The Taub-NUT Vacuum: The Universe Where Time Is a Circle

Imagine walking in a straight line across a field. Slow, ordinary steps. You never run, never approach the speed of light, never do anything a physicist would call exotic. You simply keep walking. And then, by the strict mathematics of Einstein’s theory of gravity, you arrive at a moment you have already lived. Not a copy of it. The same moment, with your earlier self standing in it.

That universe has a name. It is called the Taub-NUT vacuum, and it is not science fiction. It is an exact solution to the field equations of general relativity, written down by a mathematician in 1951 and studied by serious physicists ever since. In it, the geometry of space and time is wound together like a coil, so that motion through space carries you through time, and a loop in space becomes a loop in time.

The strangest part is not that the equations permit such a universe. The strangest part is that ours, as far as we can measure, refuses it, sitting at the one knife-edge value where time stays a straight line. This is the physics of the slinky universe, the history of how it was discovered, and the question its existence forces on us.

A universe written down in 1951

In 1951, the American mathematician Abraham Taub set out to solve a stripped-down version of Einstein’s equations. General relativity, finalized by Albert Einstein in 1915, says that spacetime is not a fixed stage but a geometric object that bends in response to mass and energy. In a region with no matter and no energy at all, a true vacuum, the equations reduce to a clean condition: the curvature, measured by something called the Ricci tensor, must vanish.

Taub looked for a vacuum universe with a particular kind of symmetry, one in which space itself is shaped like the surface of a four-dimensional ball rather than the flat space we picture. He found a valid solution. It was an empty universe, mathematically legitimate, and on its face unremarkable. It sat in the literature for twelve years.

In 1963, three physicists returned to it. Ezra Newman, Louis Tamburino, and Theodore Unti were studying generalizations of the Schwarzschild solution, the standard geometry of a non-rotating black hole. They realized that Taub’s universe and an extension of Schwarzschild could be stitched together along a shared horizon. The combined object carried two parameters. One was ordinary mass. The other was something no one had a clean physical picture for. They named it the NUT parameter, after their three surnames, and the full solution became the Taub-NUT vacuum, carrying the initials of all four men.

The slinky metric: how space and time get wound together

The whole of the strangeness lives in a single piece of the mathematics. The Taub-NUT geometry contains a term that adds the time coordinate to a spatial direction and weights the mixture by the NUT parameter. Translated out of the equations, it says something startling.

Go around a loop in space, and your position in time shifts.

This is not the time dilation of special relativity, where moving clocks tick slowly. It is not the gravitational redshift near a heavy object. It is more basic than either. The geometry itself ties motion through space to motion through time. Circle the axis of the universe, returning to the exact spatial spot you began at, and you do not return to the moment you began at. You land somewhere else in time, shifted by an amount the NUT parameter sets.

Picture an actual slinky toy sitting on a table. Seen from directly above, it is a circle. Seen from the side, it is a helix that climbs as it loops, so each turn brings you back over your starting point but higher than before. Now trade the vertical climb for time and the horizontal circle for motion through space. That is the Taub-NUT universe. Walking the circle does not bring you home to your starting instant. It winds you up, or down, the coil of time.

If the NUT parameter is zero, the coil unwinds completely and the whole solution collapses into the familiar Schwarzschild black hole. The twist exists only when the parameter does.

The NUT charge: a magnetic monopole made of gravity

Physicists interpret the NUT parameter as a gravitomagnetic charge, the gravitational version of a magnetic monopole. The comparison is worth unpacking, because it explains why the solution is taken seriously rather than dismissed.

In electromagnetism, electric charges produce electric fields, and a single isolated charge radiates its field outward like the spines of a sea urchin. Magnetism is different. Every magnet has both a north and a south pole; cut a bar magnet in half and you get two smaller magnets, never an isolated north. An isolated magnetic charge, a magnetic monopole, has never been found. Yet in 1931, Paul Dirac showed that the existence of even one such monopole anywhere in the universe would explain a deep puzzle, namely why electric charge always comes in discrete units. The monopole remains hypothetical, but it is mathematically consistent and theoretically tempting.

The NUT charge is the same idea, lifted into gravity. An ordinary mass, a planet or a star or a black hole, produces a field that radiates outward and pulls matter toward it. An object carrying a NUT charge would produce a field that does not pull. It twists. Its presence would be felt not as a downward tug but as a rotation of the local geometry, spacetime itself swirling around it. The Taub-NUT vacuum is what the gravitational field of a north pole with no south pole would look like, if such a thing existed.

Like Dirac’s magnetic monopole, the gravitomagnetic monopole has never been observed. Like the magnetic monopole, it is permitted by the mathematics. And like the magnetic monopole, it would, if real, force strange whole-number relationships onto the universe that we have no other reason to expect.

The Misner string and the bargain that turns time into a circle

There is a flaw in the geometry, and confronting it is what leads to the most disturbing feature of the whole solution.

Along the axis of the Taub-NUT spacetime, that space-time mixing term refuses to behave. At the poles it does not vanish but instead produces a singular line stretching out from the source, like a thread hanging off the geometry. This is the Misner string, named for the American relativist Charles Misner, who examined it carefully in 1967. It is not a true curvature singularity, the kind found at the center of a black hole, where physics genuinely breaks down. It is a coordinate artifact, the gravitational cousin of the “Dirac string” that trails behind a magnetic monopole. But it cannot simply be ignored. It has to be removed.

Misner found the only clean repair, and the cost was steep. If the time coordinate is made periodic, so that each moment is identified with a later one after a fixed interval, the Misner string vanishes and the geometry becomes smooth everywhere. The required interval is eight pi times the NUT parameter. The repair works. But it means time itself becomes a circle.

A unique past and a unique future dissolve. The same instant is reached, by any observer who waits long enough, a second time. Then a third. Then without end. This is the slinky in its fully wound form: a universe in which time is not an open line running from beginning to end but a closed loop that returns to itself. The cosmos repeats. The future, followed far enough, becomes the past.

Closed timelike curves: time travel without breaking a law

The technical name for a path in this geometry that returns to its own starting moment is a closed timelike curve. A timelike curve is just the worldline of a possible observer, any object moving slower than light. A closed one is a worldline that loops back and meets itself.

An observer following such a curve would notice nothing strange in the moment. Their watch would tick steadily forward. Their cells would age. They would feel no jolt, no shimmer, no sense of having jumped. They would simply walk a long path and discover, at the end of it, that they had arrived at a moment they had already lived, with their earlier self present in it.

This is worth stating plainly because it overturns an assumption almost everyone carries: that time travel into the past, if it is impossible, must be impossible because it would break the laws of physics. In the Taub-NUT vacuum, it breaks nothing. The closed timelike curves are not bolted on by a science-fiction author. They are forced by the geometry. The same equations that correctly predict the bending of starlight around the Sun and the orbits of the planets also describe the slinky universe. It is, by every formal test, a permissible solution to the theory.

What the real universe actually does

So why don’t we live there? Here the evidence is overwhelming, and it all points one way.

Gravity Probe B, a NASA satellite that operated from 2004 to 2011, carried four quartz gyroscopes machined to be among the most perfectly spherical objects ever made. It measured a subtle effect called frame-dragging, the way Earth’s rotation drags the spacetime around it. This is the same family of gravitomagnetic effect that the NUT parameter represents in extreme form. The result, 37.2 milliarcseconds of drift per year, matched general relativity’s prediction for an ordinary spinning mass, with no sign of any monopole component. The twist of the Earth on spacetime is real, measured, and exactly as small as it should be.

The LAGEOS and LARES satellites, tracked by laser to millimeter precision, confirmed the same effect independently to within a few percent. The Planck satellite, which mapped the faint afterglow of the early universe between 2009 and 2013, placed an extraordinarily tight ceiling on any overall rotation of the cosmos, far below what a large-scale slinky geometry would require. And the LIGO and Virgo detectors, which have now recorded over ninety collisions of black holes and neutron stars, find that every waveform matches templates with the NUT charge set firmly to zero. A significant gravitomagnetic monopole in any of those sources would have spoiled the match. None did.

The mathematics permits the slinky. The cosmos, measured to staggering precision, refuses it.

Chronology protection: why the loops may be forbidden after all

There is a deeper reason to think the loops cannot form, beyond the simple fact that we have never seen one. When physicists examined whether the Taub-NUT closed timelike curves are stable, they found they are not. As any matter, wave, or quantum field approaches the boundary where the time loops begin, called the chronology horizon, its energy is blue-shifted without limit. The calculations diverge to infinity. Any real attempt to form such a region would have to survive an infinite pile-up of energy precisely at the moment the loops switched on, and that pile-up would presumably tear the geometry apart before a single loop could close.

In 1992, Stephen Hawking turned this observation into the Chronology Protection Conjecture: the proposal that the laws of physics, once quantum effects are properly accounted for, always conspire to prevent closed timelike curves from forming, even though the classical equations permit them. He named the responsible mechanism, half in jest, the Chronology Protection Agency. The conjecture is widely believed but has never been proven, because a full proof would require the quantum theory of gravity we do not yet have.

Other thinkers have approached the paradox differently. The Russian physicist Igor Novikov proposed a self-consistency principle, allowing the loops but insisting that only self-consistent histories occur, so a traveler could never do anything that contradicted the past they came from. Frank Tipler described a rotating cylinder that produces closed timelike curves, but only if it is infinitely long, an unphysical requirement explored further in our article on the Tipler cylinder. Kip Thorne and his colleagues showed how a traversable wormhole could in principle be turned into a time machine, but only with exotic matter of negative energy holding it open. The pattern across all of these is the same: closed timelike curves appear readily in the mathematics and vanish the moment you ask the universe to actually build one. For more on these strange worldlines, see our companion piece on closed timelike curves.

Gödel’s universe and the question that won’t go away

The Taub-NUT vacuum is not the first universe of this kind. In 1949, for Einstein’s seventieth birthday, the logician Kurt Gödel presented an exact solution of his own: a rotating universe, uniformly filled with matter, threaded everywhere by closed timelike curves. Through every event in Gödel’s universe there passes a loop in time. Gödel, who walked home from the Institute for Advanced Study with Einstein and discussed these questions with him directly, drew an openly metaphysical conclusion. If time can run in a circle in some possible universe, he argued, then the flow of time is not a fundamental feature of reality. It is a feature of how observers happen to move through certain kinds of cosmos.

Gödel’s universe, like Taub-NUT, is not the universe we live in. Astronomical observations rule out the global rotation it requires. But the question he raised has never gone away. Closed timelike curves are not a rare pathology of the equations. They show up in Gödel’s universe, in the interior of a rotating black hole, in the Taub-NUT vacuum, in the Tipler cylinder, in wormhole constructions. They are, in a precise mathematical sense, common. And yet the universe we observe has, as far as anyone can tell, none of them. The equations are generous. Reality is strict.

The block universe, with loops

To see why this matters for the nature of time, it helps to recall the three main views philosophers and physicists hold. The block universe, also called eternalism, says past, present, and future are equally real, all of them fixed points in a single four-dimensional spacetime, and that the passage of time we feel is not a property of the structure but of how a conscious worldline traverses it. Presentism says only the present moment is real. The growing block view says the past and present are real but the future is not yet.

Among working relativists, the block universe has long been the consensus, because special relativity removes any universal “now” that presentism needs. The philosopher Hilary Putnam argued in 1967 that, under relativity, presentism collapses into eternalism. The debate continues, but the block view remains dominant.

The Taub-NUT vacuum pushes this picture to its breaking point. The block of Taub-NUT is still, formally, a static four-dimensional object. But it is self-intersecting. The same event, the same instant, exists at more than one place in the structure, reached again and again by any worldline that explores the manifold far enough. It is a block universe with loops baked into it. The question is time an illusion, posed by physicists from Einstein to Carlo Rovelli, becomes almost unbearable here. In ordinary relativity, you can at least say time does not flow. In Taub-NUT, you cannot even say the same moment happens only once.

The selection problem: the open question at the center

Here is the puzzle the Taub-NUT vacuum leaves us with, stated as cleanly as it can be.

The solution exists for every nonzero value of the NUT parameter, and closed timelike curves appear the instant that value is anything but zero. The only way to escape them is to set the parameter to exactly zero, a single point in a smooth, infinite range of possibilities. In the language of mathematics, that single point has measure zero. A blind draw from the range of possible values would miss it with probability one. And yet our universe, by every measurement we can make, is sitting precisely on it.

This is not a small tuning. Picture a combination lock with infinitely many dials, each turning smoothly across the real numbers, where almost every possible setting produces a universe of looping time, runaway energies, and no stable structures at all. The narrow band of settings that yields a livable, time-ordered cosmos is a knife-edge. Our dials are resting on it.

What does this mean? It is one of the deepest unresolved questions in physics, and reasonable people read it in different ways. Some take Hawking’s chronology protection as a hint that an undiscovered law forbids the bad configurations, so that the apparent fine-tuning is really a constraint we have not yet written down. Some appeal to a multiverse, in which every possible setting is realized somewhere and we simply find ourselves in a habitable one, though that proposal trades one mystery for another and cannot be directly tested. And some read the precision of the selection, here and across the other fine-tunings of physics, as evidence that the configuration was chosen rather than stumbled upon, pointing beyond the equations to a source that the equations themselves cannot supply.

What all of these readings share is an admission. The equations describe what is possible. They are silent on what was selected. The mathematics is permissive; the actual universe is narrow; and the narrowness is the fact that asks for an explanation.

What stays

The slinky exists in the equations. It does not exist in the sky. An exact solution to the most successful theory of gravity ever written describes a universe where walking forward in space returns you to your own past, where time coils back on itself, where the same moment is lived without end. The theory permits it without strain. The cosmos declines it without exception.

That gap, between the vast space of what general relativity allows and the thin sliver our universe actually occupies, is the real subject here. It does not close on its own. Whatever the final theory of time and gravity turns out to be, it will have to explain not only why the strange solutions exist, but why we do not live in any of them.

The Taub-NUT vacuum was permitted. The Taub-NUT vacuum was not chosen. And the question of what does the choosing is older than the equations, and very far from settled.