What happens if a relativistic space craft launches a fast probe? Can the probe travel faster than the speed of light?
Consider the following setup:
Sulaco launches a fast probe with speed u as measured by Sulaco. The probe passes Nostromo which, according to Sulaco, is streaking to the left with speed v.
The probe happens to nick a pair of sensors on Nostromo as Nostromo streaks past; the flashes from these minor collisions are noted by observers in Sulaco's rest frame. Not at all surprising, Sulaco's observers see that the ratio of the distance and the time interval between the flashes is just Dx/Dt = u. (Nostromo is moving, but Sulaco's observers know that their probe [which travels with speed u] was at the site of the first spark and, later, at the site of the second spark.)
From the Nostromo's frame, things look somewhat different: the moving Sulaco (moving with speed v) launches a probe which, in turn, moves with speed v' as shown in the figure. If Sulaco is moving close to the speed of light, and if the probe's speed in Sulaco's rest frame is also close to the speed of light, will Nostromo see the probe moving faster than c?
Let's use the Lorentz transformations to relate Dx and Dt as seen by Sulaco (recall that Dx/Dt = u) with Nostromo's measurement of v' = Dx'/Dt' where Dx' is the spacing between the sensors in Nostromo's rest frame and Dt' is the time interval between the flashes as measured by Nostromo.
You have to be careful about the sign on the velocity to be used in the Lorentz transformations; it's positive here.
According to Nostromo, the probe's speed is
Even if the probe is very fast (so that u is almost as big as c), and Sulaco is moving relativistically (so that v is almost as big as c), this relativistic velocity addition formula indicates that the probe velocity, according to Nostromo, is never as large as c.
In the limit that u and v approach c,