First, it doesn't matter which frame you begin in. All of the kinematic laws are frame invariant. Your insistence on beginning in the ground frame complicates the math but doesn't change any results.2inquisitive said:When beginning in the ground frame, why would Hooke's law have anything to do with proper rest length spacing of the cleats? The cleats are not moving relative to the ground, thus they will have their proper length spacing. The total length of the track would be contracted by a factor of two at the bottom and seven at the top because it is moving wrt the ground, but the cleats, or distance markings are not.

Why does Hooke's law come in? For simplicity consider a relativistic cartwheel. Let's say that the spokes are made of a much stronger material than the rim and actively controlled so that the radius of the wheel remains constant at all RPM regardless of any forces generated in the rim. Let's further attach a bunch of 1mm measuring rods around the rim in 1mm increments. The rods will only be attached to the rim on one end of each rod, not on both ends. At rest there is no gap between rods. As we spin the wheel up to relativistic speeds the rods length contract by γ and thus there is a gap between one rod and the next indicating that the rim has become physically strained by γ with accompanying stresses according to Hooke's law. In other words, there are two effects here, the geometric effect of length contraction as governed by the Lorentz transform and the stretching effect of strain as governed by Hooke's law.

Now, as the wheel rolls along the ground let's have it make marks every time the attached part of a rod strikes the ground. At low speeds the marks are 1mm apart, but at relativistic speeds the distance between marks is increased by γ. In other words, although the geometric effect does not occur at the ground the strain effect does. Therefore, even though there is no length contraction for the bottom tread that does not imply that the distance between cleats on the bottom tread is equal to the proper rest distance. Objects can certainly be strained at rest.

-Dale