# Strange behavior from non-zero origin in relativity

This is from Purcell's E&M book, appendix G.

I take the photo in (a) as a given. In frame F, the letter E is stationary, and in frame F', the letter L is stationary. This assumes that $$\beta = \sqrt3/2$$, $$\gamma = 2$$, and that at the start in both frames ($$t=0$$ and $$t'=0$$), both letters E and L are lined up with their left sides in the same spot. In other words, the ICs are $$x(t=0) = 1$$, $$x'(t=0)=2$$

I wanted to clarify how Purcell went from (a) to determine the coordinates in (b). I assume you can argue there's time dilation, so the origin moves 4 units to the right in (a) and $$4/\gamma = 2$$ units to the left in (b). Is this right?

Assuming this is right, is there a deeper meaning to how events not defined at the origin have strange behavior like this?

Edit: For instance, if the E and L were both located in their respective frames at x=x’=0, the times would be equal, right?

In case (a), we know that because of length contraction, the position of the origin on $$F'$$ in $$F$$ is $$x = 0$$, since it is displaced by $$-2$$ from the left edge of the $$L$$ in $$F'$$, which corresponds to a displacement of $$-1$$ in $$F$$. Then, because we know the origin of $$F'$$ ends up at $$x = 4$$, $$t = \frac{4}{\beta} = 4.62$$ ns.
In case (b), we know that the position of the origin of $$F$$ in $$F'$$ is $$x' = 1.5$$, since it is displaced by $$-1$$ from the left edge of the $$E$$ in $$F$$, which corresponds to a displacement of $$-0.5$$ in $$F'$$. Then, because we know the origin ends up at $$x' = -2$$, $$t' = \frac{2 - (-1.5)}{\beta} = 4.04$$ ns.
• Just to clarify, are the diagrams (a) and (b) unrelated? For instance, the time corresponding t' for diagram (a) would be $t/\gamma$, and not 4.04 ns, right? Jul 20, 2021 at 21:19