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You can exert a constant force on an object, but since F = ma and m increases as the object accelerates, the acceleration will decrease when constant speed is applied.
But ot maintain the constant acceleration we have to increase F (numerically, delta_F/delta_t = m_0 * a^2). Theoretically I don’t see why we can’t achieve constant acceleration this way, but in practise it will take too much force I guess. I would say that while Gracie is moving, both perspectives are correct (Gracie says George is younger and vice versa). But if Gracie decides to stop to compare her age with George, she will need to deccelerate, so she will no longer be moving with constant velocity. Which means that only George’s perspective can be correct.
However I like Abdulrahman’s idea that Lorentz transformation will not work as before. According to the Pole perspective, it does not change (as it is grabbed simultaneously). According to the Barn perspective, the pole gets stratched by factor gamma^2. Before the grab the length of the Pole from the Barn’s perspective was L_0 / gamma (due to usual Lorenz contraction).
Aterthe grab we get something like this. Let the Barn coordinates be unprimed and the Pole coordinates – primed. delta_t’ = 0, delta_x’= L_0, as the Pole is grabbed simultaneously from its perspective, and we use rest length for delta_x’. Using Lorenz transformation we get: delta_x = gamma * (delta_x’ + 0) = gamma * L_0 – that is the distance from Barn’s perspective.
So we get L_0 / gamma before the grab and L_0 * gamma after, therefore the factor is gamma^2, just as in the compression case. I’d say he skates across the gratings because from the perspective of a skateboard the gap is small enough. The observers on the sidewalk, however, will see something quite strange as if Bart was floating for some time over the gap, but they can figure out why he does not fall just as team barn figured out why team pole thinks that the pole does not fit into a barn.
No paradox here. There is only one dynamite. And it will or will not explode depending on its perspective. From its perspective (the pole perspective) it doesn’t fit. So it does not explode. And the observer at rest (the barn perspective) will not see the dynamite explode.
If there are two dynamites (at rest relative to the barn and moving with the pole), the first one will explode and the second one won’t. But it’s absolutely normal, so no paradox here either. From our perspective it is indeed Lorenz contracted, the clock is moving in a horizontal direction. So the trajectory of a light ball is not very straightforward. So it is not ticking symmetrically right-left as if the clock was stationary, but only now its length is shorter. If the clock is contracted, by doing some calculations we get that the tick-tock time is the same in the moving horizontal clock as in the moving vertical clock. Actually, we used that fact to derive that the clock is Lorenz contracted. But if we forget this method of derivation and use the fact that it is contracted (because we derived it using another method), we get the same tick-tock times for horizontal and vertical clocks. I see it this way.
No, the angles will be different (more acute for George). The mountain is moving from George’s perspective, therefore it is Lorenz contracted for him. The height of the mountain stays the same, but the length of the base of the mountain will be shorter for George, which results in a more acute angle – a steeper slope.