skycastlefish said:
Next is the part I don't understand. Imagine the same spaceship scenario as above only this time, in addition to the two light beams fired by the ships, a light beam is also fired at the space ships from directly ahead and in the line of travel. So now the moving ship is chasing a light beam and moving to meet a light beam that is also racing toward him. How can he not measure the light racing toward him as faster than the one he's chasing?
Precisely because he synchronizes his clocks differently--that's what "relativity of simultaneity" means! The
Einstein clock synchronization convention is that each observer defines what it means for two clocks at different locations to be "synchronized" using the
assumption that light travels at the same speed in all directions relative to themselves. One way of putting this is that each observer says two clocks at rest relative to them are "synchronized" if, when they set off a flash at the exact midpoint of the two clocks, each clock will have the same reading at the moment the light from the flash strikes that clock. But consider what that would mean if the procedure was carried out by an observer in motion relative to you--say, an observer on a ship which is moving forward at relativistic speed in your frame, who is trying to "synchronize" two clocks at either end of the ship. If he follows the Einstein synchronization convention, he will set off a flash at the midpoint of the ship, and each clock will be set to the same time, like 11 AM, at the moment the light from the flash reaches them. Since the light was set off at the midpoint of the ship, in the ship's rest frame the light naturally had the same distance to travel from the midpoint to either end, so setting the clocks to read the same time when the light reached them guaranteed that
by definition the light must have traveled at the same speed in each direction (since for each of the two light beams traveling to the two clocks, speed = distance traveled/[time light reached the clock - time light was emitted from flash], and the flash happened at a single point in spacetime so naturally the time it was emitted would be the same for both beams).
But now think about how this would look in your frame where the ship is moving. If you have synchronized all your clocks in such a way as to guarantee that both beams travel at the same speed in
your frame, then naturally since the back clock is moving towards the point where the flash was set off while the front clock is moving away from that point, according to
your measurements the light must reach the back clock at an earlier time than it reaches the front clock (you could measure this by having a set of synchronized clocks at rest in your frame, and you could note the time on whichever one of your clocks happened to be right next to the back of the ship at the moment the light reached it, and likewise note the time on whichever one of your clocks happened to be right next to the front of the ship at the moment the light reached it). So from your point of view, the guy on the ship has set his two clocks in a way that makes them
out-of-sync, by just the right amount that he gets the "illusory" result ('illusory' from your point of view) that the light beams both took the same amount of time to reach either end of his ship, when "in fact" (again from your point of view), the light traveling towards the back took less time. Of course the guy on the ship would say that it's your clocks that are out-of-sync, and your claims about the times are the "illusory" ones! And since the laws of physics work exactly the same in both your two coordinate systems (whose time coordinates are defined by clocks at rest in each frame which have been 'synchronized' in this way), there's no objective physical basis for saying one of your definitions of simultaneity and time is "right" while the other is "wrong".
For a numerical example of how it all works out, here's something I came up with on an older thread:
Say there's a ruler that's 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor (which determines the amount of length contraction and time dilation) is 1.25, so in my frame its length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by vx/c^2 = (0.6c)(50 light-seconds)/c^2 = 30 seconds.
Now, when the back end of the moving ruler is lined up with the 0-light-seconds mark of my own ruler (with my own ruler at rest relative to me), I set up a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler.
Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.
