Kurt Gödel was, by common consent, the greatest logician since Aristotle. His two incompleteness theorems broke ground in logic, mathematics, and philosophy; his monograph on the axiom of choice and the continuum hypothesis is a masterpiece of modern mathematics. He also published extraordinary results in cosmology, demonstrating that a rotating universe was among the solutions to the field equations of general relativity. What made these results extraordinary is just that they permit time travel. In Gödel’s rotating universe there are paths in space-time that return to their starting point. Such are the closed timelike curves. This is a very strange result, but it is not one that violates special relativity. An object traveling along a closed timelike curve never travels faster than the local speed of light. Such curves remain confined locally to their future light cones, and so represent possible paths for material objects. Within a rotating universe, a traveler can journey into the future but arrive in the past.

Gödel’s rotating universe is infinite, nonexpanding, and filled with an idealised, homogenous perfect fluid. To stay balanced against gravitational collapse, Gödel’s universe rotates, the angular velocity outward juxtaposed to the gravitational pull inward. “Matter everywhere,” Gödel explained, “rotates relative to the compass of inertia with the angular velocity 2(πκρ)½.” In this expression, ρ is the mean density of matter, and κ, Newton’s gravitational constant. Unlike a spinning top, Gödel universes do not rotate about a privileged geometrical axis. In each universe, the local inertial frames are rotating with respect to frames determined by distant galaxies. Every observer sees himself at the centre of rotation. This is hardly unusual. A rotating universe would no more require a unique center about which it rotates than an expanding universe would require a unique center from which it expands.

Time travel requires the long haul. The world lines that allow time travel are not the shortest distance between two points in space and time. But time travel needs no special time machine. Closed timelike curves thread themselves through every single space-time point in Gödel’s universe. The traveler with time enough to spare can visit any point. These ubiquitous closed timelike curves mean that the universe in which they are embedded cannot be foliated into universe-wide and space-like hypersurfaces. They have no well-defined universal present. While subsections of Gödel universes can have well-defined histories, Gödel universes as a whole do not. Gödel universes neither begin nor end. They are infinite space-time blocks.

Closed timelike curves obey special relativity’s prohibition forbidding any material object to travel faster than the speed of light. Viewed globally, however, Gödel universes offer viable frames of reference in which closed timelike curves begin before they end or end before they begin. The twins figuring in special relativity reappear in rotating universes. Suppose that one twin is co-moving with the matter that surrounds him; the other, and apparently more adventuresome twin, is flying off along any of the accelerated paths that space-time permits. The traveling twin can have overall journeys that seem superluminally fast; he may even boast that, with respect to his twin, the duration of his travel is negative. Time travels can appear to go faster than light or backward in time.

The curiously permissive structure of Gödel’s rotating universes reflects the difference between special and general relativity, the place where they diverge with respect to the local and global properties of space-time. In special relativity, light cones are parallel at every point; no one can move into the future while arriving in the past light cone without exceeding the speed of light. General relativity is otherwise. The presence of mass affects the local structure of a light cone, and assigns light cones different alignments from point to point. Any massive object makes the light cones of nearby points tilt towards the mass. General relativity also predicts that the movement of mass creates an additional potential, so that rotating masses make local light cones tilt in the direction of rotation. For many objects, such as the earth, cone-tilting effects are practically negligible. Colossal masses or great rotational velocities yield substantial cone tilting.

Any object that strays within the black hole’s event horizon must eventually intersect the singularity at the hole’s center and be destroyed. A critical radius exists around the Tipler cylinder where cone-tilting produces locally future-directed journeys that wind back into their own past. But a Tipler cylinder facilitates access to the past over the period in which it itself exists; it makes possible only a local closed timelike curve, and so supports only a finite model of time travel. Gödel universes are infinite and fully accessible throughout.

Time travel in a Gödel universe would require a great deal of fuel. “[T]he time traveler,” Paul Nahin noted, “would have to move at least as fast as nearly 71% of the speed of light.” Even if his rocket ship could transform matter completely into energy, his fuel would weigh more than his rocket. This objection comprised Gödel’s own reason for discounting the possibility of time travel, but it still left time travel physically possible. The feasibility of time travel changes once electromagnetic forces play a role. It changes dramatically. If they are electrically charged, John Earman observed, time travelers can use the Lorenz force to travel along close timelike curves. No rocket ship is required.

A Gödel space-time exhibits cone-tilting effects across the universe as a whole. Below, M symbolises the space-time manifold. If p is any arbitrary space-time point, J–( p) designates its causal past—the set of all points q in M such that there is a causal curve from q to p. In the same way, I–( p) designates its chronological past—the set of all points q in M such that q chronologically precedes p.

Because past and future light cones at each point effectively defocus and refocus, the whole manifold lies within the causal and chronological past and the causal and chronological future of every single point.

There are two critical radii around each point in Gödel space-time. The interior radius marks the boundary permitting closed lightlike curves to form. It is at this boundary that it first becomes possible to send a light ray into a point’s future, but have it arrive at the precise point in space from which it departed. The outer radius marks the boundary at which an object traveling below the speed of light can revisit its starting point in space-time. Observer B’s journey, it is important to note, involves backward time travel. The traveler experiences time flowing forward; it has a positive duration, but viewed from the outside, it is the other way around: his travel goes backwards in time; it has a negative duration.

Closed timelike curves are intrinsic and irreducible features of Gödel space-times. If they are possible, so is time travel. And with time travel, certain paradoxes arise. Imagine a traveler arriving in the past and killing his own grandfather. Would he survive the encounter if he broke the causal chain leading to his own existence?

There are two popular views about how these kinds of paradox might be managed. The first is committed to an ensemble of equally concrete but different versions of the physical world. Travelers into the past arrive in worlds that are distinct from those that they left. They are free to kill their grandfather secure in the knowledge that their grandfather is not really their grandfather, but something like his counterpart. David Deutsch and Michael Lockwood think that restrictions posed by classical systems on the actions of time travelers imply that time travel must must displace travelers into different worlds. It is by no means clear that time travelers under such a scheme are really following a closed timelike curve. Closed timelike curves are paths that return to, or very close to, their own spatiotemporal starting points.

On quite another view, time travel really does return an agent to his very own past. A temporal rerun is possible only if everything he does in the past is already in place in his history. This means everything. Consistency might be maintained through the most ordinary of physical processes: the gun misfires, or the bullet dribbles out inconclusively, or at the very last moment your grandfather ducks to tie his shoelaces. Your efforts can make the past what it was but they cannot make the past different from what it was.

In some tantalising remarks in draft, Gödel observed that the constraints imposed on time travelers are not unique to time travel:

This state of affairs seems to imply an absurdity. For it enables one e.g. to travel into the near past of those places where he has himself lived. There he would encounter a person which would be himself so and so many years ago. Now he could do something to this person which he knows by his own memory has not happened to him. This and similar contradictions, however, presuppose not only the practical feasibility of the trip into the past (velocities very close to that of light would be necessary for it) but also certain decision[s] on the part of the traveler; whose possibility one concludes only from vague conviction of the freedom of the will. Practically the same inconsistencies (again by neglecting certain “practical” difficulties) can be derived from the assumption of strict causality and the freedom of the will in the sense just indicated. Hence, as far as the paradoxical situation under consideration is concerned, an R-world [rotating] is not any more absurd than any world subject to strict causality.

There is yet another way of interpreting the paradox of time travel—the third. Seth Lloyd has outlined an analysis of closed timelike curves that differs markedly from the analysis that Deutsch suggested. Lloyd proposed a logic-gate test of model incorporating a grandfather paradox circuit. Consistency is achieved, and catastrophe prevented, through quantum teleportation and postselection. Both Lloyd and Deutsch aim to preserve consistency, but their theories lead to different and distinct predictions. And Deutsch’s theory, Lloyd argues, is weird. A time traveler emerging from one of Deutsch’s closed timelike curves, “enters a universe that has nothing to do with the one she exited in the future.” Lloyd’s own theory, Lloyd observes favorably, is less weird. “[T]he time traveler returns to the same universe that she remembers in the past.”

Physicists generally agree that Gödel’s universe is possible in general relativity. If it is possible, is it actual as well—a description of the universe in which we live? The answer is no. Gödel’s universe is nonexpanding and rotating; ours is apparently expanding and nonrotating. But while this is true enough, it does not fully capture Gödel’s argument, which is, in essence, metaphysical. Time, Gödel argued, is not real. This is the conclusion of his brief note, “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy.” Closed timelike curves in a rotating universe are evidence of an etiolated concept of time. “[F]or every [emphasis original] possible definition of a world time one could travel into regions of the universe which are past according to that definition.” This is quite sufficient to show that “an objective lapse of time would lose every justification in these worlds.” For an observer uninterested in time travel, time might well appear to be passing; but, Gödel notes, “there will always exist possible observers to whose experienced lapse of time no objective lapse corresponds.” There is no objective sense of time that might encompass one observer moving into the future, and another, into the past.

Gödel at once asked the obvious question: “Of what use is it if such conditions prevail in certain possible worlds?” One good question deserves another. “Does that mean anything for the question interesting us whether in our world there exists an objective lapse of time?” It does, Gödel is persuaded, and for two reasons. The first is a reminder. We do not live in a rotating universe; nevertheless, such universes are physically possible. And more. They represent solutions to the equations of general relativity. This is a fact of some importance, for it means that whether time in the ordinary sense exists “depends on the particular way in which matter and its motions are arranged in the world.” From this it follows that duration is an accidental or contingent property of time. But time cannot accidentally or contingently possess the property of duration. Time, Gödel asserts, is necessarily a process of absolute becoming, one now after the other. Duration is of the essence, and so the objectivity of time must be necessary. Lapsing in some universes and present in others, time itself could not be a necessary feature of physical existence. If it is not a necessary feature of physical existence, then time must be ideal in the Kantian sense, perceptually inescapable but physically unreal.

Unlike the Eleatics, Gödel thought the unreality of time was a physical, and not a logical, discovery.

There is something of René Descartes in this argument. Descartes argued that even if the external world did not exist, an evil demon could easily persuade us that it did. Our experiences, after all, would be the same whether an external world existed or not. It follows that any certainty we might have in the existence of an external world cannot be derived from the evidence of our senses. Gödel’s argument is similar. Suppose that in a world lacking physical becoming, we were nonetheless persuaded that time is quite real, and that, as Ecclesiastes insists, the sun also rises. What warrant would we have for our belief? If time did not exist, no part of our evidence would change. If genuine and experiential temporal passage can come apart, this fatally undermines our principal reason for believing in the existence of time. What else, Gödel might ask, could underwrite our belief that time passes apart from our experience?

Gödel thought that his work completed a revolution in our understanding of time that special relativity began. For the moment, we have two incompatible candidates for a fundamental physical theory. They are general relativity and the Standard Model of particle physics. What do more recent theories say? It seems that Gödel’s universe loses its closed timelike curves if modeled in string theory. Gödel’s universe conforms to general relativity. This is, by itself, interesting and unexpected; but from this it does not follow that Gödel’s universe is a realistic physical model of any world, let alone our own. It would seem that in order to make it physically more realistic, physicists must remove chronology violations from its counterintuitive core. Since 1949, some steps have been made towards reconciling general relativity and quantum field theory, but among the casualties of progress are Gödel’s universal closed timelike curves.

Letters to the Editors Beyond Gödel’s Time by Peter Riggs

Closed Timelike Curves and Singularities by Jean-Pierre Luminet