2inquisitive said:

Let me state my viewpoint again. DaleSpam gave a correct explaination in his post** if one begins the exercise in the TANK frame**. ... My scenario above, to which Pete responded, was from **beginning in the ground frame**, **not** the tank frame. Special Theory states there are no preferred frames, so I believe it is fine to begin from the ground frame, ... It is when one begins in the ground frame of reference when odd things happen, not when one begins in the tank frame and **transforms ** into the ground frame.

DaleSpam said:

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.

I finished the analysis starting from the ground frame. The most difficult part was figuring out a formula to use for the worldline of an arbitrary material point of the track in the ground frame. I finally found a reference on MathWorld that described the Fourier Series expansion of an asymmetric triangle wave. Using that I developed an analytical expression for the point's worldline in the ground frame:

s = {ct, vt + ∑(b sin[k(vt-x0)])}

where

b = 4 (-1)^n c^4 L sin[π n (c^2 + v^2)/(2c^2)]/(n^2 π^2 (c^4 - v^4))

k = n π/(L γ^2)

L is the length of the tank tread in the ground frame

v is the velocity of the tank in the ground frame

x0 is a term indicating the initial position and direction

and the summation is from n=1 to n=infinity

With this expression for s we can differentiate to get the four-velocity, u, and we can transform to the tank's rest frame to get s' and u'. We can select two different material points by choosing different values for x0, and we can calculate numerically the proper distance between these two points by following the same procedure as above.

For c=1, L=2, t=0, and v=.6 we get the following:

x0a=.5 and x0b=.6 are material points on the bottom of the tread, the plane of simultaneity from a intersects the worldline of b at t=0 in the ground frame, the two corresponding events are {0,.5} and {0,.6} and the proper distance is .1. Transforming to the tank's rest frame the plane of simultaneity from a intersects the worldline of b at t'=-.45, the two corresponding events are {-.375,.625} and {-.45,.75} and the proper distance is again .1.

x0c=2.5 and x0d=2.6 are material points on the top of the tread, the plane of simultaneity from c intersects the worldline of d at t=-.1875 in the ground frame, the two corresponding events are {0,.294} and {-.188,.082} and the proper distance is .1. Transforming to the tank's rest frame the plane of simultaneity from c intersects the worldline of d at t'=-.296, the two corresponding events are {-.221,.368} and {-.296,.243} and the proper distance is again .1.

In summary, both frames agree on the proper distance between two material points therefore both frames agree on the strain, both for material points on the top of the tread and for material points on the bottom. It doesn't matter which frame you begin in, the results are the same. Beginning in the ground frame complicates the math a lot, but does not change the results at all.

-Dale