There come times, in the course of human events, whereby an official-unofficial OTI Staff Scientist such as myself feels compelled to write about pop-cultural issues which he or she (statistically: he) finds irksome on a professional level. Back to The Future is a nearly endless font of these issues for me—from the obligatory complications surrounding causality in time-travel paradoxes to hoverboards which don’t work above water to the continued reminder of the sheer mathematical improbability of Huey Lewis and the News. But as my esteemed colleagues have already touched on these with great aplomb, I’ve got some another problem to Overthink™ today.

And oh yes, there will be math involved.

Pop culture has envisioned a multitude of time–travel devices, and these give rise to the usual scientific musings (physical and metaphysical alike): questions of causality, internal paradoxes, and the “inconveniences” associated with the creation a billion alternate universes or self-referential circular time lines. Also, there’s usually lightning and dramatic music.

However, there’s a much smaller–though fundamentally difficult–caveat to time travel that’s always kinda’ bothered me. With the notable exceptions of Doctor Who and Bill and Ted’s Excellent Adventure, the spatial aspects of time travel are almost universally overlooked when characters decide to go gallivanting through temporal mayhem. I was reminded when questioned by a fellow OTI™ writer in a recent email; so as to protect his/her (statistically: his) anonymity, I’ll keep the author’s identity a secret :

Q.

“Watthew Mrather” writes: The DeLorean can’t go anywhere in space, except in the conventional driving/flying sense. Your [sic] always in the same place you left, but in a different time.

A.

Well, for starters, that’s not a question, “Mr. Mrather,” but I’ll go along with it anyway. Remember, though: without proper punctuation and grammar, we are no better than the animals.

In fact, in order to pull off the kind of time travel we see in the Back To The Future trilogy–the kind where the traveler is transposed in time, but remains stationary in the same relative position to where he/she left–the DeLorean would have to be an outstanding space ship, in addition to its already laudable work as a time-ship. A major issue of freely traveling within time while limiting one’s self to a local reference frame–say, a California mall parking lot–is that the reference frame itself isn’t stationary. As an illustration, let’s figure out how far the DeLorean would have to travel in order to stay in sync with the Earth over a relatively small time-jump. We’ll look at the simplest example (and the first one, diagetically speaking) of the whole BTTF trilogy. You all remember the scene, right? (Spoiler alert: Professor Plum and Alex P. Keaton send a dog one minute into the future.)

http://www.youtube.com/watch?v=BytKSy8M4bk

(Why no one in sci-fi movies ever names their pet Smoluchowski, I’ll never know.)

As most of us know, the Earth both rotates about its axis (accounting for the regular cycle of day and night, and hence a good bit of poetry) and revolves around the sun (accounting for the seasons, and hence the rest of poetry). However, the litany of additional factors affecting Earth’s absolute position in space makes it fantastically difficult to calculate where a given point on its surface would be after some interval of time has passed. A little professional lingo, here: we in the science biz refer to this level of complexity as a “colossal mind-fuck.” So, to make my life easier, I’ll first lay down the standard boilerplate warning we employ for all Overthought(™-pending) mathematical meta-transmutations:

For all calculations pertaining to a period of time less than one hour, the Overthinking It™ writers reserve the right to disregard effects due to the nonperpendicular axial-tilt of the Earth relative to its plane of revolution, the precession of the Earth about its axis, the gradual decay of Earth’s orbit into the sun, the rotation of the Milky Way, analogous rotational and linear velocities within the Virgo Supercluster, and the general expansion of the Univese.

Also, we won’t factor in Daylight Savings Time.

Got it? Good. Let’s get this party started.

SO, according to Doc Brown’s stopwatch, Einstein (the dog, not the Princetonian–or the grossly mispronounced Russian film director) travels precisely one minute into the future on this first jump, arriving, relative to their frame of reference, at the same location he left. But how far has this reference frame itself traveled during that one minute? Let’s calculate and see.