That I shall do.
Before we get there, one thing about the clock experiment. That is not the sole proof. Rather, we also have sub-atomic particles that are detected far lower than they should (example are muons). They should decay very quickly, and thus not reach the ground in the numbers they do. However when we take into account the fact they experience time more slowly than we do, the numbers add up correctly.
Anyways, onto the step by step process (and I did a search for your original thread so I know what to say here, and I am guessing it was the one called What is the time?).
First, we have to get out of the concept of an absolute sense of motion. Let us say you are the ONLY object in space. You are moving at a constant velocity. Is there anyway to determine your velocity? No. There is nothing with which to measure your frame of reference, nor anyway to tell if you are actually moving because there is no acceleration, and so no force.
Now, Let us say there are two people, Bill and Nicole. They are floating in space with constant velocity. Who is moving? From Bill's perspective since he feels no force, he can claim Nicole is the one moving. But Nicole can say the same about Bill. There is no way to determine who is actually moving. They both could, or only one could. IF however, we toss in a third observer, they could determine who is moving where, but then again, the same problem exists only with three people (is the observer moving?).
Now that we have defined an important concept. Velocity is relative. Both Bill and Nicole are equally valid in claiming the other is moving.
Now let us move to the next part. Light. Light travels at a constant velocity as per the Maxwell equations on electromagnetism. But a mathematical result is not enough. This was verified in the famous experiment to detect an ether by its effects that failed to produce them (and subsequent effects). Light seems to be constant no matter which direction and how fast we move. Einstein as a young lad of 16 is supposed to have asked himself what a lightwave would look like if one could catch up with it. He eventually figured out he couldn't but it set in motion his thoughts.
According to the above mentioned items, the speed of light is always the same. No matter who is moving how or when every observer will agree the speed of light is 299,792,458 meters per second. Thus both Bill and Nicole will agree light travels at c. What does this mean?
Well for one it means length must contract in the direction of motion, a property called Lorentz contraction, but the more important one you are questioning is about time.
The other consequence is that time slows down as you move faster and faster. A simple way to illustrate this is to use something called a light clock. Simply put, you have two mirrors facing each other with a photon bouncing between them. Each tick we can say represents for the sake of simplicity, one second (a tick being defined as making a cycle from one mirror to the other and back again). We are sitting at rest in our setting observing a clock also at rest. Next, we have a second light clock slide past us. As we observe it the path the photon has to take is diagonal since the whole system moves as the photon is in transit. However, as was stated earlier, in the Bill/Nicole part, the clock has equal footing to say it is at rest, and so it will still seem as though it is at rest and merely traveling up and down. We measure it to have taken longer than the other clock (say 2 seconds), but it measures itself to have taken 1 second to complete it, and that we are the ones moving more slowly. Why can it say we are moving slowly? Because at the velocity it would have to travel to attain a time dilation such as that, it would take that long for light from us, and thus information about us, to catch up with it and tell it we are doing that. Hence it is because of the relativity of observers in motion we detect time to pass more slowly for objects in motion because each has a valid claim that they are at rest (remember SR deals ONLY with constant velocity).
Now, what back to the effect on length. Say Nicole is traveling in a spaceship. Before taking off on a test course that is 2 million meters long, Bill measures the spaceship to be 10 meters long. Nicole enters the track and for the entire duration that she is on the 'track' she travels at a constant velocity. Bill has sophisticated equipment he can use to also measure the length of the spacecraft . He starts a timer at the instant the front of the ship passes a point and stops it when the tail passes the same point. Since this is all constant velocity, he can use simple d=vt. BUT using the new knowledge we have from the time bit, we know that Nicole can claim she is at rest and that Bill is moving. And since she observes Bill to be moving however fast she was, she will note his clock running slow, and thus his measurement of her craft using the slower time rate will result in a shorter length. He has a watch that will only elapse a shorter time in other words. Hence the spacecraft will have contracted.
So what does this all have to do with spacetime? Well let us use this as an example. Say at the start it always takes Nicole exactly 10 seconds to traverse the 2 million meters. However, she starts to run the distance at a slant. Bill measures a longer time for her to have completed the run. At first he might think maybe the ship was no longer traveling at the pre-determined velocity as always. But then Nicole tells him she traveled at a slight diagonal. We can then clearly see what has happened. If before she was traveling solely in the x direction, then her velocity vector was totally in that direction and hence it took 10 seconds. But as she went in a diagonal in the y direction, some of her constant velocity now had to be split. Since only the x direction would wind up at the end of the track, she has less velocity in that direction, but the magnitude of the resulting vector will be her pre-determined velocity (simple s2 = x2+y2). The same thing happens in spacetime as I stated above. If we are at perfect rest in space, then our velocity in time will be the most it can be, c. The most we can be in space is c, and perfectly at rest in time. But remember, they are not seperate, but rather united in spacetime. Thus our spacetime velocity is always c. To put it more mathematically, s2 = x2+y2+z2-c2t2. The -c2t2 part is a conversion of the temporal dimension to a spatial one so that it can be added, and it is negative due to the treatment of time as having an imaginary parameter, since it is not exactly the same as a spatial dimension. One can see that as the sum of the spatial velocity vectors' magnitudes approaches c2 (thus an actual velocity in spacetime of c) the temporal one, t, must become zero. This can be stated more effectively in calculus with rates of change.
ds2 = dx2+dy2+dz2-c2dt2
We know that the first derivative of position vectors is the velocity vector and hence we see the change in spacetime is equal to the sum of the velocities of space and time. As dx, dy, and dz increase, dt must decrease and vice versa.
Hope that helps, and any other PF people may expand or clairify points as needed. Also thanks to Brian Greene for some assistance in some of the explination via [bold]The Elegant Universe[/bold].
