A scene from the TV series “Timeless.” The one behind the protagonists is — or should be — a time machine

The Space Problem of Time Travel

A science-fiction cornerstone is time travel. Thanks to some wonderful machinery invented in the future, human beings can go back in time, disappearing from their present to find themselves in the same place in any past era. But is such spatial precision physically possible?

One of the most prolific and compelling themes of science-fiction literature and cinema is undoubtedly time travel. An incalculable number of films — from Terminator to Back to the Future, from 12 Monkeys to X-Men: Days of Future Past — bases its plot on the return of the protagonists to previous eras, with the idea of doing something that, in one way or another, alters the course of history and creates a different future (or rather a different present, from the point of view of those who come from the future).

But time travel inevitably produces paradoxical situations, such as the possibility that the time traveler will encounter a younger copy of herself or that she kills her parents before they conceive her. However, there’s another enormous theoretical difficulty that makes the invention of a time machine that can do what is shown in movies highly unlikely. And it is a difficulty that is rarely thought about — the problem of spatial precision. Caleb Scharf, a well-known British astronomer and science writer, director of Columbia Astrobiology Center, spoke about it in an article published some time ago in Scientific American.

Let’s take it for granted — says Scharf — that someone invented a technology that allows time travel. However, moving through time is only part of the problem. For every shift in time, there is always, in fact, also a shift in space, which a hypothetical time machine should calculate with absolute precision, if it really wanted to guarantee the traveler the possibility of finding herself, exiting the machine, in the same place she got in.

In science-fiction films, things are much simpler than that. You usually see the time traveler setting on some futuristic dashboard the date on which she wants to return to as if it were the only thing that matters. The problem of where is deliberately left in the shadows. It is as if there were in the background the unspoken idea that time travel and space travel are two separate and independent things. The spectator thus remains with the belief that the time machine has the sole function of making a time jump, without physically moving in space. The passenger exits the vehicle, and miraculously finds herself in the same place.

In reality, even travel back in time a single month can be an insoluble problem concerning the spatial coordinates that should be set at the start. Each point in space, in fact, is in constant and unstoppable movement within the Universe. To establish where that point was a month earlier, one must take into account how it has moved with respect to different reference systems.

First, the Earth turns on itself and drags with it all the points on its surface with speed proportional to latitude. At the equator, the speed is maximum, with a displacement of over 1,600 km/h. It means that if one activated the time machine in a place on the equator, only an hour before departure that same location would have been 1,600 km further west. Finding oneself, therefore, in the same point of the Earth precisely one month before, requires a complex calculation, which must take into account even the small fluctuations in the duration of the day. In fact, missing the place of arrival in the past, even just for a few kilometers, could mean reappearing in the belly of a mountain or the open sea. Or suspended in the air.

But the Earth’s rotation is only the first of the spatial problems to be solved. The second is the revolution around the Sun. The Earth travels its annual orbit at an average speed of 110,000 km/h. In a month, it covers an arc of orbit equal to approximately 78 million km. We must also take this shift into account, if we want to hope that, going back a month in the past, we will find ourselves at least on Earth. And the calculation cannot be based on the average speed but must be much more precise. It must take into account the fact that, according to Kepler’s second law, Earth’s orbital velocity is variable. It is higher when our planet is close to perihelion and lesser when it is close to aphelion.

The calculation is then made even more complicated by the fact that the eccentricity of the Earth’s orbit varies slightly over time. It affects the shape of the orbit and the orbital speed. If we go back only a month in the past, the effect would be negligible. But, if instead we wanted to go back in time to the age of the dinosaurs, it would become a huge and almost insoluble problem. The variations of the Earth’s orbit over millions of years are determined, in fact, by a complex of gravitational influences that cannot be reconstructed backward with the necessary precision. The time machine would thus risk completely missing Earth’s position, condemning the unwary traveler who returned tens or hundreds of millions of years in the past, to die in interplanetary space. (A big deal, if you want to go back 85 million years as in the Terra Nova TV series.)

But there are still other and more severe problems with spatial coordinates. The Earth does not move only on itself and around the Sun, but together with the whole solar system around the galactic center. It does so with a speed of around 828,000 km/h, which means that in a month our planet and the entire solar system have moved nearly 600 million km with respect to the center of the Milky Way. Skipping a month in the past with your time machine, therefore, requires taking into account this shift as well, knowing that a mistake of only 1 part in 1,000 would be enough to miss the Earth by hundreds of thousands of miles (very probable error, since we know the path traveled by the solar system around the galactic center in a rather approximate way).

And it’s not over. The Milky Way with us inside, along with all the other galaxies of the Local Group, is racing at a speed of 2.4 million km/h compared to the cosmic microwave background, the radiation at about 3 degrees Kelvin that fills the Universe in every direction. Concerning this radiation, which can be considered as an optimal reference system for different observers to agree on what is stationary and what moves in the Universe, we move in one month, with our galaxy, of something like 1.7 billion km. And let’s not forget, finally, that the Universe is continuously expanding. Space between galaxies grows with a speed of about 70 km/s per megaparsec in every direction.

In short, for a time machine to bring its passenger back by even a month in the past, unloading her in the same place on Earth from which she started, precision in the calculation of the spatial coordinates is required in the order of 1 part per trillion! It is an astounding level of accuracy that, at least in the current state of knowledge, is simply unattainable.

The little-considered problem of spatial coordinates is probably one of the reasons why we don’t see time travelers around. At least that’s what Caleb Scharf thinks:

It’s also fun to consider that this could provide an answer to the question of why, if time travel is ever invented, we haven’t been visited by beings from the future (you know, right before certain presidential elections, or other key moments). Perhaps the reason is that no one has (ever) solved the spatial problem, and the cosmos is littered with time travelers adrift between the stars and galaxies.

Science writer with a lifelong passion for astronomy and comparisons between different scales of magnitude.

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