A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2017 Jan 13, 19:41 -0800
Gary LaPook, you wrote:
"I had been deaming about that movie and I woke up and said "they didn't allow for the Lorentz tranformation!" I can't believe that came to me in my dream."
Heh. :} We all know that neither science fiction movies nor dreams can be held to an exact scientific standard so anything goes. Nonetheless, for our nav-nerd record...
The Lorentz transformation could enter into a story like this in one of two ways: 1) they tell us how long the trip is going to take as seen by observers on Earth, and we reduce that by the factor gamma that is fundamental to the Lorentz transformation to account for time dilation for the people aboard ship, or 2) they tell us how far away the destination is as measured by observers on Earth and we reduce that distance by the factor gamma for length contraction as seen by observers aboard ship. Either way, you get a shorter travel time on-board. The catch here is that the movie does not seem to have provided either of these pieces of information. The colony planet orbits a fictional star. I haven't seen the movie, but I read through some sections of a subtitle file (presumably for pirated versions of the film since it isn't available outside theaters yet). There's no reference to the distance to the "colony planet" named "Homestead II" in the film. Instead we're told that the trip will take 120 years. But if that's the shipboard time, then it's already been adjusted for relativistic effects. There's also a reference in the subtitle file to a speed of "50% of the speed of light". If that's their speed, which would be consistent the whole "sleeper ship" motif (popular in science fiction for decades), then the relativistic factor gamma wouldn't amount to much difference anyway. The equation for the time dilation factor is
gamma = 1/sqrt(1-beta2)
where beta is the fraction of the speed of light. At 99% of the speed of light (beta=0.99), the gamma factor is just about 7, and for every "pair of nines" you add to the fraction, you get a factor 10 more. So at 99.99% of c, gamma is about 70, and at 99.9999% of c, gamma is about 700 (actually sqrt(2)·500). Time slows down by a factor 700. In other words, if a trip to a star takes 700 years as seen by observers back on Earth, it only takes one year aboard ship (there are other huge problems with this, but we can ignore them for now). At those speeds, so very close to the speed of light, the Lorentzian-Einsteinian time dilation or equivalently the shortening of the distance to the destination by Lorentz contraction is a huge effect, but at 50% of the speed of light, if that is in fact the relative speed quoted in the film, it's only about 15% --relatively unimportant. If the speed is instead 75% of c, then the time dilation factor is 1.53 (100 years aboard ship is 153 years in the frame of reference of the Earth and the other "fixed" stars).
Another astronomical clue: there is a so-called "slingshot" by the star Arcturus a couple of years after Jim awakens. The ship has been in transit for 30 years when he first asks a computer for that detail, and roughly two years have elapsed after that when the Artcurus flyby apparently occurs. So that's 32 years aboard ship while Arcturus is located about 37 lightyears from Earth. This gives an implied speed of about 75% of c. There is also a conversation about sending a message back to Earth, and Jim learns he will have to wait 55 years for a reply. This is in the right ballpark, but I haven't checked the details. I'll nitpick and point out that there are no "slingshots" (no gravitational slingshots) to speak of at such high speeds. The gravitational field of a star barely affects an object moving at 75% of the speed of light. This is a common problem in science fiction stories about interstellar flight. But here I feel we have to grant some artistic license since otherwise an interstellar "sleeper ship" flight is probably incredibly dull. It's more fun to pretend that there might be a crew with a "Captain" and a "Navigator" or at least plans being executed by a computer that are important to the task of reaching the destination. Point-and-shoot just doesn't have the same dramatic tension.
PS: off-topic, yes, but it's been a quiet week.