Jarle said:
That's a long and informative post, thanks. but I only understood parts of it, some things I don't understand, maybe you could explain?
Sure.
Jarle said:
"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 \gamma
is 1.25, so in my frame its length is 50/1.25 = 40 light seconds long."
What does that sign mean, and why does it get 1.25?
I don't know if you caught it, but I did explain a little what it meant in the first paragraph of that post:
But in relativity, each observer sees other observer's rulers and clocks as giving distorted readings, since each observer measures moving rulers squashed along their direction of motion by a factor of \gamma = 1/\sqrt{1 - v^2/c^2}, and moving clocks to be slowed-down by a factor of \gamma and also out-of-sync with one another.
To explain in a little more detail, that sign is the greek symbol "gamma", and as mentioned above it's equal to 1/\sqrt{1 - v^2/c^2}. It's given its own symbol because this factor appears in a number of equations in relativity, so it's useful to have a shorthand. For example, if an object is moving at speed v in my frame, and if its length in its direction of motion is L in its
own rest frame, then in my frame its length along this axis is given by l = L*\sqrt{1 - v^2/c^2} = L/\gamma. And if a clock is moving at speed v in my frame, and between two ticks it elapses a time of T in its own rest frame, then in my frame it elapses a time of t = T/\sqrt{1 - v^2/c^2} = T*\gamma.
Jarle said:
"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."
Also this I didn't understand. IS the ruler rotating? is it just moving forward?, and what does it read that is 0 seconds? And why is the clock 30 seconds behind yours?
This was something I had also given a quick explanation for earlier in the post when I said:
(it's important to note that different frames disagree on 'simultaneity' in relativity, so clocks which are synchronized in their own rest frame will appear out-of-sync in other frames--if the clocks are synchronized and have a separation of x in their own rest frame, then in another frame where they're moving at speed v along the axis between them, the back clock will be ahead of the front clock by a time of vx/c^2)
The 30-second time difference between the two clocks as seen in my frame is based on that equation vx/c^2 -- the two clocks are on either end of the ruler, and the ruler is 50 light-seconds (ls) long in its own rest frame, and in my frame it's moving at 0.6c, so if the clocks are synchronized in the ruler's frame then they must be out-of-sync by vx/c^2 = (0.6 ls/s)*(50 ls)/(1 ls/s)^2 = 30 s in my frame.
Again, the "relativity of simultaneity" is a basic feature of relativity--if two events happen simultaneously (at the same time-coordinate) in one frame, they will have happened at different times in all other frames. For example, if two clocks at different locations both tick 12:00 at the same time in the clocks' rest frame, so they're in sync in that frame, then in other frames they will have ticked 12:00 at different times, so they're out-of-sync.
The relativity of simultaneity can be understood as a consequence of the procedure for "synchronizing" clocks in relativity--according to what's called the "Einstein synchronization convention", clocks should be synchronized using light-signals, making the assumption that light travels at the same speed in all directions. If you make this assumption, then you can synchronize two clocks at rest in your frame by setting off a flash at the midpoint between them, and then setting them to read the same time at the moment the light from the flash reaches each one. But this procedure automatically leads to disagreements about simultaneity. Suppose I am on a ship which is moving forward at high speed in your frame, and I set off a flash at the midpoint of the ship to synchronize two clocks at the front and back. In
your frame, the back of the ship is moving towards the point where the flash was set off, while the front of the ship is moving away from that point, so if you assume light travels at the same speed in all directions in your own frame, then you should say that the light will catch up with the clock at the back before it catches up with the clock at the front. So if I set the clocks to both read the same time when the light catches up with them, you will see the back clock being ahead of the front clock.
Jarle said:
"since the clock at the front read -30 seconds, it now reads 50 seconds."
I didn't understand this, where did you get the fifty seconds from?
From the fact mentioned at the top of that paragraph that "80 seconds have passed on the clocks at the front and back of the moving ruler." The clock at the front read -30 seconds at the time the light was emitted at the back (in my frame), and the clock elapsed 80 seconds between that moment and the moment the light reached the front (again in my frame, although 100 seconds elapsed between these moments according to my own clocks), so when the light reached it the clock would read -30 + 80 = 50 seconds.
Jarle said:
"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"
Wouldn't you have to shoot the light beam at the front of the ruler for the rest of it to pass it?
You could just imagine a spherical flash sending light in all directions, but if you want to think of it in terms of a beam being sent in a particular direction, then yes, it would have to be aimed in the direction of the front. In my scenario my ruler and the moving ruler are moving in parallel, and the flash is set off at the point in space and time where the back of the moving ruler and the back of my ruler are lined up, so you can assume it's aimed in the direction of the fronts of both rulers. The whole problem can be assumed to be in one dimension, the other spatial directions aren't important here.
Jarle said:
Well, I have another question too, so this can make sense to me. If an object is moving 0.5c, then the time goes 0.5 of the normal?
No, as I said at the beginning of that post, "and moving clocks to be slowed-down by a factor of \gamma", and with v = 0.5c, \gamma would equal 1/\sqrt{1 - 0.5^2} = about 1.1547.
Jarle said:
And at the last bit, is it taken as a factor that you can not observe something before the light from it reaches your eyes? Or do you think of it as what we see is happening excactly there and then. (like: a sound we hear is not produced the excact same time we hear it, the sound-waves have to move through the air to reach our ears first)
The usefulness of having rulers with multiple synchronized clocks at different points along their length is that they allow you to assign coordinates to events using only
local measurements. For example, if I look through my telescope in 2006 and see a distant explosion, and I see that it happened next to a mark on my ruler which is 3 light-years from where I am, then I can look at the clock sitting on that mark and see that it read a date of 2003 at the moment the explosion happened, so that would be the time-coordinate I'd retroactively assign to the event, not the time that the light from the event actually reached me.
Jarle said:
Does it matter which DIRECTION we move to slow down time?
No, any clock moving at speed v will be slowed down by a factor of 1/\sqrt{1 - v^2/c^2} in my frame, regardless of its direction (although note that in that clock's own frame, it will be
my clock that's slowed down by this amount--time dilation is relative, there is no true answer to which clock is 'really' running slower as long as both are moving inertially at constant speed and direction).
Jarle said:
Sorry for all the questions, but this is kind of foggy to me, yet so disturbingly interesting.
No problem, and keep asking questions as they come up, it's the best way to learn about this stuff.
