If an observer at the middle of a spaceship shines laser in both directions and the ship changes on contact with laser, what will earth observer see?

Yet another SR "paradox". A spaceship is moving w.r.t someone on earth. From the POV of a ship observer standing in the middle, he shines a laser beam in both directions. Assume that the part of the ship the laser touches explodes, and that the explosion imparts some force to the ship (normal to the surface at the point of explosion pointing towards the inside of the ship).

From the ship guy's perspective, both ends of the ship explode at the same time. So the same force is imparted at both ends of the ship in opposite directions - no acceleration. But from the earth guy's perspective, the rear end explodes first. So the ship will get some boost in one direction - hence some acceleration.

How can this apparent contradiction be resolved?

Following Will's comment, I tried to resolve it with a spacetime diagram but I still can't figure it out. Sorry if I sound dense but i'm new to the subject.

This is a spacetime diagram from the pov of earth guy. The 45 degree lines represent worldlines for the laser beams sent by rocket guy in both directions. The "mid" worldline is that of rocket guy. $$t_1$$ and $$t_2$$ are the times of the events corresponding to rear and front side explosions.

Initially I was thinking that earth guy could just take a photograph of the rocket between $$t_1$$ and $$t_2$$, in which the rear end of the rocket would be destroyed and the front end intact. Based on your comments, I also have to consider these events - "light from the rear explosion reaches earth guy", "light from the front explosion reaches earth guy".

Here's my pathetic attempt at doing that:

Obviously I'm doing something so obviously wrong that it's painful for people familiar with the subject to look at. So at this point I'd be grateful for pointers on how to resolve the contradiction, because even these two events of light reaching earth guy are not simultaneous.

• Isn't this kind of a variation of the ladder paradox? Aug 22, 2023 at 20:46
• "So the same force is imparted at both ends of the ship in opposite directions - no acceleration." No acceleration of what exactly? Each end of the ship feels a net force, so each end accelerates. Aug 22, 2023 at 20:47
• @KyleKanos It might be ladder paradox plus collision with barn door, since Born Rigidity might be required to handle th explosions.
– JEB
Aug 22, 2023 at 23:07
• Related meta question: physics.meta.stackexchange.com/q/14498/2451 Aug 23, 2023 at 6:14
• "This is not at all a homework question". I think it's actually a great homework question; the answer reveals itself as soon as you correctly draw and analyze the spacetime diagram. Aug 23, 2023 at 14:19

This is a case where the finite propagation of all effects, including the impulse from the explosions, must be taken into account. In the Earth frame, the rear explosion happens first, and a shock wave begins propagating forward through the spaceship. Note that the whole ship does not (and cannot) feel the effects of the explosion simultaneously: the force travels through the ship at the speed of sound in the spaceship's metal. Long before the whole ship feels the effect of the rear explosion, the front explosion happens. So even in the Earth's frame, there is never a case where the whole ship feels the effect of just one of the explosions.

Also note that in the Earth's frame the body of the spaceship is moving forward, so the explosion propagation is (relatively) slower in the forward direction from the rear explosion. So the two explosion waves will still meet in the center of the spaceship.

The black axes are the earth frame.

The two thick green broken lines are the worldlines of the ends of the ship.

The thinner green lines are the worldlines the ends of the ship would have followed if the explosions hadn't happened.

The blue lines are lines of simultaneity for the ship frame.

The two explosions happen simultaneously in the ship frame, hence both on one blue line.

The left-side explosion imparts some leftward velocity to the left side of the ship, so that its net velocity to the right (as seen from earth) is reduced. The right-side explosion imparts some rightward velocity to the right side of the ship so that its net velocity to the right (as seen from earth) is increased.

(These additions to velocity could shrink pretty quickly as the ends of the ship are pulled inward by neighboring parts of the ship that weren't pushed out; I haven't drawn any of that, because it's not relevant to the relativity question.)

The red lines are the shockwaves traveling from the explosions to the opposite ends of the ship. Depending on all sorts of details about how the ships are constructed, these will tend to carry forces that pull the ends of the ship inward. I've drawn this so that the velocities are returned to the original velocities and the ship is permanently expanded. More complicated things could happen, bringing the ship back closer to its original length.

To see what's happening in the earth frame, look along any horizontal line. There is a time when no explosions have happened yet, then a time at which only the first explosion has happened and the ship has stretched leftward, then a later time when it has also stretched rightward, etc.

Your attempt to include the light from the explosion reaching the earth guy seems to serve no purpose (unless I misunderstand your purpose). Usually in relativity, "the earth guy's perspective" refers to the earth guy's frame --- that is, the times and locations that he assigns to various events after he has sorted everything out, corrected for the time it takes signals to reach him, etc --- not to events where he receives those signals.