Time Travel. A History

Admit it: you didn’t help build the pyramids. That’s a fact, but is it a logical fact? Not every logician finds these syllogisms self-evident. Some things cannot be proved or disproved by logic. The words Hospers deploys are more slippery than he seems to notice, beginning with the word time. And in the end, he’s openly assuming the thing he’s trying to prove. “The whole alleged situation is riddled with contradictions,” he concludes. “When we say we can imagine it, we are only uttering the words, but there is nothing in fact even logically possible for the words to describe.”

Kurt Gödel begged to differ. He was the century’s preeminent logician, the logician whose discoveries made it impossible ever to think of logic in the same way. And he knew his way around a paradox.

Where a logical assertion of Hospers sounds like this—“It is logically impossible to go from January 1 to any other day except January 2 of the same year”—Gödel, working from a different playbook, sounded more like this:

That there exists no one parametric system of three-spaces orthogonal on the x₀-lines follows immediately from the necessary and sufficient condition which a vector field v in a four-space must satisfy, if there is to exist a system of three-spaces everywhere orthogonal on the vectors of the field.

He was talking about world lines in Einstein’s space-time continuum. This was in 1949. Gödel had published his greatest work eighteen years earlier, when he was a twenty-five-year-old in Vienna: mathematical proof that extinguished once and for all the hope that logic or mathematics might assemble a complete and consistent system of axioms, powerful enough to describe natural arithmetic and either provably true or provably false. Gödel’s incompleteness theorems were built on a paradox and leave us with a greater paradox.^*2 We know that complete certainty must always elude us. We know that for certain.

Now Gödel was thinking about time—“that mysterious and self-contradictory being which, on the other hand, seems to form the basis of the world’s and our own existence.” Having escaped Vienna after the Anschluss by way of the Trans-Siberian Railway, he settled at the Institute for Advanced Study in Princeton, where he and Einstein intensified a friendship that had begun in the early thirties. Their walks together, from Fuld Hall to Olden Farm, witnessed enviously by their colleagues, became legendary. In his last years Einstein told someone that he still went to the Institute mainly um das Privileg zu haben, mit Gödel zu Fuss nach Hause gehen zu dürfen, to have the privilege of walking home with Gödel. For Einstein’s seventieth birthday, in 1949, his friend presented him with a surprising calculation: that his field equations of general relativity allow for the possibility of “universes” in which time is cyclical—or, to put it more precisely, universes in which some world lines loop back upon themselves. These are “closed timelike lines,” or as a physicist today would say, closed timelike curves (CTCs). They are circular highways lacking on-ramps or off-ramps. A closed timelike curve loops back on itself and thus defies ordinary notions of cause and effect: events are their own cause. (The universe itself—entire—would be rotating, something for which astronomers have found no evidence, and by Gödel’s calculations a CTC would have to be extremely large—billions of light-years—but people seldom mention these details.)^*3

If the attention paid to CTCs is disproportionate to their importance or plausibility, Stephen Hawking knows why: “Scientists working in this field have to disguise their real interest by using technical terms like ‘closed timelike curves’ that are code for time travel.” And time travel is sexy. Even for a pathologically shy, borderline paranoid Austrian logician. Almost hidden inside the bouquet of computation, Gödel provided a few words of almost-plain English:

In particular, if P, Q are any two points on a world line of matter, and P precedes Q on this line, there exists a time-like line connecting P and Q on which Q precedes P; i.e., it is theoretically possible in these worlds to travel into the past, or otherwise influence the past.