n8trix said:
Thanks for replying, everyone.
I think what you're saying may be slowly sinking in. Bear in mind I just started reading about the Lorentz transformation and I have the mathematical prowess of a sixth-grader.
1) What is the synchronization convention assumed in the Lorentz transformation?
The physical basis for the coordinate systems used in SR is the idea that each observer has a set of rulers at rest relative to himself with a bunch of clocks attached at different locations along the rulers, and that to assign space and time coordinates to an event, he looks at local readings on the ruler and clock that were right next to the event when it happened. This leaves the problem of how to synchronize clocks at different locations, though--you can't just bring them together, synchronize them in a common location, and then move them apart, since the clocks will slow down when you move them. So Einstein suggested in his 1905 paper that a natural way to synchronize clocks would be using light-signals, with each observer making the
assumption that light travels at the same speed in all directions in his frame. For example, you could set off a flash at the midpoint of two clocks, and then set them both to read the same time at the moment the light from the flash reached them. But if you think about this a little, you'll see that if each observer synchronizes his own clocks in this way, it will necessarily lead to each observer seeing every other observer's clocks as being out-of-sync. For example, imagine I'm in a rocket traveling by you at high velocity, and I synchronize two onboard clocks at the front and back of the rocket by setting off a flash in the middle of the rocket and setting them to read the same time when the light hits them. In your frame, though, the back of the rocket is moving towards the point where the flash was set off while the front of the rocket is moving away from that point, so if you assume those two light beams move at the same speed in your frame, then in your frame the light will catch up with the back clock before it catches up with the front clock, so if both clocks read the same time when the light hits them this means the back clock must be ahead of the front clock in your frame.
n8trix said:
2)
Can you (or somebody) elaborate on this?
Sure, an alternate synchronization convention would be for one special observer to synchronize his clocks using Einstein's procedure, but then for every other observer to set his clocks in such a way that they will still be seen as synchronized in this special observer's coordinate system (this will require them to know their velocity relative to the special observer). If the special observer uses coordinates x,t and some other observer moving at velocity v in his frame uses coordinates x',t', then this synchronization convention will result in the following coordinate transformation:
t' = t / \gamma
x' = \gamma (x - vt)
where \gamma = 1/\sqrt{1 - v^2/c^2}
Compare with the Lorentz transformation used when you use Einstein's synchronization convention:
t' = \gamma (t - vx/c^2)
x' = \gamma (x - vt)
That factor of vx / c^2 in the parentheses of the time transformation insures that clocks at different locations along the x-axis which are in sync in the first frame will not be in sync in the second.
n8trix said:
BTW, for anyone interested in enlightening me, the original question was: How is the conventionality of the one-way speed of light related to the isotropy of space, especially in connection with symmetry and conservation laws? Are these connections surprising or interesting? Can anyone help me here? Thanks!
I think pervect addressed this above, if you use an alternate synchronization convention the speed of light will be different in different directions, but also two objects with the same mass and equal and opposite momentums will
not have equal and opposite velocities, so the relation between the momentum of the speed of an object with a given mass will depend on its direction, introducing an anisotropy in space.
