Mustakafine said:
Some friends and I were out at a pub last night and this question came up.
If you were in space and you had two balls. You threw one ball in one direction at half the speed of light. You through the other ball in exactly the opposite direction then relative to each other they would be traveling the speed of light.
As the observer in the center I would be able to see light from both object but it would red shifted quite a bit. But what would an observer on each of the balls see of the opposite ball?
Does light propagate through space at the same speed regardless of the speed of it source relative to another object?
You're right about the light from the moving balls being redshifted. In fact, you can tell the speed of an object just from the amount of redshift. The faster an object travels, the more the redshift. Redshift is caused by the increasing distance between two objects moving away from each other and thus the increasing time it takes for signals to travel between them. At 0.5c the redshift is so great that the balls would become invisible. But let's say the balls and the launcher have some sophisticated electronics in them so that they can emit a radio pulse at, say, once per second, and they have receivers in them to detect these pulses from the other objects and measure how far apart they are in time. Repetitive pulses like these are the same effect as redshift, which is a reduction in the frequency of light caused by increasing relative motion, but it is easier for us to talk about the time intervals between pulses which is the reciprocal of the frequency. So a redshift in frequency is the same as a lengthening of time intervals.
So let's start at the beginning and assume the launcher emits a pulse at the exact moment that it launches the two balls in opposite directions at 0.5c, one half the speed of light. We'll call this "Pulse0". First, let's focus on how one of the balls will detect the pulses from the launcher. Where will it be one second later when Pulse1 is emitted? It will be half way between Pulse0 and the launcher, correct? What about when Pulse2 is emitted from the launcher? Now the ball will still be half way between Pulse0 and the launcher but Pulse1 will have reached the ball, correct? So what will the electronics on the ball measure as the time interval between the first two pulses emitted by the launcher? Well, if it weren't for relativity, it would measure two seconds, but we know that moving clocks tick slower than normal when they are traveling at high speeds. At 0.5c, the timing circuitry in the ball will take 1.1547 seconds per tick. Now if you have two clocks ticking at different rates, the slower one will measure time intervals as shorter than the other clock. As a result, the ball measures the time interval between the pulses coming in from the launcher at 2 seconds divided by 1.1547 which is 1.73205 seconds.
We could repeat the same analysis for the other ball but since the geometry is symmetrical we can say right off the bat that both balls make the same measurement.
Now, what about the launcher's measurement of the "redshift" from the two balls? Let's assume the first pulse was emitted at launch time. The second pulse will be emitted 1.1547 seconds later because the timing circuitry in the balls is running slow by that amount. How far away from the launcher will each ball be when this happens? That's easy, 0.5 times 1.1547 or 0.57735 light seconds away. How long will it take for this pulse to travel back to the launcher? That's easy too, 0.57735 seconds. That means the total time will be 1.1547 plus 0.57735 or 1.73205 seconds.
So we see that the balls both measure the same "redshift" for the pulses coming from the launcher as the launcher measures of the "redshift" coming from the two balls. This illustrates the symmetrical relationships that are characteristic of Special Relativity. Two objects in relative motion emitting the same signal will be measured by each other identically. Note that we have done our analysis on just the first two pulses but we would get the same results if we continued for any number of other pulses.
Now let's see what each ball measures of the other ball's pulses. We already know when and where each ball is when it emits its second pulse but now we have to figure out where the other ball will be when it receives that second pulse because this determines the amount of "redshift" that each ball sees of the other ball. That was your question.
First, we can realize that we have already solved part of the problem as we know how long that second pulse has taken to get from the first ball to the launcher, 1.73205 seconds and we know where the second ball is at that time traveling at half the speed of light, 0.866025 light seconds from the launcher. All we have to do is figure out how long it will take for the pulse traveling at c to catch up to the second ball traveling at 0.5c. That's really pretty easy to do because we know that the pulse is traveling twice as fast as the ball so in another 1.73205 seconds the ball will have traveled another 0.866025 light seconds.
So the total time for the second pulse from the first ball to reach the second ball is 1.73205 plus 1.73205 or 3.4641 seconds. Now how long will this second ball measure this time to be? Since its clock is running slow by 1.1547 it is 3.4641 divided by 1.1546 or 3 seconds (rounded off). If we take the reciprocal of this, we get the conventional factor for redshift of 0.33333.
So instead of the redshift being zero (what you get at the speed of light), which is what you and your friends in the pub thought, it is only a factor of 1/3.
Now this is probably too much to share with your friends but maybe what will satisfy them is for you to point out to them that when you are considering redshifts between a "stationary" observer and a "moving" observer, the time dilation factor is applied just once but between two observers moving in opposite directions it is applied twice so the speed of light is never reached.
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