Mathematics suggested that time travel is physically possible – and Kurt Gödel proved it. Mathematician Karl Sigmund explains how the polymath did it.

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There may be no better way to get truly lost in space-time than to travel to the past and fiddle around with causality. Polymath Kurt Gödel suggested that you could, for instance, land near your younger self and “do something” to that person. If your action was drastic enough, like murder (or is it suicide?), then you could neither have embarked on your time trip, nor perpetrated the dark deed. But then no one would have stopped you from going back in time and so you can commit your crime after all. You are lost in a loop. It’s no longer where you are, but *whether* you are.

Gödel was the first to prove that, according to general relativity, this sort of time travel can be done. While logically impossible, the equations say it is physically possible. How can that actually be the case?

Widely hailed as “the greatest logician since Aristotle”, Gödel is mainly known for his mathematical and philosophical work. By age 25, while at the University of Vienna, he developed his notorious incompleteness theorems. These basically say that there is no finite set of assumptions that can underpin all of mathematics. This was quickly perceived as a turning point in the subject.

In 1934, Gödel, now 28, was among the first to be invited to the newly founded Institute for Advanced Study in Princeton, New Jersey. During the following years, he commuted between Princeton and Vienna.

After a traumatic journey around a war-torn globe, Gödel settled in Princeton for good in 1940. This is when his friendship with Albert Einstein developed. Their daily walks became legendary. Einstein quipped: “I come to my office just for the privilege to escort Gödel back home.” The two strollers seemed eerily out of their time. The atomic bomb was built without Einstein, and the computer without Gödel.

When Einstein’s 70th birthday approached, Gödel was asked to contribute to the impending *Festschrift* a philosophical chapter on German philosopher Immanuel Kant and relativity – a well-grazed field. To his mother, he wrote: “I was asked to write a paper for a volume on the philosophical meaning of Einstein and his theory; of course, I could not very well refuse.”

Gödel began to reflect on Kant’s view that time was not, as Newton would have it, an absolute, objective part of the world, but an a priori form of intuition constraining our cognition. As Kant said: “What we represent ourselves as changes would, in beings with other forms of cognition, give rise to a perception in which… change would not occur at all.” Such “beings” would experience the world as timeless.

In his special relativity, Einstein had famously shown that different observers can have different notions of “now”. Hence, no absolute time. (“Newton, forgive me!” sighed Einstein.) However, this theory does not include gravitation. Add mass, and a kind of absolute time seems to sneak back! At least, it does so in the standard model of cosmology. There, the overall flow of matter works as a universal clock. Space-time is sliced in an infinity of layers, each representing a “now”, one succeeding another. Is this a necessary feature of general relativity? Gödel had found a mathematical kernel in a philosophical problem. That was his trademark.

At this stage, according to cosmologist Wolfgang Rindler, serendipity stepped in: Gödel stumbled across a letter to the journal *Nature* by physicist George Gamow, entitled “Rotating universe?”. It points out that apparently most objects in the sky spin like tops. Stars do it, planets do it, even spiral galaxies do it. They rotate. But why?

Gamow suggested that the whole universe rotates, and that this rotation trickles down, so to speak, to smaller and smaller structures: from universe to galaxies, from galaxies to stars, from stars to planets. The idea was ingenious, but extremely vague. No equations, no measurements. However, the paper ended with a friendly nudge for someone to start calculating.

With typical thoroughness, Gödel took up the gauntlet. He had always been a hard worker, who used an alarm clock not for waking up but for going to bed. He confided to his mother that his cosmology absorbed him so much that even when he tried to listen to the radio or to movies, he could do so “only with half an ear”. Eventually, Gödel discovered exact solutions of Einstein’s equations, which described a rotating universe.

However, while Gamow had imagined that the centre of rotation of our world is somewhere far away, beyond the reach of the strongest telescopes, Gödel’s universe rotates in every point. This does not solve Gamow’s quest for the cause of galactic rotations, but yields another, amazing result. In contrast to all then-known cosmological models, Gödel’s findings showed that there is no “now” that’s valid everywhere. This was exactly what he had set out to achieve: vindicate Kant (and Einstein) by showing that there is no absolute time.

“Talked a lot with Gödel,” wrote his friend Oskar Morgenstern, the economist who, together with John von Neumann, had founded game theory. He knew Gödel from former Viennese days and reported all their meetings in his diary. “His cosmological work makes good progress. Now one can travel into the past, or reach arbitrarily distant places in arbitrarily short time. This will cause a nice stir.” Time travel had been invented.

In Gödel’s universe, you don’t have to flip the arrow of time to go back to the past. Your time runs as usual. No need to shift entropy in return gear. You just step into a rocket and take off, to fly in a very wide curve (very wide!) at a very high speed (but less than the speed of light). The rocket’s trajectory weaves between light cones, never leaving them but exploiting the fact that in a rotating universe, they are not arrayed in parallel. The trip would consume an awful amount of energy.

Gödel just managed to meet the editorial timeline. On his 70th birthday, Einstein got Gödel’s manuscript for a present (and a sweater knitted by Kurt’s wife Adele). He thanked him for the gifts and confessed that the spectre of time travel had worried him for decades. Now the spectre had materialised. Einstein declared Gödel’s paper “one of the most important since my own”, and stated his hope that time travel could be excluded by some as yet unknown physical law. Soon after, Gödel received the first Albert Einstein award. It went with a modest amount of money which Gödel, as it turned out, could use well.

Next, according to philosopher Palle Yourgrau, “something extraordinary happened: nothing”.

For several decades, the mind-bending discovery of Gödel, far from causing “a nice stir”, got very little attention. When Harry Woolf, the director of the Institute for Advanced Study, arranged the eulogies to be given at Gödel’s funeral in 1978, he listed the topics to be covered: set theory and logic, followed by relativity, which he noted was “not worth a talk”.

Only by and by did eminent cosmologists, such as Stephen Hawking, Kip Thorne or John Barrow, convey an area of respectability to the field. Today, it is mainstream. With time, it transpired that, years before Gödel’s breakthrough, several other cosmological models had exhibited both rotation and the possibility of time travel. However, this aspect had never been noticed, not even by the engineers of these universes.

Many physicists are happy to leave the paradoxical aspects of time travel to philosophers. They invoke a “chronology protection law” that would step in to prevent the worst. It sounds like whistling in the dark but helps to overcome the problem of haunting your own present as a revenant from the future.

And does our universe rotate? Gödel was equivocal on that issue. Sometimes he claimed that his model only served as a thought experiment, to display the illusionary character of time, which cannot depend on accidental features of the place we happen to inhabit. Cosmologist Freeman Dyson, however, reported that Gödel, near the end of his life, had shown dismay when told that evidence for a rotating universe is lacking.

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*Credit for article given to **Karl Sigmund***