Time Dilation is a physical effect that is a direct consequence of
Einstein's Special Theory of Relativity. It is a result of the fact that the
speed of light is the same in all
reference frames. To see this, we will need a small amount of
algebra and a little bit of
ASCII art, but for the most part this is a
conceptual argument.
First, ask yourself this: How do we tell time? How can we be sure we have a system for telling time which makes sense when converting from one reference frame to another? Since Einstein tells us that the speed of light is the same no matter what reference frame we are in, it seems like this would be a good candidate to ensure our system for measuring time was proper. Let's construct a clock, then, which measures time by determining how long it takes a beam of light can travel some fixed distance. The precision in our measurement of time will then be exactly equivalent to our precision in our measurement of distance. If our measurement of distance is invariant from one reference frame to another*, then our measurement of time should be accurate no matter what reference frame we are in.
So now we have a clock, which looks something like this:
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What you're looking at is a pair of
mirrors reflecting a beam of light up and down between them. For each one-way trip, the beam of light ticks off an amount of time equal to:
Δt0 = d/c
Where d is the distance between mirrors, and c is the speed of light.
Great clock, huh? Well, now let's see what this clock tells us when we look at it in a moving reference frame. If we put the clock on a train moving along the ground at a velocity v, the picture we see from the ground looks something like this:
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The length of the trajectory has increased, and since the speed of light is
invariant, the travel time has also increased! The length of time between events changes when in a moving reference frame. An experimenter on the train will measure the same Δt
0 as before, but an observer on the ground will calculate a different Δt, which we can determine by using the
pythagorean theorem:
(length of travel distance)2 = (height)2 + (amount it's moved in time t)2
(cΔt)2 = d2 +(vΔt)2
Δt2(c2-v2) = d2
Δt = d/√(c2-v2)
Now we use our equation above for the time Δt0 the observer on the train measures:
Δt = cΔt0/√(c2-v2) = Δt0/√(1-v2/c2)
Thus, the clock on the train ticks more slowly than one on the ground. The exact ratio has been calculated.
Immediately, a paradox presents itself. Einstein says that all inertial reference frames are "equally right"; there is no universal stationary frame. From the perspective of the observer on the train, the train is stationary while the guy on the ground is moving along with velocity equal to -v. Thus, the clock that the grounded observer is carrying should be ticking slowly from the perspective of the observer on the train! Each observer accurately calculates that the other's clock is ticking more slowly than his own clock! How is this possible?
The answer has to do with the relativity of simultaneity. Read that node if you haven't yet, because I'm not going to explain any more than I have to in one node. There are two main points that should be understood:
(a) Simultaneity of events is relative, depending on your reference frame, and
(b) The time-ordering of events can be relative, depending on your reference frame.
Given this, it should be no surprise that the question of whether two clocks "agree" or which is "faster" or "slower" should depend on our reference frame. To ask which clock reached 12:00 first is a question of the ordering of events in time, which we see is a relative question when the events are at different locations in space. "Comparing clocks" doesn't make any real sense unless the clocks are brought together to the same point in space. For further discussion of this issue, see twin paradox.
In addition to time dilation and the relativity of simultaneity, Special Relativity also predicts an effect known as length contraction.
*It isn't, but that doesn't matter for this argument, the reason being that the light-ray trajectory we are going to set up is perpendicular to the velocity of the other reference frame.