http://www.straightdope.com/classics/a2_095.html
Your equation for accumulated velocities is enlightening.
If there were no bodies in space, no reference points, speed would not exist. Right?
What is the speed of a comet if there are no telephone phone poles zipping by?
Vince
OK, but here’s a scenario that still bothers me.
A string of ships traveling a near light speed toward a distant planet. The ship farthest from the planet sends a laser signal to the ship in front of him, and that ship in turn passes the message on to the next ship… the last ship beams the message to the planet.
At the same time as the very first message was sent, the first ship sends a laser message directly to the planet.
Do both messages arrive at the same time? How does this appear to an observer who is motionless relative to the planet?
I think I can clear this up for you a little.
I believe the laser sent directly to the planet would arrive first, because the planet is moving away from the ship much more slowly than the other ship. To wit: if you are chasing someone who is running away from you, it will take you much longer to catch him than if you are chasing someone who is stationary. If the ships are moving near the speed of light, it will take some time for the lasers (moving at light speed) to catch them. Planets, meanwhile, do not move anywhere near the speed of light, so the laser will make up ground much more quickly, and arrive earlier.
As for how this would appear to an observer on the planet, it would depend on the length of the laser beams and the duration of time they exist. A short laser beam in motion moves too fast to be seen, so the laser would have to be large enough and exist for a long enough duration to be visible. If that was the case, there would still be no discernible difference unless the distances were incredibly great, because at short distances, the difference in speed between an object traveling near the speed of light and one at the speed of light is negligible. At large enough distances, the difference could be noticeable, but the observer would need some pretty impressive equipment to detect it.
I hope that cleared things up for you. If anyone sees any flaws in my logic, I’d appreciate the feedback!
Here’s my take. Assuming that there’s no delay when when the ship receives and transmits the signal, it’ll arrive at the same time. However, the key is, an observer on the planet will disagree with an observer on the ship when asked how much time elapsed between the sending and receiving of the signal.
You don’t even need to consider relativity if all you care about is what the observer on the planet sees. Think about it like this.
Let’s say there are 3 ships traveling towards the planet at .9 C, 1 light year apart (i.e. ship 1 is 100 light years away from the planet, ship 2 is 99 ly, ship 3 98 ly). Ship 1 fires the signals. The laser directed to the planet take 100 years to reach it. Now, since the laser directed at ship 2 is only going .1 C faster than the ship is, RELATIVE TO THE OBSERVER ON THE PLANET, it takes 10 years to catch up to ship 2, which has traveled 9 light years by this time, so that it’s 90 ly away from the planet. Ship 2 immediately fires a laser at ship 3, again taking 10 years to reach it. By this time, ship 3 has traveld 18 ly, and is 80 ly away from the planet. Thus the laser from ship 3 will arrive in another 80 years, making a grand total of 100 years from start to finish, exactly the same as the laser fired directly at the planet.
Aren’t both messages traveling the same distance at the same speed?
If you assume instantaneous retransmission, then yes. Any even slight real-world delay would mean that the laser arrives first.
Understood. I should have specified; I just meant that the velocity of the ships, whether or not a significant fraction of c, is not relevant.