The Novikov Principle: Why You Cannot Change the Past

In 1990, a group of physicists published a paper in Physical Review D with a sober, almost bureaucratic title: “Cauchy problem in spacetimes with closed timelike curves.” John Friedman, Michael Morris, Igor Novikov, Fernando Echeverria, Gunnar Klinkhammer, Kip Thorne, and Ulvi Yurtsever, between them most of a field, stated a single conjecture about time travel. In a universe permitting paths that loop into their own past, the only events that can happen are the ones that remain globally self-consistent. The probability of a paradox is exactly zero.

This is the Novikov self-consistency principle. It does not stop the traveler with a force. It does not punish the paradox. It simply removes the paradoxical histories from the set of solutions that physics permits, so they never occur. The grandfather is safe. The weapon jams, the hand slips, some small accident intervenes, not because the universe is fighting back, but because the only version of events that exists is the one in which it does.

The principle sounds like science fiction. It is in fact a serious idea, born from a serious problem: relativity does not forbid time travel. It permits it. And once the equations permit it, someone has to say what actually happens.

Closed timelike curves: time travel inside relativity

General relativity, our best and most precisely tested theory of gravity, allows paths that loop back into their own past. The technical name is a closed timelike curve, often abbreviated CTC. A worldline is the track an object traces through spacetime. A timelike worldline always moves slower than light. A closed timelike curve does so while looping back and reconnecting with its own past. Locally, everything is ordinary. Globally, the path is a loop. Follow it far enough forward and you arrive at your own earlier moment.

These curves are not contrived. They keep surfacing in exact solutions of Einstein’s field equations.

• In 1924, Kornel Lanczos wrote down a rotating-dust solution. In 1937, Willem van Stockum analyzed an infinite rigidly rotating cylinder of dust whose deep interior contained closed timelike curves, though no one recognized them as such at the time.
• In 1949, Kurt Gödel presented an entire rotating universe, as a birthday gift for Albert Einstein, in which a closed timelike curve passes through every event.
• In 1974, Frank Tipler showed a massive infinitely long cylinder, above ten solar masses, near neutron-star density, spinning at a sizeable fraction of the speed of light, could twist spacetime into closed timelike curves around its axis, an object now known as the Tipler cylinder.
• In 1988, Kip Thorne, with his students Michael Morris and Ulvi Yurtsever, showed that a traversable wormhole could be turned into a time machine if one of its mouths moved at high speed or sat in a strong gravitational field.

The Kerr metric, the most realistic description of rotating black holes, also contains closed timelike curves in its deep interior. So does the spacetime of two infinitely long cosmic strings sliding past each other, a configuration analyzed by J. Richard Gott in 1991. Loops in time are not a single exotic loophole. They surface again and again, in solution after solution, in the deep structure of general relativity.

The wormhole route has one well-known cost. Holding the throat open requires exotic matter that violates the weak energy condition, matter with negative energy density. Nature does provide a trace of negative energy, in the Casimir effect measured between two metal plates, but whether it can be concentrated enough to build a working time machine is an open question.

The takeaway: relativity does not forbid time travel into the past. It permits it, as a recurring feature of its exact solutions, and that is the problem the Novikov principle was written to address.

The grandfather paradox and the Cauchy problem

Once you allow the loop, you inherit the paradox. A traveler kills his grandfather. He is never born. He never makes the trip. The grandfather lives. The traveler is born and goes back. The chain eats its own tail. For most of the twentieth century this was treated as a proof, by contradiction, that backward time travel must be impossible.

But the equations would not cooperate. The solutions kept appearing. So the question sharpened: if a closed timelike curve can exist, what actually happens, physically, when a traveler tries to create a paradox on one?

This sharpening became precise through the Cauchy problem. In ordinary physics, you specify the state of a system at one moment and the laws evolve that initial data forward uniquely. The Cauchy problem is the deep reason physics can predict anything. Closed timelike curves break it. At the boundary called the chronology horizon, initial data on an earlier surface no longer determine the evolution uniquely, because events in the looping region depend on themselves.

The question of consistency stops being philosophical. It becomes a mathematical one. Given a spacetime that loops, which physical histories are even well-defined on it?

Novikov’s answer

Igor Novikov, the Russian astrophysicist who also proposed white holes in 1964, gave the first sketch of the answer in books from 1975 and 1983. His phrasing was modest. The closing of time curves, he wrote, does not necessarily imply a violation of causality, since the events along such a closed line may be all self-adjusted.

Self-adjusted. The events on the loop do not contradict each other, because the only loops that exist are the ones whose events already fit. The 1990 Physical Review D paper turned that intuition into a formal statement: the only solutions to the laws of physics that can occur locally in the real universe are those which are globally self-consistent. A local solution is admissible only if it extends to a complete global solution well-defined on all nonsingular spacetime.

There is no force in this principle. Nothing reaches in to stop the traveler. The histories that contain contradictions are simply not among the histories the equations admit, so they never occur. The Stanford Encyclopedia of Philosophy entry on time machines lays out the formal background in detail.

The billiard ball that delivers a glancing blow

To strip the paradox of its biology, the physicists replaced the grandfather with a billiard ball, in a thought experiment the string theorist Joseph Polchinski proposed and which Thorne came to call Polchinski’s paradox. A ball rolls into a wormhole, emerges in the past, and strikes its earlier self so that the earlier self never enters. Thorne set two Caltech students, Fernando Echeverria and Gunnar Klinkhammer, to find out what actually happens.

They could not build the paradox. Every paradoxical trajectory failed to close. But for every starting condition they tried, they found self-consistent solutions in which the returning ball delivers a softer, glancing blow. That blow nudges the earlier ball into the wormhole at a slightly different angle, so that it returns and delivers exactly that glancing blow. The collision that seemed to create the paradox turns out to be the thing required to produce it. The whole history closes on itself with no loose end.

The self-consistent solution is not a near-miss or a compromise. It is a complete history in which the time-traveling ball causes the exact deflection that sends it back to cause that deflection.

There was a result they found troubling, and we will return to it. For a single starting trajectory of the ball, they could find not one self-consistent solution but many, in fact infinitely many, and the classical physics they were using contained no rule whatsoever to say which one happens. We will come back to this gap.

The takeaway: under Novikov’s principle, the grandfather paradox does not generate a contradiction. It generates a fan of self-consistent alternatives, of which one actually occurs. Why this one rather than another, the equations do not say.

The quantum versions: Deutsch, Lloyd, Hawking

The world is quantum, and the quantum versions of the time loop disagree.

In 1991, David Deutsch proposed that a quantum system on a closed timelike curve must satisfy a fixed-point condition on its density matrix: the state entering the loop must equal the state coming back. He proved that such a self-reproducing state always exists. The grandfather paradox dissolves into a quantum superposition that loops back on itself consistently. Writing with Michael Lockwood in Scientific American in 1994, Deutsch put the upshot bluntly. Common sense, they wrote, may rule out such excursions, but the laws of physics do not.

Deutsch’s model has unsettling consequences. It is nonlinear, in tension with the linearity of standard quantum mechanics, and it would let a traveler perform feats the rest of physics forbids. It would defeat the no-cloning theorem. In 2014, Martin Ringbauer, Timothy Ralph, and colleagues at the University of Queensland published, in Nature Communications, a photonic simulation of Deutsch’s condition, with a success probability of one in nine. They built it using entangled photons and post-selection. They did not, they were careful to note, build a real closed timelike curve.

A rival, the post-selected model of Seth Lloyd and Lorenzo Maccone, was tested at the University of Toronto in 2011, sending a photon, as the experimenters put it, a few billionths of a second back in time to try to kill its former self. The two quantum models are physically inequivalent and disagree about what closed timelike curves would actually do. Physics does not yet know which, if either, is right.

A deflating caveat sits over the whole project. Jürgen Tolksdorf and Rainer Verch, in 2018 and 2021, showed that the Deutsch fixed-point condition can be satisfied to arbitrary precision in ordinary quantum field theory with no closed timelike curve anywhere, and even in classical statistical mechanics. The condition is not unique to time travel at all. Headlines announcing that the 2014 photon experiment had validated the physics of time travel were claiming more than the mathematics can bear.

A third position holds that the loops never form. Stephen Hawking, in 1992, proposed the chronology protection conjecture: the laws of physics prevent closed timelike curves from forming on macroscopic scales. His mechanism was the vacuum back-reaction at the chronology horizon, where Frolov, and separately Kim and Thorne, had shown the renormalized stress-energy of quantum fluctuations diverges. Hawking famously named the supposed enforcer the Chronology Protection Agency, and added that there was strong experimental evidence in favor of the conjecture from the fact that we have not been invaded by hordes of tourists from the future. His joke was funnier than the argument was decisive. Hawking himself later conceded that the back-reaction does not necessarily enforce protection, and Li-Xin Li showed in 1996 that the divergence can be smoothed away. The full resolution, on every account, awaits a theory of quantum gravity we do not yet have.

The bootstrap paradox, and the gap at the centre

The strangest object in this whole subject is not the grandfather paradox but its quiet opposite. In the bootstrap paradox, a traveler reads a proof of a theorem in a book, carries it into the past, and dictates it to the mathematician who later writes the book the traveler read. The proof is correct. The loop closes. And the simple question, who first discovered the proof, has no answer. The information exists because it exists. It was never created.

The bootstrap paradox is fully self-consistent. Under Novikov, under Deutsch, under the consistent-histories framework Robert Griffiths, Roland Omnès, Murray Gell-Mann and James Hartle have developed for quantum mechanics, such loops are not forbidden. They simply close. The Russian physicist Sergei Krasnikov has argued that objects that appear from a consistent loop are no less physical than objects that appear from a singularity at the start of the universe.

Yet a strain runs through the whole arrangement, and a fair account names it. The loop sits in tension with the conservation of information, a principle quantum mechanics treats nearly as sacred, and with the second law of thermodynamics, which expects ordered structure, a proof, a watch, a complex pattern, to trace back to a source. Information from nowhere is not a solved problem in physics.

And there is the deeper gap, the one Echeverria and Klinkhammer flagged for billiard balls. The classical formalism admits infinitely many self-consistent histories for a single initial condition, and contains no rule to pick which one is real. The sum-over-histories assigns each consistent extension a well-defined probability, but probability is not selection. Something settles on one history. The physics describes the consistent loops. It is mute about why this loop, rather than any of the others, is the one that occurs.

This is the open question at the heart of the subject, and it has more than one possible reading. One holds that the gap is purely physical, a sign that a future theory of quantum gravity will name the selecting mechanism. Another holds that the gap is the kind no how-question can close, and that a reality which presents infinitely many consistent alternatives and realizes exactly one is, at minimum, deeply unlike anything blind matter is built to produce. The companion video takes the second reading further than this page does. Either way, what is settled is narrower and still remarkable. A largely forgotten 1990 paper proposed a principle under which time travel is permitted and the paradox is impossible, and three decades of follow-up have not dislodged it.

What is also clear is that Novikov’s principle sits naturally inside a particular picture of time, the block universe of relativity, in which the past is fixed and finished. The compromise between that block and the felt openness of becoming is captured in the growing block universe. Whichever picture is right, the loop, once allowed to exist, is constrained to close. The gun was always going to jam. The blow was always going to glance. The story was always whole.