Adding subluminal and superluminal velocities.

[mrspeedybob]

Suppose we have a long row of rods, each of which is equipped with a clock and a solenoid which can raise and lower the rod. All the clocks are synchronized. Each rod is programed to raise and lower at a slightly different time so that in the rest frame of the apparatus the rods rise and fall in a wave that moves at 1.5 c. What you would actually see from the midpoint of the row would be the wave moving away from you in both directions at different speeds, but after accounting for the travel time of the light you would realize that what you saw was a single wave traveling past you at 1.5 c.

Now suppose a spaceship travels along the row of rods at 0.5 c in the opposite direction of the wave. The ship observes the wave, accounts for the travel time of the light in its own frame of reference, and calculates the speed of the wave. Using the formula for the addition of relativistic velocities I get that it should observe a speed of 1.14 c.

So the ship moving in the opposite direction of the wave observes a lower wave velocity then the observer standing still?

I then computed the relative velocity of the wave for a ship moving in the same direction at 0.5 c and came up with a result of 4 c.

This seems completely counterintuitive. Is this just one of those counterintuitive results of SR or am I applying the formula incorrectly? Does the formula for adding relativistic velocities not apply to superluminal velocities?

 

[mfb]

It is counter-intuitive, but it is not wrong. With the right speed in the direction of the wave, the wave will even get an "infinite speed" (all rods move at the same time) - and if you increase the speed of the spaceship, the wave will go in the opposite direction.

I guess (but I did not check) that the velocity addition formula still works.

 

[DrGreg]

If you visualise how the Lorentz transform changes the slopes of lines, if a speed lower than c is increasing towards c, then a speed faster than c is decreasing towards c.

Image credit: Jonathan Doolin, Wikipedia, CC BY-SA 2.5