contradiction

It helps to be precise about what the bootstrap paradox is not.

The grandfather paradox describes a history that cannot exist: the traveler both does and does not prevent their own journey. The bootstrap loop describes a history that can exist, at least as a matter of logic. There is no point in the story where something both happens and does not happen.

The Novikov self-consistency principle captures this. Friedman and colleagues proposed that, in spacetimes with closed timelike curves, a local sequence of events can occur only if it forms part of a globally consistent history. ^[3] That principle excludes the grandfather paradox. It does not exclude bootstrap loops. A bootstrap loop satisfies the self-consistency condition by construction. It is not a violation of the principle but one of the solutions the principle allows.

In formal terms, the self-consistency condition for anything traveling around a loop is a fixed-point condition. If $S$ is the state that emerges from the past end of the loop, and $F$ describes how ordinary physics transforms it before it re-enters at the future end, consistency requires

A grandfather-type setup is one where $F$ has no fixed point. A bootstrap loop is a fixed point that contains structure, such as a blueprint, which does not come from anywhere outside the loop.

For an object carried around a loop, the condition is stricter. Let $\Psi(\tau)$ be the complete physical state of the object, including every atom and every scratch, as a function of its own elapsed time $\tau$. If the loop takes proper time $T$, then

In words: after completing the loop, the object must be exactly what it was at the start, down to the last detail. This single equation is where most of the physical difficulty lives.

So the bootstrap paradox does not break the logic of CTCs or the self-consistency principle. What it raises are three different kinds of doubt: about explanation, about knowledge, and about thermodynamics.

Objection 1: Information From Nowhere

The first objection is the most intuitive. Surely information, especially complex information, must come from somewhere. A design that no one designed seems to violate some principle of accounting.

David Lewis addressed this in his 1976 analysis of time travel. ^[2] He accepted that a causal loop is possible and that each event in it is explained by the one before. What has no explanation is the loop as a whole: why this loop, with this content, rather than another or none. Lewis argued that this is strange but not impossible. Physics already accepts events without a sufficient cause. In standard quantum mechanics, for example, there is no deeper reason why a particular radioactive atom decays at a particular moment. ^[1,2]

The Stanford Encyclopedia of Philosophy entry on time travel surveys the continuing debate. Some philosophers hold that causal loops are impossible; others argue that the objections fail, or that loops are no less explicable than other brute facts. ^[1]

Physics adds a useful angle. Deterministic, reversible physical laws neither create nor destroy information. They carry it forward, transform it, and never produce it from nothing. In ordinary spacetime this means that information present now was present, in some form, at earlier times. On a closed loop there is no "earlier" outside the loop. The information is simply carried around it. Viewed this way, a bootstrap loop does not violate the physical bookkeeping of information. It violates our expectation that every item of information has a first appearance.

Status: philosophical interpretation. No physical law is known to require that information have a temporal origin.

Objection 2: Which Loop Happens?

A deeper problem appears once the loop is written as a fixed-point equation. The engineer's loop is consistent. But so is a loop in which the envelope contains a different design. So is one in which the envelope contains nonsense, one in which it is blank, and one in which no envelope ever appears. If the laws of physics admit all of these, why would nature select the one carrying a working design?

This is the same issue that appeared in the classical billiard-ball analysis of wormhole time machines, where many initial conditions allowed several, sometimes infinitely many, consistent histories. ^[4,5] Classical physics gives no rule for choosing among them.

The quantum models of closed timelike curves do supply rules, and those rules are unkind to the engineer.

David Deutsch's 1991 model treats the system on the loop as a quantum state required to be consistent in a statistical sense. ^[6] Deutsch discussed a version of the bootstrap problem, sometimes called the unproved theorem paradox: a mathematician finds a proof in a book, which a time traveler brought from the future, copied from that same book. When the consistency condition has several solutions, Deutsch added a rule selecting the one with maximum entropy, meaning maximum disorder. ^[6,7] Applied to the book, that rule favors the least structured possibility. A meaningful proof becomes vanishingly improbable.

Seth Lloyd and colleagues reached a similar conclusion in their post-selected model of time travel, without an extra rule. Because nothing in the circuit favors one proof over another, their model returns an unbiased mixture of all possible "proofs." ^[7] Almost all of those are gibberish.

In both models, then, the loop is allowed but the valuable content is not favored. The engineer is far more likely to find an envelope of random marks than a working blueprint.

Status: mathematical result within specific quantum models of CTCs. Whether nature follows either model is unknown.

Objection 3: Knowledge Must Be Created

Deutsch raised a sharper objection about knowledge specifically. In his 1991 paper he identified two troubling features of classical time travel: the possibility of inconsistency, and the possibility of knowledge being created in a way that conflicts with the philosophy of science. ^[6] He held that knowledge comes into existence only through evolutionary processes: variation and selection, conjecture and criticism, trial and error. A blueprint that works encodes the results of such a process. A bootstrap blueprint encodes those results without the process ever taking place.

On this view, the bootstrap paradox is not a logical contradiction but a violation of a principle about how knowledge arises. That principle is broadly consistent with how complex information is understood elsewhere. Genomes are explained by a long history of gradual accumulation (see Origin of Life); mathematical proofs are explained by the work of mathematicians. A bootstrap loop would be the one place where such structure had no history of construction.

Deutsch argued that in his quantum model this pathology is mitigated, largely because of the entropy rule above: the loop does not favor knowledge-bearing states. ^[6]

There is an interesting complication. Aaronson and Watrous proved that a computer using Deutsch-type time loops could efficiently solve any problem in the complexity class PSPACE, which includes problems believed to be far harder than anything conventional or quantum computers can handle (for a sense of ordinary quantum speedups, see Grover's Algorithm). ^[8] The answer seems to arrive from the loop. But it does not arrive from nowhere. It is forced by the consistency condition acting on a carefully designed circuit: the only consistent state is the one that encodes the answer. One reading is that the consistency condition itself does the work of computation. On that reading, even the most powerful CTC computers do not create knowledge from nothing. They use a strange new kind of physical process to create it.

Status: Deutsch's principle is a philosophical position, not a physical law. The Aaronson–Watrous result is a mathematical theorem about a hypothetical model.

Objection 4: Entropy and Memory

The strongest physical objection concerns object loops, and it comes from thermodynamics.

The second law of thermodynamics says that in an isolated system, entropy, a measure of disorder, does not decrease. Objects wear out. Paper yellows. Watches lose their polish. But the loop condition $\Psi(\tau_0+T) = \Psi(\tau_0)$ requires the object to return to exactly its starting state. An isolated object that aged would fail that test.

Novikov and his collaborators confronted this directly. Lossev and Novikov proposed the possibility of systems with closed worldlines, which they called the "Jinn of the time machine." Their conclusion was that such systems are possible only if they interact with the outside world and gain energy from it to regenerate their internal structure. ^[9] The watch could exist on a loop only if something outside it, such as a watchmaker, restores it to its original state before it is handed back. The disorder is not removed; it is exported to the environment, where the ordinary second law continues to hold.

Related work examined the billiard ball with inelastic collisions, in which some energy turns to heat. Self-consistent solutions still existed for generic initial conditions. ^[10] Friction and heat do not automatically destroy consistency. They constrain which histories are consistent.

Information loops fare better than object loops on this point, because information can be copied onto fresh material. The engineer does not need the original sheets of paper to survive; she only needs to deliver a copy. That is one reason information loops seem more plausible than object loops.

More recent theoretical work points in a stranger direction. Carlo Rovelli argued that the apparent paradoxes of CTCs disappear once irreversible processes along them are analyzed carefully, but that CTCs then do not allow travel to the past in the thermodynamic sense, such as arriving in the past with a memory of the future. ^[11] Lorenzo Gavassino studied an idealized spaceship traveling on a closed timelike curve and found, in that model, that memories formed during the journey must be erased by its end, and that any entropy produced must decrease back to its starting value. ^[12]

If results like these generalize, which has not been established, they would bear directly on information loops. A traveler who carries knowledge around the whole loop in her own memory might be unable to do so. The loop would permit only histories in which records are consistently erased or restored.

Status: accepted theory for the second law in ordinary settings; theoretical proposals, based on specific models, for how it applies on CTCs.

Does the Bootstrap Paradox Break Anything?

It is useful to collect the verdicts.

• Logical consistency: not broken. A bootstrap loop is a consistent history. ^[1,2]
• The Novikov principle: not broken. Bootstrap loops are among the solutions it allows. ^[3]
• Closed timelike curves: not ruled out by the paradox. If CTCs are impossible, the reasons are elsewhere, such as the need for exotic matter or Hawking's chronology protection conjecture. ^[13]
• Explanation: strained. The loop as a whole has no external cause. Whether that is acceptable is a philosophical question. ^[1,2]
• Knowledge: strained. Complex content without a creative process conflicts with a widely held view of how knowledge arises, and quantum CTC models suppress such content. ^[6,7]
• Thermodynamics: strained for objects, less for information. Objects on loops must be continually restored by their environment. ^[9,12]

A different family of models avoids bootstrap loops altogether. If travel to the past leads into a different history, rather than one's own, then the plans in the envelope could have been designed by someone in an earlier history. The loop becomes a spiral. Hauser and Shoshany analyzed such multiple-history models mathematically and discussed how they treat both consistency and bootstrap paradoxes. ^[14] Those models give up the single-history picture that the Novikov principle assumes.

Evidence and Limitations

The evidence here is entirely theoretical.

• Established fact: No closed timelike curve has ever been observed, and no bootstrap loop has ever been documented. ^[13]
• Mathematical possibility: General Relativity admits spacetimes with closed timelike curves, and the self-consistency principle permits causal loops within them. ^[3]
• Model-dependent result: In Deutsch's and Lloyd's quantum models, loops that carry meaningful information are not favored. ^[6,7]
• Theoretical proposals: Thermodynamic analyses suggest that object loops need external regeneration, and that memories on a loop may be erased. ^[9,11,12] Some of this work is recent and not yet broadly tested by other groups.
• Philosophical interpretation: Whether an unexplained loop is acceptable, and whether knowledge can exist without being created, are questions about explanation and epistemology rather than experiment. ^[1,2]

Nothing currently known could settle which picture is right, because no one can build a loop to test it. Laboratory experiments with photons have simulated the mathematics of quantum CTC models, but they do not send anything into the past. ^[7]

What the Paradox Exposes

The bootstrap paradox is valuable because it isolates a question that the grandfather paradox obscures. Contradiction is not the only thing that might make time travel unacceptable. A perfectly consistent history can still be troubling if it contains design without a designer or knowledge without discovery.

Examining that trouble shows what our explanations normally assume. We expect every structure to have a history of assembly, from the complexity of living things to the content of a theorem. Physics, it turns out, does not contain a law guaranteeing this. What it does contain is thermodynamics, which makes structure costly, and, in quantum models of time travel, probability rules that make free structure unlikely. Those constraints do much of the work that a missing law of origins would otherwise do.

The topic also connects to how we think about time itself. In a block universe, a loop is simply part of spacetime's shape, and its content is a fact about that shape. Whether facts of that kind need further explanation is the same question raised, in other forms, by debates over determinism and first causes.

Bootstrap loops are logically consistent and compatible with the self-consistency principle. If closed timelike curves exist, physics can describe causal loops, while thermodynamics and particular quantum models sharply constrain what they might carry. No observation tells us that such a loop exists, which quantum model would govern it, or whether a history can contain knowledge that no one created.

The engineer can keep the blueprint. Particular quantum models suggest that a useful one would be extraordinarily unlikely; whether nature would use any such model remains unknown.

References

[1] Smith, N. J. J. "Time Travel." Stanford Encyclopedia of Philosophy (first published 2013; revised 2024). https://plato.stanford.edu/entries/time-travel/

[2] Lewis, D. (1976). "The Paradoxes of Time Travel." American Philosophical Quarterly, 13(2), 145–152. https://www.jstor.org/stable/20009616

[3] Friedman, J., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., & Yurtsever, U. (1990). "Cauchy problem in spacetimes with closed timelike curves." Physical Review D, 42, 1915–1930. https://doi.org/10.1103/PhysRevD.42.1915

[4] Echeverria, F., Klinkhammer, G., & Thorne, K. S. (1991). "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory." Physical Review D, 44, 1077–1099. https://doi.org/10.1103/PhysRevD.44.1077

[5] Smeenk, C., Arntzenius, F., & Maudlin, T. "Time Travel and Modern Physics." Stanford Encyclopedia of Philosophy (first published 2000; revised 2023). https://plato.stanford.edu/entries/time-travel-phys/

[6] Deutsch, D. (1991). "Quantum mechanics near closed timelike lines." Physical Review D, 44, 3197–3217. https://doi.org/10.1103/PhysRevD.44.3197

[7] Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V., Shikano, Y., Pirandola, S., Rozema, L. A., Darabi, A., Soudagar, Y., Shalm, L. K., & Steinberg, A. M. (2011). "Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency." Physical Review Letters, 106, 040403. https://doi.org/10.1103/PhysRevLett.106.040403

[8] Aaronson, S., & Watrous, J. (2009). "Closed timelike curves make quantum and classical computing equivalent." Proceedings of the Royal Society A, 465, 631–647. https://doi.org/10.1098/rspa.2008.0350

[9] Lossev, A., & Novikov, I. D. (1992). "The Jinn of the time machine: nontrivial self-consistent solutions." Classical and Quantum Gravity, 9, 2309–2321. https://doi.org/10.1088/0264-9381/9/10/014

[10] Mikheeva, E. V., & Novikov, I. D. (1993). "Inelastic billiard ball in a spacetime with a time machine." Physical Review D, 47, 1432–1436. https://doi.org/10.1103/PhysRevD.47.1432

[11] Rovelli, C. (2019). "Can we travel to the past? Irreversible physics along closed timelike curves." arXiv preprint 1912.04702. https://arxiv.org/abs/1912.04702

[12] Gavassino, L. (2024). "Life on a closed timelike curve." Classical and Quantum Gravity, 42, 015002. https://doi.org/10.1088/1361-6382/ad98df

[13] Hawking, S. W. (1992). "Chronology protection conjecture." Physical Review D, 46(2), 603–611. https://doi.org/10.1103/PhysRevD.46.603

[14] Hauser, J., & Shoshany, B. (2020). "Time travel paradoxes and multiple histories." Physical Review D, 102, 064062. https://doi.org/10.1103/PhysRevD.102.064062