The Limit of Speed

I'll show you why. Suppose Astro tried to push his ship all the way up to the speed of light. Well, we've already seen that the energy of an object is proportional to that "gamma" factor, which is so prevalent in relativity calculations. But you've also learned by now that gamma becomes infinitely large when the speed of an object is equal to the speed of light. So, in order for Astro to accelerate his ship to the speed of light, he would require an infinite amount of energy, which is clearly impossible. So any object with mass can never reach the speed of light, though there's no limit to how close to the speed of light an object can come. (An object that doesn't have mass must, in fact, travel exactly at the speed of light, for reasons I won't go into. But the only objects with no mass are particles of light (called "photons") and maybe neutrinos.)

There are other reasons an object's speed cannot exceed the speed of light. One of these involves "causality", the relation between cause and effect. Suppose I launch a baseball, and it breaks a window. My throwing the baseball is then the cause of the window breaking. If speeds faster than light were possible, then there could actually be some frames of reference in which the window breaks before the ball is thrown. This leads to all kinds of logical contradictions (especially if someone intercepts the ball in flight, preventing the window from breaking, after the window has already broken!) so we rule out the possibility of things moving faster than light. Furthermore, causality rules out not just objects traveling faster than light, but any kind of communication that travels faster than light. The speed of light, to the best of our knowledge, is an uncrossable barrier.

Now if you're a science fiction fan, as I am, this may come as bad news. After all, there's almost certainly no intelligent life in our solar system outside of earth, and the distance between stars is huge. Even the nearest star would take four years to reach at light speed. So without traveling faster than light, it's pretty much impossible to go flying around the galaxy, meeting alien civilizations, fighting wars for galactic empires, and so forth. Bummer, eh?

On the other hand, it may not be as hopeless as all that, due to length contraction. Suppose you climb into a spaceship and head for a star 10 light years away, at a speed close to the speed of light. From the earth's frame, the trip has to take at least 10 years. However, to the passengers on the ship, the length is contracted, and so the trip can take less than 10 years. And the closer to the speed of light the ship travels (relative to the earth and the star) the more the length is contracted. (You can also deduce this by considering time dilation.)

To illustrate this, here's a table showing some travel times to various destinations, at various speeds. Let me explain what all this means. First of all, in order to get serious length contraction, we need to get very close to the speed of light. So I'm considering a trip in which we have a spaceship able to generate a constant acceleration. That is, the people inside the ship experience a continuous acceleration of, for example, 1 g for half the trip, and then turn around and decelerate at 1 g the rest of the way. (I'm almost positive I've calculated all this correctly.)

The second column gives the distance to the star or galaxy of our destination, in light years. (A light year is the distance light travels in a year: about six trillion miles.) I've included calculations for three different accelerations, one smaller, one greater, and one equal to earth's gravitational acceleration. The 2g trip would be pretty uncomfortable, and you can probably forget anything much faster than that. The fourth column shows the maximum speed (at the halfway point, just as the ship is turning around to decelerate) as a fraction of the speed of light. The entries near the bottom have more 9's than I felt like including.

The final two columns give the amount of time the trip would take, first in the earth's frame of reference, and then in the frame of reference of the spaceship. The difference is important. It means, for example, that if you are on a spaceship traveling to Betelgeuse with an acceleration of 1 g, about 6.8 years will pass on the ship before you get there. (The "ship times" increase very slowly despite the big increases in distance, because the greater the distance, the closer to the speed of light you can get before you have to slow down again, and so the more length contraction you get!) But when you arrive there, over 500 years will have passed on earth. And any message you send to earth when you get to Betelgeuse will take over 500 years to get there, as will any reply. So even if humankind one day spreads out across the galaxies, the different settlements will be very isolated. People on earth won't be talking to people near Betelgeuse on any regular basis.

Now of course there are enormous technical difficulties to building a spacecraft that can accelerate indefinitely like this. These difficulties may prove insurmountable, and we're out in fantasy land. But if they're not insurmountable, and if our species survives long enough to surmount them, then what I've just described represents the logistics of far space travel, according to the special theory of relativity.

Of course, many works of science fiction still involve faster than light travel. But they always have to invent some weird concept to go with it, like "warp" or "hyperspace". The bottom line is this: given everything we know about space and time today, traveling faster than light is impossible. But if you like, you can always cling to the hope that some sort of loophole or whole new branch of physics will be discovered that will allow objects to travel faster than the speed of light. Then we can get to work on that big galactic empire.