[repost] What is the speed of laser spot?

Question

Rapidly rotating guidance laser installed on board
of satellite projects a spot on the ground. The spot
moves with constant speed v = 600,000,000 m/s with
respect to the ground.

What is the speed of the spot observed by tracking
system installed on board of jet fighter flying with
speed s = 1000 m/s in the same direction
(i.e. chasing the spot)?

Answer 1

This one is rife with ambiguities over the question of how the "observations" necessary to determine the speed of the moving laser spot are conducted. If the jet flighter relied on markings on the ground and sychronized its clocks with the ground, then of course it would "observe" the spot as going 600,000,000 m/s as well, because it is borrowing the "proper" time and distances instead of using its own. As another example, if the jet fighter relies on the ground markings, but uses its own clocks to tick off the time it takes the laser spot to cover a distance on the ground, then this figure is reduced by the gamma factor, and so we get a figure that's not normally found in relativity theory becuase the pilot is "mixing dimensions". If the jet fighter is somehow able to conduct its own independent time and space measurements without relying on anything on the ground, and assuming that it is making those observations that aren't skewed by relativistic angles of sight (which only adds to more complications to this question), then the simple relativistic addition of velocities should work----maybe? But the spot is moving than the speed of light, are we going to run into an erroneous result here?

Let c = speed of light, T = proper time interval, V = 600,000,000 m/s = proper velocity of laser spot, t = jet time interval, W = jet measured velocity of laser spot (to be determined), and v = 1000 m/s. Then we know that because of the invariance of the spacetime interval:

(ct)² - (Wt)² = (cT)² - (VT)²

and, after a little careful graphical analysis of the problem, that

cT - (v/c) VT = (√(c²-v²))t

Solving the 2 equations for t and W does end up with the relativistic addition of velocities after all:

W = (v-V)/(1-(vV)/c²) = 600,003,000 m/s

which seems like a paradoxical result. However, in fact, if the laser spot was moving on the ground at speed c²/v = 9 x 10^13 m/s, then to the jet, it would appear to be moving infinitely fast, and for laser spot speeds higher than that, it would actually seem to be going the OTHER direction! Funny things that you run into with relativity things.

Answer 2

My gut instinct is that to leading orders, the velocity of the spot isn't going to change much. Nobody except the spot (which isn't really an entity) is moving at relativistic velocities. Are you assuming that the satellite is stationary (at least wrt ground)?


I have to travel to CERN tomorrow AM, so I won't have time to follow up--I'm curious, though, what else is supposedly going on here.

Are you assuming that the jet's tracker isn't smart enough to correct for the time it takes for the light to get to it from the spot? That, of course, would screw with the perceived speed, which would then be infinite for a spot traveling at exactly c (all the spot light would arrive at one time) and fall off in way which would be straight-forward to calculate.

Answer 3

299,792,458 metres per second is the speed of light.

Your premise exceeds this by 300%, and so, is impossible.