Layman said:
Peter, I'll try to answer this the best I can, but I'm not a mathematician.
Then you shouldn't be making claims about what the math of SR says or doesn't say.
Layman said:
According to Einstien (actually the Lorentz transformations) a moving clock runs slower than one that isn't moving, right?
This is the way it's often described, yes, but IMO it's a bad description, because it leaves out too much information.
Also, it's not a good idea to talk about some vague, unspecified scenario. Let's talk instead about a concrete scenario: the train and the embankment. In the Einstein example that I think you were referring to in the other thread, we had the following, IIRC:
(1) An observer at the center of the train, which is moving along a track parallel to the embankment.
(2) An observer standing on the embankment. The two observers pass each other (i.e., they are co-located, since we are ignoring all but one spatial dimension) at some particular instant.
(3) Two lightning strikes that hit particular points on the embankment at particular instants.
(4) Light from those two lightning strikes travels to the observer on the embankment; the light from both strikes reaches him at the same instant. This happens *after* the two observers pass each other.
So we have four events of interest: event A, the first lightning strike hitting the embankment; event B, the second lightning strike hitting the embankment; event O, the two observers passing each other; and event L, the light from the two lightning strikes reaching the observer on the embankment.
Mathematically, the way we model this is to first pick a frame of reference, and then assign each event of interest coordinates in that frame of reference. I'll do that below.
Layman said:
The LT tells you that, right? What the LT does NOT, and CAN NOT, tell you is which of two objects is relatively motionless. The math can't tell you that.
I'm not sure what this means. There will be some velocity ##v## in the math, which represents the relative velocity of two observers or objects. In our train-embankment scenario, it represents the relative velocity of the train and the embankment; and we would use that ##v## in the Lorentz transformation formulas to convert coordinates of events in the embankment frame to coordinates of events in the train frame. If you are saying that that process does not pick out either frame as "motionless", you are correct. If you're saying something else, you'll need to clarify what it is before I can respond to it.
Layman said:
either way, one clock will be slower than the other.
But which clock it will be depends on which frame you choose. More precisely, that's true as long as the relative motion between the two frames is the same, which is the case for our train-embankment scenario. In other scenarios, where that's not the case, there might be an invariant sense in which one clock runs slower than the other. It depends on the scenario.
Layman said:
If the guy on the train is moving, his clock will be slower. If the guy on the embankment is moving, then, per the LT, his clock will be slower.
This is just another way of saying what I said just now, that which clock runs slower depends on which frame you choose. Choosing a frame amounts to choosing which object you are going to assume to be "motionless" for purposes of your calculation. But the physics doesn't care which frame you choose; they're all equivalent. The only reason to choose one at all is to do a particular calculation. See below.
Layman said:
Which one is slower? The one which is moving. To me, that is a question of physics, not mathematics.
No, it's a question of properly understanding what "moving" means.
Let me work through the train-embankment example more explicitly, to illustrate what I said above. Suppose that we know, from actual measurements made by the observer on the embankment, that the coordinates of the four events of interest, in the embankment frame (i.e., the frame in which the embankment is motionless) are:
Event O: ##(t, x) = (0, 0)##
Event A: ##(t, x) = (0, -1)##
Event B: ##(t, x) = (0, 1)##
Event L: ##(t, x) = (1, 0)##
Note that I am using units in which the speed of light is 1; for example, distance could be in feet and time in nanoseconds, or distance in kilometers and time in light-kilometers (the time it takes light to travel 1 kilometer, or 1/300,000 of a second), or whatever. The specific units don't matter for this discussion; I'll just call them "units".
Note also that, as you can see from the above, 1 unit of time elapses in the embankment frame between event O and event L. Physically, this means that 1 unit of time elapses on the embankment observer's clock between the train observer passing him and the light from the two lightning strikes reaching him.
Now, if we know the velocity ##v## of the train relative to the embankment, we can use the Lorentz transformation to obtain the coordinates of these three events in the train frame (i.e., the frame in which the train is motionless). The transformation equations are:
$$
t' = \gamma \left( t - v x \right)
$$
$$
x' = \gamma \left( x - v t \right)
$$
where ##\gamma = 1 / \sqrt{ 1 - v^2 }##.
Applying these to the above coordinates gives:
Event O: ##(t', x') = (0, 0)##
Event A: ##(t', x') = (\gamma v, \gamma)##
Event B: ##(t', x') = (- \gamma v, \gamma)##
Event L: ##(t', x') = (\gamma, - \gamma v)##
To see how these numbers show that "moving clocks run slower", observe that, in the train frame, the observer on the embankment is moving, so the elapsed time by his clock (1 unit) between event O and event L is less than the coordinate time between those two events in the train frame (##\gamma## units). So, relative to the train, the embankment clock does indeed show "time dilation". (And notice how I reversed things, by showing how the *embankment* clock runs slow relative to the train frame, rather than how the train clock runs slow relative to the embankment frame? That was to illustrate that which clock is "moving" and runs slow is a matter of choice of frame; the physics is the same either way.)
But notice how much that description leaves out. First, it leaves out the fact that event L has a different *spatial location*, in the train frame, than event O (because in the train frame, the embankment is moving). Second, it leaves out relativity of simultaneity: notice that events A and B have *different* time coordinates, in the train frame, whereas they have the same time coordinate (they are simultaneous) in the embankment frame. (Physically, this is because we specified that light from the two lightning strikes reach the observer on the *embankment* at the same instant. If we had specified that light from the two lightning strikes reached the observer on the *train* at the same instant, we would have obtained different coordinates for events A and B.) To really understand what's going on, you have to include *all* the physics, not just time dilation.
