We saw with time dilation that durations in time are not the same in different reference frames. For instance, we saw that the lifetime of an astronaut will never vary in their own frame, but will be observed to be longer from any observer not in the proper frame. The lifetime gets dilated (made longer) by a factor of [math]\gamma=\frac{1}{\sqrt{1-v^2/c^2}}[/math] which can get very large as v approaches the speed of light.[br][br]While all that is true and has been experimentally verified - although with particles and not astronauts - there is a necessary corrolary to time dilation, which we will see is length contraction. Before we go into the derivation, let me explain why length contraction must follow from time dilation.[br][br]Consider the muons that get created in the upper atmosphere and have a life time of [math]2.2\mu s[/math] in the proper frame. We saw that in earth's reference frame they are observed to live around [math]\gamma=150[/math] times longer since they travel at such a high speed. So in earth's frame they are seen to make it through 100km of atmosphere to the ground since they have 150 times longer than their proper life time to exist before decaying. [br][br]We must, however, ask what the situation looks like from the muon's frame. In its frame there are only [math]2.2\mu s[/math] allotted to make it to the ground. This would be an impossibility if there are 100km to travel since simple math (x=vt=ct) tells us that even at the speed of light it'd only make it 660m. [br][br]Recall that Einstein's first postulate of special relativity states that the laws of physics must give us the same description of nature (conclusions) from all inertial reference frames - one of which is the muon's frame. Here is where we find that since the laws of physics must give the same outcome (that the muons reach the ground) from all inertial frames (the muon's included), that the distance must be shorter to accommodate the shorter lifetime. How much shorter? Exactly as much shorter as the proper lifetime is shorter than the dilated lifetime, or 150 times. This shortening of distances and lengths is called [b]length contraction[/b].

[center][math][br]\Delta t_0=\frac{2x_0}{c} \\[br]\Delta t = \Delta t_{out}+\Delta t_{back} \\[br]\Delta t_{out}=\frac{x}{c(1-\tfrac{v}{c})} \\[br]\Delta t_{back}=\frac{x}{c(1+\tfrac{v}{c})} \\[br]\Delta t = \frac{x}{c(1-\tfrac{v}{c})}+\frac{x}{c(1+\tfrac{v}{c})} \\[br]\Delta t = \frac{x}{c}(\frac{1+v/c+1-v/c}{1+\tfrac{v^2}{c^2}})=\frac{2x}{c}\frac{1}{1-\tfrac{v^2}{c^2}} \\[br]\text{Relating these two durations of time by use of the time dilation equation gives: } \\[br]\frac{2x}{c}\frac{1}{1-\tfrac{v^2}{c^2}}=\frac{2x_0}{c}\frac{1}{\sqrt{1-\tfrac{v^2}{c^2}}} \\[br]x=x_0\sqrt{1-\tfrac{v^2}{c^2}}= \frac{x_0}{\gamma}.[br][/math][/center]