A2.4 Faster than light?
If two objects are approaching one another at speeds approaching the speed of light, could the velocity of one relative to the other exceed light speed?
We all know that nothing can travel faster than light, don’t we? In Chapter 3, when we look at the implications of special relativity for energy, we’ll see why this should be so. Even so, Cormorant has a plan.
He reckons that if he and his cousin Shag go on an intensive workout programme and think really positively, they should each be able to reach a speed of 0.6c (ie 60% of light speed) relative to Nefertiti, standing on the ground and observing sceptically. If they then fly directly towards one another, their combined velocities (should – and here Galileo nods in agreement – surely equal 0.6c - -0.6c = 1.2c. This would mean that, in Cormorant’s frame of reference, Shag should be approaching him at a velocity greater than light speed:
He speculates that seabirds at the peak of their physical powers may actually have been at the heart of the warp drive of the starship Enterprise. However …….
Nefertiti, looking a little disappointed, reminds Cormorant that the velocities he envisages for Shag and himself will not be straightforwardly additive because of relativistic effects. To find out if they can actually go faster than light, they’re going to have to use the velocity transformation formula, to find the velocity of the one in the frame of reference of the other. Let’s see what they’ll find:
If you worked through this example, you'll have discovered that - contrary to what classical mechanics would suggest - breaking the speed of light barrier might not be easy. Cormorant has not given up, however. He wonders just how fast he and his cousin will have to go, to make his dream come true. Let's do a bit of algebra to explore this question:
This exercise reveals that this means that, provided the individual velocities are (as they must be: see chapter 3) less than the speed of light, then the value of the added velocities must itself always be less than the speed of light.
This is really important because, as we have established in Chapter 2, in special relativity there is not really any such thing as “stationary”: you are only stationary in your own frame of reference. So if this velocity addition formula had allowed the possibility of a combined velocity greater than that of light, it would have meant that, simply by changing your frame of reference, you could be travelling faster than light and breaking fundamental laws of the universe. But you can’t, so there.
A disappointment for Cormorant but hold on a minute. Nefertiti has some consolation for him: think back to question A2-6, where we realised that if we remain in Nefertiti's frame of reference there are no tricky relativistic issues. Since both seabirds' velocities were measured in this same reference frame they are simply additive and hence, if each is flying at 0.6c, their convergence rate, as Nefertiti observes it will be 1.2c. Speed of light exceeded! No universal laws have been broken here because this is just a rate of convergence - neither bird can claim to have an individual velocity in excess of c in this (or any other) inertial reference frame. Still, Cormorant will take it.
