The terms "length contraction" and "time dilation"
Is there a particular reason why we say length contraction but time dilation? A Lorentz transformation [tex]\Lambda=\gamma\begin{pmatrix}1 & v\\ v & 1\end{pmatrix}[/tex] takes [tex]\begin{pmatrix}1\\ 0\end{pmatrix}[/tex] to [tex]\gamma\begin{pmatrix}1\\ v\end{pmatrix}[/tex], which dilates the time coordinate by a factor of [itex]\gamma[/itex], but the same [itex]\Lambda[/itex] also takes [tex]\begin{pmatrix}0\\ 1\end{pmatrix}[/tex] to [tex]\gamma\begin{pmatrix}v\\ 1\end{pmatrix}[/tex], which dilates, not contracts, the position coordinate by a factor of [itex]\gamma[/itex]. Of course a dilation by k is a contraction by 1/k, but the terminology still sounds weird to me.
(The upper component of the 2x1 matrices is the time coordinate). 
Re: The terms "length contraction" and "time dilation"
If you have two events at the same position but different times in one frame S, then in a different frame S' moving relative to S, the time between them will be greater, so that's time dilation. It is likewise true that if you have two events at the same time but different positions in one frame S, then in a different frame S' moving relative to S, the distance between them would be greater as well. But the "length" in length contraction does not refer to the distance between a single pair of events which may or may not be simultaneous depending on your choice of framerather it involves looking at two worldlines representing the front and back of an inertial object, with each frame defining the object's "length" by looking at the distance between a pair of events on the two worldlines that are simultaneous in that frame (i.e. simultaneous measurements of the distance between the front and back of the object).

Re: The terms "length contraction" and "time dilation"
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[tex]dx'=dx/\gamma[/tex] [tex]dt'=dt\gamma[/tex] 
Re: The terms "length contraction" and "time dilation"
An object is moving relative to me. The time interval I measure is greater than the elapsed proper time for that object (dilated).
The length I measure for that object is less than its proper length (contracted). 
Re: The terms "length contraction" and "time dilation"
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Re: The terms "length contraction" and "time dilation"
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Re: The terms "length contraction" and "time dilation"
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[tex]dx'=\gamma(dxvdt)[/tex] [tex]dt'=\gamma(dtvdx/c^2)[/tex] To measure [tex]dx'[/tex] you need to mark the endpoints of the object in the comoving frame F' simultaneously, so, you need to make [tex]dt'=0[/tex]. This means [tex]dt=vdx/c^2[/tex]. Substitute back in the expression for [tex]dx'[/tex] and you get [tex]dx'=\frac{dx}{\gamma}[/tex] For time dilation, we want to get [tex]dt'[/tex] as a function of [tex]dt[/tex] when [tex]dx=0[/tex]. You get immediately [tex]dt'=\gamma dt[/tex] 
Re: The terms "length contraction" and "time dilation"
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Re: The terms "length contraction" and "time dilation"
Thanks guys. I think Jesse "wins" this thread, because #2 got me to draw a spacetime diagram and see the difference between what I did in #1 and length contraction. The calculation I ended up doing looked like this:
[tex]L=x_Avt_A=\gamma(vt_A'+x_A'vt_A'v^2x_A')=\gamma(1v^2)x_A'=\frac{L_0}{\gamma}[/tex] (Sorry, I'm too lazy to draw the diagram that really explains what I'm doing. The point is that length contraction is more than just a Lorentz transformation). I still think the terms "time dilation" and "length contraction" are pretty weird, because it sounds like length contraction would be to the x axis what time dilation is to the t axis, when in fact length contraction has an "extra feature" that time dilation doesn't have. But it's not the only thing in physics that has a misleading name, and I think I can live with it. 
Re: The terms "length contraction" and "time dilation"
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Re: The terms "length contraction" and "time dilation"
We had an epid thread about this last year! I think we more or less agreed that it can be a bit misleading when they're paired together in introductory texts as if contraction is the thing that space does, and dilation the analogous thing that time does.

Re: The terms "length contraction" and "time dilation"
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In this way you can have the symmetry x'=γx t'=γt 
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