NCC Civilization Spaceship

The NCC Civilization spaceship, having the length of 221 m is approaching the star gate SG-7 (having the same length 221 m) at the speed $0.9c$. A paint bomb on the command room (which is the first room of the ship) is going to explode (and paint the commander with green paint) when the front of the spaceship will be at the far end of the star gate (event FF) . Fortunately, when the rear of the spaceship goes by the near end of the star gate (event RN) , a radio emitter present at the end of the  spaceship will send a signal to the command room to deactivate the paint bomb.

Spaceship view: a) What is the star gate length? b) Which event occurs first, FF or RN? c) Will the paint bomb ever explode?

Star gate view: d) What length does the spaceship have? e) Which event is first, FF or RN? f) Will the paint bomb ever explode?

If there is a difference between the two situations, please explain why. If not, please explain why.

The relativity says that for any standing observer the length of the object in motion ($L$) is contracted from the original zero speed length ($L_0$) so that

$L=L_0sqrt{1-(v/c)^2}=221sqrt{1-0.9^2}=96.33 m$

From the reference of a person situated on the star ship (star ship view) the star gate is moving towards the star ship with the speed $v$ and the star ship is standing. Therefore, the length of the star ship is $L_0=221 m$ and the length of the star gate is $L=96.33 m$. The event FF is first. The time difference between FF and NR events is

$Delta t=(L-L_0)/v=(221-96.33)/(0.9c)=0.462 mu s$.

In this situation the paint bomb will explode.

From the reference of a person standing next to the star gate (star gate view), the spaceship is moving towards the star gate with speed $v$ and the star gate is standing. Therefore the length, seen by standing person, of the spaceship will be $L_0=221 m$. The NR event will be first to happen. The time difference between FF and NR events is the same $Delta t =0.462 mu s$. In this situation the paint bomb will be deactivated.

The difference between the space ship view and the star gate view comes from the fact that all interactions have a finite speed of propagation. Thus the signal emitted from the end of the spaceship will need a time to travel towards the command room and deactivate the paint bomb. This time necessary for the signal to travel from the end of the spaceship to the command room is (light and radio waves have the same speed for all systems, and thus in the two situations they need to travel 221 m from spaceship end to command room):

$Delta t’=L_0/c=221/c=0.737 mu s$.

So that the command room will have passed by the front end of the star gate, when the signal emitted from the end of the star ship will arrive to deactivate the paint bomb. So the paint bomb will explode even in the reference system of the star gate view.