I'd like to expand upon the concepts of time dilation already covered in this node, and offer, what I think is, a more comprehensible proof of it that only involves the pythagorean theorem and the fact that c, the speed of light, is constant no matter the reference frame. I'll let the upvotes/downvotes be a judgement of the clarity of my explanation.
Time dilation is a result of Einstein's theory of special relativity, which was first published in 1905. It's a mathematical statement that relates the time elapsed in two different reference frames, moving at speeds relative to each other. The bestknown example of time dilation is the famous twin paradox. Time dilation itself can be simplified to a trite statement like, "Time slows down when you move faster," but this is not quite true. Really, it shows how time is NOT an absolute thing, and time measured in one frame is not necessarily the same as time measured in another frame. It shatters the idea of simultaneity and can lead to many situations that, to us, seem paradoxical (see: the twin paradox).
Why does this occur? Essentially, it is a result of the constancy of the speed of light as measured from ANY reference frame. The speed of any particular photon, a beam/wave of light, they measure the exact same speed, no matter how much the difference in speeds between the two reference frames. This is odd, for one, because you would expect that the motion of light is similar to that of cars: you can perhaps move at a speed relative to the speed of light, and light will appear to "slow down," but this is not the case. The result of this? Time slows down in a
moving reference frame. I'll show mathematically how this is so in a bit, and it only
requires the constancy of c and knowledge of the pythagorean theorem to prove.
The standard illustration of time dilation involves trains. The reason for this is because during the times
during which Einstein developed Special Relativity (the early 20th century), the the fastest
conceivable mode of transportation was, in fact, a train. People commuted on them and they became the ultimate symbol of speed, being the fastest land transport
around. This tradition has largely stuck among physicists, even though we have jets, airliners, and really fast cars.
Okay, here's the example: Dick is standing on a train platform. Jane is standing on a
train which moves past the platform at a velocity v; for the sake of argument, assume v is
large enough (greater than 10% the speed of light) for the effects of special relativity to be detected.
There is a clock on the train that both Dick and Jane can see, as well as a clock on the
platform. When the train was not moving, the clocks were calibrated and basically ticked
perfectly in unison. However, When the train is moving a paradox occurs: Dick observes the clock on
the platform and the clock on the train, and finds the clock on the train to be running slow
(note that the clock on the train is moving WITH the train at speed v). When 2 seconds
elapses for Dick, he sees less time elapsed elapsing for Jane simply by looking at the clock on the train. Jane observes the exact
same phenomenon in reverse: because Dick is moving relative to herself, she see the clock on
the platform to be slow while the clock with her on the train is perfectly fine. it's easy to see how this alone seems, to our feeble minds and our slowpaced lives, is a paradox. So who is right? Well, they both are, from their own reference frames. This discrepancy can be easily shown mathematically.
Okay. The set up for this proof is very similar to the one above. On the train are two
mirrors, and a photon bounces back and forth between these two mirrors. Let t be time
measured on Dick's clock for the light to go from the bottom top top mirror, and t' be time
measured on Jane's watch for the same phenomenon, who is moving at velocity v relative to
Dick. The mirrors on the train are oriented vertically and set a distance d apart. Here's
what the experiment looks like to Jane, showing the path of the beam during one bounce:

^


 d = c t'



In time t' (measured on Jane's watch) the light beam travels a distance d = c t'. This
is simply because distance = speed * time. Easy enough, right?
The next question is, What does Dick see? Well, consider this: the time Dick measures for
the the light beam to hit the top mirror from the bottom mirror is t, which is presumably a
longer amount of time than t'. However, he also notices that the mirrors themselves travel a
distance v t horizontally. Thus, the light beam also has this extra distance to travel. To
Dick, the light moves in a slanted path, forming the hypotenuse of a right triangle. The
length of this hypotenuse is c t, the speed at which light travels * the time of its motion
(according to Dick). Let's look at a diagram: (note my rather shitty attempt at a diagonal.
Just assume it's straight)
 
^
/
/
D= c t 
/

/

/ x = v t
 > 
Initial Final
position position
of of
mirror mirror
Notice that the light still travels at the speed of light, only now it travels on a
diagonal path. So Dick still measures c the same value as Jane would measure it on the
train (he finds distance traveled, divides by time on his watch, and it remains c). One
more thing to note: we can construct a right triangle from this. The vertical distance
traveled by light remains the same even though the train is moving. We have a
measure for this already: y = d = c t':
Diagonal line  path light travels according to Dick
vertical line  path light travels according to Jane
 
^
/
/ 
D= c t  
/  y = c t'
 
/ 
 
/ x = v t 
 > 
Initial Final
That looks like a right triangle to me. And it is! So now we can find a relationship
between t and t' using the pythagorean theorem.
(v t)^{2} + (c t')^{2} = (c t)^{2}
c^{2}t'^{2} = c^{2}t^{2}  v^{2}t^{2} = t^{2}(c^{2}  v^{2})
t'^{2} = t^{2}(c^{2}  v^{2})/c^{2} = t^{2}(1  v^{2}/c^{2})
t' = t √(1  v^{2}/c^{2})
Or, as it is normally written:
t = t' * 1 / √(1  v^{2}/c^{2}) = t'γ
Where γ is the Lorentz factor, equal to 1 / √(1  v^{2}/c^{2}). This is given its own symbol
because it appears so much in relativity.
So there you have it. A simple explanation of time dilation. Let's confirm the results: t should always be greater than t', except when v = 0. What happens as v approaches c? Well, the denominator of gamma becomes smaller and smaller, making gamma overall larger and larger. Thus, t increases as v increases, and is thus always greater than t' unless v = 0. So we got what we expected. Nifty, eh?
The implications of this idea are pretty amazing. Once scientists got used to the idea, it really made them question whether the idea of "simultaneity" is a valid one, and really had some interesting philosophical ideas. There is NO absolute reference frame, but the laws of physics hold true for ALL reference frames (that speed of light will always be c, &c.). There is no absolute marker for truth or true speed or anything, you can only designate things relative to each other. The fact that time isn't absolute was really an astounding idea.
But I digress. You can read about this in other articles, try special relativity. :)
Quod erat demonstrandum, biznatch.