Collision and simultaneity in special relativity

Question

An almost rigid rectangle with length $D$ can rotate around its center. Assume that rectangle is horizontal and does not rotate before collision. According to observer $S$ two balls collide at the same time at two ends points of rectangle $R$ & $L$, in $t=0$ and before collision two balls have same speed $u$ in $y$ axis therefore there is no rotation for rectangle at all after clash. For these collisions we can assign two events in spacetime, that is $E_1=(-ct,r)=(0,-D/2,0,0)$ and $E_2=(0,+D/2,0,0)$. Note that $S$ is at rest relative to rectangle.

Now let's consider another observer $S'$ who moves at speed $-v$ relative to rectangle in $x$ axis. By using Lorentz transformation we can see that $E_1'=(-\gamma v D/2c,\gamma (-D/2+ vt),0,0)$ and $E_2'=(\gamma v D/2c,\gamma (+D/2+ vt),0,0)$. Because two events $E_1'$ and $E_2'$ are not simultaneous in this frame ($\Delta t'=\gamma vD/c^2$) it's logical to deduce that the rectangle will rotate after first collision. While it's not possible for obvious reasons (one can put a bomb under rectangle for example). Where am i mistaken? Does system wait for a signal that is coming from two end points to the center?

PS: I am fully aware that 100% rigid bodies are not possible in SR, by almost rigid i meant that we have an elastic collision and rectangle tends to rotates than bends around it's center.

Answer 1

It looks like you’ve almost answered your own question with your postscript. The rectangle is not rigid, so it deforms slightly all along it’s length to transmit the momentum of each collision. In the rectangle’s frame, these deformations propagate simultaneously from both ends to the center. In other reference frames the deformations occur in a different sequence, but still all occur and transmit momentum equivalently.

Perhaps of special interest is the event of the deformations reaching the center. This event is localized in both space and time, so it is a proper relativistic “event”, so the deformations will cancel simultaneously at the center in all frames (assuming they cancel rather than producing a standing wave in any frame, which is dubious but perhaps possible).