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I've found a derivation of the formula for the relativistic momentum where they considered a car crashing into a wall in the system of the car and in an inertial system that moves parallel to the wall (and therefore perpendicularly to the movement of the car). They argue that since both observers see the same damage done to wall and car, the component of the car's momentum towards the wall needs to be the same in both systems. Since the car moves slower seen from the parallel moving system (because of time dilation), the Lorentz factor needs to be introduced in the formula for relativistic momentum.

I don't really find this reasoning (same damage implies same component of momentum towards the wall) satisfactory. At this point, everything known about collisions is non-relativistic. Is there a better argument why a component of momentum should be invariant for Lorentz boosts perpendicular to that component?