Y'know, I always THOUGHT I understood relativity...

Okay, I’m pretty sure I still understand it, I just wanna clear something up regarding the “speed of light”. I was reading a Scientific American article dealing with time travel, lightspeed, and other funky physics stuff that I eat up like a starving child in a candy store. One of the things mentioned was this:

They’re dealing with the whole “if you travel faster than light you go back in time” thing. If you go faster than light, says those with bigger brains than mine, you’ll be able to look behind you and see yourself before you left. That much I can understand. However, I’m not getting something… just because you can SEE into the past, that doesn’t mean you actually ARE in the past. If you travelled around the universe as described in the article, on the return arc, wouldn’t everything you see simply appear to be radically blue-shifted and appear to move quicker, right up to the point where you return to your starting point?

Now, that ain’t my question, 'cuz I know that I’m not smarter than those big-name physics guys that get all the cool quantum particles named after them. I’m assuming that the mechanism for the aforementioned travel-faster-than-light time travel scheme has nothing to do with the photons themselves… that there’s an overarching law of space/time that both governs the speed of light as well as time. Well, if so, what is it? How has it been determined that, if you go faster than light, you go back in time? Sure, sure, time slows down as you get closer to lightspeed, light appears the same from all reference points, etc. etc. I guess I just don’t automagically see how that translates into time travel. What am I missing?

Well, I’m confused. I thought relativity theory essentially said that you could never exceed (or even attain) the speed of light at all, because as you accelerate approaching light speed, mass increases to infinity and time slows to zero from the standpoint of a stationary observer. This was explained by Cecil in the memorable phrase, “the faster you go, the slower you go.”

What Dr. Kaku is describing in the interview is a solution to Einstein’s equations called the “Gödel Universe”, and the fact that it contains objects called “closed timelike curves.” Google on either one of these phrases if you want to try & find more information, but in a nutshell:

There’s no faster-than-light travel involved here, but rather an odd consequence of global properties of the curvature of spacetime. In general relativity, when you allow space to be curved, there’s the possibility of finding curves that never go faster than light (as defined in each small region of space the curve passes) but have the property that if you travel along one of these curves, you’ll eventually come back to the same place and time as you started.

The simplest possible example of this: Suppose space were one-dimensional, so our space-time was two-dimensional (one space direction, one time direction.) This is the spacetime of special relativity. We could, however, take the “strip” of this spacetime lying between t=0 sec and t=1 sec (or any values you’d care to name), discard the rest of the spacetime, and “glue” the edges of the strip together. Now suppose a spaceship decides to stay in one place in space, just travelling along a curve that takes him from (x = 0 m, t = 0 sec) to (x = 0 m, t = 1 sec.) Since we’ve glued the edges of our spacetime together, these two points are the same; so the spaceship has managed to arrive back at the same time & place he started without ever travelling faster than light.

This is a highly artificial spacetime to be thinking about, to be sure, but closed timelike curves arise more naturally in some other spacetimes (including the Gödel Universe.) Usually, they’re taken to be a sign that the spacetime you’re looking at isn’t physically realistic, although nobody has proven that closed timelike curves cannot be created in a “physically reasonable” spacetime.