De facto superluminal travel?

Question

It seems that the limitation to subluminal travel can be practically circumvented (so to speak...not breaking any physical laws here) if we consider the viewpoint of the traveler, not some outside observer:

• As your ship approaches $c$ (as measured from some inertial observer) the proper time of the trip decreases. Hence, if you measured the distance to your location as 10 light years, you would reach your destination having aged less than 10 years, given that you travel at a sufficient fraction of $c$. Thus, to you, you traversed the distance faster than light.

Is this a valid interpretation, or will something intervene to prevent this apparent superluminal travel?

Answer 1

It's a perfectly valid interpretation. If you have a look at the question How long would it take me to travel to a distant star? you'll find it's quite possible to cover a distance greater than $ct$ in an elapsed time $t$. But this doesn't mean special relativity is wrong - indeed the calculations done in that Q/A were done using special relativity.

The problem is that when teaching students special relativity we start by using simplifications that apply only to inertial frames i.e. situations where there is no acceleration (or indeed spacetime curvature, as in GR). Students then apply these simplifications to situations where there is acceleration, and as a result run into problems like the apparent FTL travel you describe, or most notoriously the twin paradox.

To understand what's going on you need to know that the invariance of the speed is light is a local property. That is, if you measure the speed of light at your position you'll get the value $c$, and if you measure the speed of any massive object at your position you'll get a value less than $c$.

If a measurement is not local neither of these results is guaranteed. For example right now there are distant galaxies moving at faster than light relative to us. This is a result derived using general relativity, but a related effect happens even in special relativity. If you accelerate at a constant acceleration for long enough a horizon will develop behind you (the Rindler horizon), and the horizon develops because everything behind it is travelling faster than light relative to you.

So you're quite correct that superluminal travel is possible in the sense you describe, but superluminal travel is still impossible in the sense Einstein described.

Answer 2

One can in principle travel a given distance along a ruler in arbitrary short time. The relevant velocity definition is proper velocity: the distance measured by an observer at rest with respect to the ruler, divided by the time passing on the wristwatch of the traveller.

Note that proper velocity deploys a mixture of reference frames, whereas ordinary velocity (bounded by $c$, the speed of light) is defined using distances and durations as observed from one single reference frame. Both velocities are identical in the non-relativistic limit of low speeds.

Proper velocities larger than the speed of light are not in conflict with the relativistic notion that superluminal motion is physically impossible: a photon has unbounded proper velocity and can not be overtaken.