jbmolineux said:
Imagine a stream of photons moving from the sun to the Earth (A, B, C, D, E, F, G, etc.).
Now imagine 2 different men measuring the speed of the photons, and (somehow) recording which photon is hitting them. They are in the same position, but one is traveling 1/2 the speed of light (relative to the earth) toward the sun, and the other is traveling 1/2 the speed of light away from the sun. Say they are giving each other a high-five as they pass and doing the measurement at the same time that their hands slap.
Before we get into the details of your scenario, we have to be clear on how each man is going to measure the speed of a photon. If you're going to measure the speed of anything, you have to detect when that thing passes by a clock and then detect when it passes by a second clock some measured distance away. Unfortunately, we can't detect when a photon passes by unless we stop it and that messes up the measurement. Instead, let's consider, as russ watters did in post #2, a burst of photons so that all we have to do is detect a small percentage of them and let the rest pass by on their way to the second clock. This means a steady stream of photons coming from the sun is not going to do it. We have to block that stream and then briefly let some photons through and then block it again. Since you had the men giving high-fives to each other we can let their hands do this blocking process.
Well that's just one problem. A second problem is that they still can't do the measurement the way I described because it requires two clocks that have to be set to the same time and that is not a trivial task so for the time being, let's change the way they do the measurement from a one-way measurement to a two-way measurement. Now, instead of a clock at some measured distance away, let's put a mirror there and reflect the photons back to the first clock where the man is.
Here is a spacetime diagram showing the rest frame of a man located along the blue line with a clock ticking off 1 nsec intervals of time depicted by the dots and a mirror located 6 feet away depicted by the thick black line. Photons depicted by the thin red line coming from the sun off to the left pass by the man and his hands and his clock when it happens to read 0. Then it hits the mirror 6 nsecs later and reflects back to him when his clock reads 12 nsec:
Since the round-trip distance is 12 feet, he concludes that the speed of the photons is 1 foot per nsec.
Now we have to transform all the coordinates of the events in the above diagram to one in which one man is moving at 0.5c towards the sun:
We see that because of length contraction (the mirror is closer to him) and time dilation (his tick marks are farther apart) and relativity of simultaneity (events that were simultaneous in his rest frame are not in this frame), he still measures 12 nsecs as the time for the round-trip so he comes to the same conclusion that the photons traveled at 1 foot per nsec.
We can do another transformation for the second man moving at 0.5c away from the sun but we'll show him in red with a green mirror:
jbmolineux said:
The question I want to ask, is which photon in the stream are each of the measuring? For the guy who is moving toward the sun, the photon-stream is moving towards him at the speed of light (but 1.5 the speed of light relative to the earth, since he's moving 1/2 the speed of light toward the sun). But for the guy moving away from the sun, the photon stream is moving toward him also at the speed of light, but only 1/2 the speed of light relative to the earth.
If you look at the individual diagrams for each man, you can see that the time it takes for the photon to go in each direction has the 3x ratio (1.5/0.5 = 3) that you are considering but it is in the opposite order for the two men since they both are making round-trip measurements. Here is a spacetime diagram showing both of their measurements together:
jbmolineux said:
Therefore, aren't they going to be measuring two totally different photons to be in the same place at the same time as they pass each other? Won't the first guy be measuring a much earlier-in-the-stream photon, while the guy moving away measuring a later-in-the-stream photon? How is this explained?
Actually, they are measuring the same burst of photons, it's just that the photons reflect off the first man's mirror earlier than the second man's mirror.
Now I'm going to show you how the men can make one-way measurements with two clocks that have been synchronized. They have to synchronize their two clocks before they make their measurements and they do it using the same technique that they used for making the round-trip measurements. Look at this diagram:
At some time, the blue man sends a burst of photons to his second clock and then when he sees the reflection off the clock, he also sees what time is on the second clock. He averages the time on his first clock when he sent the burst with the time he sees the reflection and he adjusts the second clock so that he will see that time when he repeats the measurement. Eventually, he will get the two clocks synchronized as shown in the example above. The other man does the same thing.
Now we can see what happens when the two men pass as shown in this diagram:
And once again, they both measure the speed of the photons to be 1 foot per nsec even though the light takes three times longer for one man than the other.
