My understanding of the spacetime metric is as follows: If Alice and Bob witness two light flashes $E1$ and $E2$, and Alice and Bob measures the distance between the position of the two light flashes as $\Delta x_A$ and $\Delta x_B$ respectively, and they also measure the time that elapsed between witnessing the two events happening as $\Delta t_A$ and $\Delta t_B$ respectively, then
$$(\Delta x_B)^2 – c^2(\Delta t_B)^2 = (\Delta x_A)^2 – c^2(\Delta t_A)^2.$$
I have been able to derive time dilation from nothing but this equation, as follows:
$$c^2(\Delta t_B)^2 -(\Delta x_B)^2 = c^2(\Delta t_A)^2 -(\Delta x_A)^2 $$
$$c^2(\Delta t_B)^2\left(1 – \frac{(\Delta x_B)^2}{c^2 (\Delta t_B)^2}\right) = c^2(\Delta t_A)^2\left(1 -\frac{(\Delta x_A)^2}{c^2\Delta t_A^2}\right) $$
$$(\Delta t_B)^2\left(1 – \frac{(\Delta x_B)^2}{c^2 (\Delta t_B)^2}\right) = (\Delta t_A)^2\left(1 -\frac{(\Delta x_A)^2}{c^2\Delta t_A^2}\right) $$
$$(\Delta t_B)^2\left(1 – \frac{v_B^2}{c^2}\right) = (\Delta t_A)^2\left(1 -\frac{v_A^2}{c^2}\right) $$
\begin{equation}
\Delta t_B\sqrt{1 – \frac{v_B^2}{c^2}} = \Delta t_A\sqrt{1 -\frac{v_A^2}{c^2}}
\end{equation}
Letting $\gamma_B = \frac{1}{\sqrt{1 – \frac{v_B^2}{c^2}}}$
we therefore find
\begin{equation}
\Delta t_B = \Delta t_A \frac{\gamma_B}{\gamma_A}
\end{equation}
In the case when the events are at the same position in Alice's reference frame, $\gamma_A$ becomes 1, and this equation is the familiar time dilation equation.
Here is where the real question begins: When I try to exploit this same line of reasoning for length contraction, I run into issues.
Lets assume that Alice observes these flashes at each end of a measuring stick, which is at rest with respect to Alice, both at the same time in her reference frame. Then $\Delta t_A = 0$, and we get
$$(\Delta x_B)^2 – c^2(\Delta t_B)^2 = (\Delta x_A)^2$$
We assume now that Alice (and, hence, the measuring stick) are moving at a non-0 velocity $v_B$ when observed by Bob. Now we do the same sort of manipulations we did in the previous case, factoring out a $\Delta x_B^2$
$$(\Delta x_B)^2\left(1 – \frac{c^2}{v_B^2}\right) = (\Delta x_A)^2$$
Immediately I start to see something has gone wrong, since $(1 – \frac{c^2}{v_B^2})$ must be negative (by what I have heard about physics, $v_B^2 < c^2$), thus one of the $\Delta x$s must be imaginary. Indeed if we continue
$$(\Delta x_B)^2 = (\Delta x_A)^2\frac{1}{(1 – \frac{c^2}{v_B^2}) }$$
$$(\Delta x_B)^2 = (\Delta x_A)^2\frac{v_B^2}{(v_B^2 -c^2) }$$
$$(\Delta x_B)^2 = (\Delta x_A)^2\frac{v_B^2}{-c^2(-\frac{v_B^2}{c^2} + 1) }$$
$$(\Delta x_B)^2 = -(\Delta x_A)^2\frac{v_B^2}{c^2}\gamma^2$$
But this implies that
$$\Delta x_B = i \Delta x_A\frac{v_B}{c}\gamma$$
That is, we have gotten an imaginary value for the observed length of the measuring stick by Bob! That doesn't make any sense!
I know there are other derivations of length contraction, but I am just not sure why this one isn't working. I have gone over the reasoning several times but I can't seem to find a flaw in it. Specifically, why should this work for time dilation but not for length contraction? Where did I go wrong?
Best Answer
What your analysis is doing is finding the spatial separation in Bob's frame of two simultaneous events in Alice's- ie of two events in Alice's frame that happen at different times in Bob's.
What you need to do instead is to compare the spatial separation of the two ends in Alice's frame with their separation at two simultaneous moments in Bob's frame. The way to do that is to consider the worldlines of the two ends of the object in Alice's frame (they will be parallel lines in the direction of her time axis) and find their spatial separation along a line of constant time in Bob's frame- that will give you the length of the object in Bob's frame.
The reason why length contraction occurs is that from Alice's perspective time Bob is noting the positions of the two ends of the object at two separate times- specifically, he's measuring the position of the leading edge of the object earlier than the position of the trailing edge, which allows the trailing edge a bit of time to move forward in the intervening period, thus giving a shortened result for the length.
You can see how it happens of you consider two people a distance apart on a platform, who are trying to measure the length of a passing train. They decide to do it by one of them noting the position of the front of the train as it passes and the other noting the position of the rear, then measuring the distance along the platform between the two positions they have noted. Clearly that only works if they note the position of the two ends at exactly the same time- if they note the positions at two different times, the train will have moved between the two measurements and they will get the wrong result. That's in effect what happens with length contraction.