EngineeredVision said:
Ok, so I understand why clocks would appear to operate at different speeds between an observer and traveler in different inertial frames, but what actually causes the difference in the so-called simultaneity? Does the actual curvature of space-time change as something approaches c?
The idea of different observers having different definitions of simultaneity is part of special relativity, so it doesn't have anything to do with curved spacetime, which only appears in general relativity. The relativity of simultaneity shouldn't necessarily be thought of as a "physical effect" at all, it's just a consequence of how different observers define their coordinate systems, although the choice of coordinate systems used in SR is the most "natural" one to use physically because it insures that the laws of physics will obey the same equations in each observer's coordinate system.
When Einstein originally came up with relativity, he imagined that each observer would define the coordinates of events using purely local measurements, to avoid the issue of light-speed delays. Imagine I have a large grid of rulers that reach throughout space and are at rest with respect to me, with clocks attached to each marking on the ruler. Then if I see an event such as an explosion through my telescope, I can just look at the marking on the ruler that was right next to the explosion when it happened, and look at the reading on the clock that was attached to that marking at the moment the explosion happened, and this will give me the space and time coordinates I assign to the event.
For this to work though, I have to make sure that clocks at different locations along the ruler are "synchronized" with each other in some sense. Because of time dilation, I can't just synchronize them by taking them to the same point in space, making sure they read the same time, and then moving them apart--the act of moving them will cause them to change speeds. So Einstein suggested that each observer could synchronize different clocks in his system using the
assumption that Maxwell's laws of electromagnetism are valid in his coordinate system, which means that light must travel at c in all directions in his system. So if you make this assumption, you can synchronize clocks using light signals--just set off a flash at the midpoint of two clocks, and if they both read the same time at the moment the light from the flash reaches them, they are defined to be "synchronized".
Now, there's nothing that says you
have to synchronize your clocks using Einstein's synchronization convention. But if you do, you find an elegant result--if different inertial observers all synchronize their own set of clocks using this convention, then the all laws of physics will obey the same equations regardless of which observer's coordinates you use to express them. This is because all the known fundamental laws of physics have a mathematical property called "Lorentz-symmetry", which insures they will have the same form in different coordinate systems which are related by the Lorentz transformation--if we ever turned up a fundamental law that was not Lorentz-symmetric, it would no longer be true that all the fundamental laws obey the same equations in all coordinate systems constructed this way. But most physicists would consider this unlikely, it would seem a strange coincidence that all the most fundamental laws have happened to have this symmetry if it wasn't a symmetry that was built into the most fundamental laws out there waiting for us to discover (from which all our current 'fundamental' laws can presumably be derived). The fact that all fundamental laws of physics are Lorentz-symmetric also insures that the rate of actual physical clocks must match the rate that coordinate time passes in the clock's own rest frame--if there were some frames where physical clocks ticked at the same rate as coordinate time and others where they didn't, then clearly the laws of physics cannot be obeying precisely the same equations in both coordinate systems. To put it another way, if you were to create a computer simulation of a universe with different laws of physics than our own, as long as the equations you programmed in had the mathematical property of Lorentz-invariance, this would be enough to absolutely guarantee that you'd see relativistic phenomena like the twin paradox within your simulated universe--you wouldn't need to program in any extra "laws of relativity" or anything, in fact if you just programmed in the equations without noticing they had this particular mathematical property, you might be surprised by such phenomena (consider the fact that Maxwell's laws of electromagnetism are Lorentz-symmetric, even though they were discovered well before relativity was understood, and people were certainly surprised by things like the fact that light seemed to have the same speed in all directions regardless of which direction the Earth was moving).
So Einstein's synchronization convention is the most physically natural way to set up the coordinate systems of different inertial observers. But as a consequence of this convention, different coordinate systems will disagree about simultaneity. Imagine I am traveling past you on a rocket, and at the midpoint of the rocket I set off a flash, and set clocks on either end of the rocket to read the same time when the light from the flash reaches them, so that they are synchronized in my rest frame. But if you assume that light travels at constant velocity in all directions in
your rest frame, then in your coordinate system the light from the flash cannot have hit both clocks at the same time--after all, the back of the rocket is traveling towards the point where the flash was set off while the front of the rocket is traveling away from that point, so the light should take longer to catch up with the front than with the back. So, if I have set the clocks so that they read the same time when the light hits them, in your frame my clocks must be out-of-sync. That's pretty much all there is to it.
EngineeredVision said:
It makes sense that a stationary observer would observe that the clock of a fast traveling vessel is ticking slower because of the delay between ticks that is the result of the amount of time light takes to travel between observers.
That would just be the classical Doppler effect (The relativistic Doppler effect follows a different equation, because it is a consequence of
both the fact that the distance to the source changes between signals, so the signals have further to travel,
and the fact that the source is genuinely emitting signals slower in your frame). Time dilation can be seen even if you make purely local measurements to avoid the issue of signal delays, as on the scheme using a grid of rulers and synchronized clocks that I described above. I provided some diagrams of a specific example of two rulers sliding alongside each other with clocks attached to the markers on each on on
this thread, if you're interested.
