If the the length of the pole is (L) from Team Pole’s perspective, it will remain (L) from their perspective after Team Barn simultaneously grabs hold out of it. Team Barn, however, measures the pole’s length to be (L/gamma) before their members grab it, and (L*gamma) after they grab it.

Explanation: Let us think about the two members of Team Barn who grab the front and the rear of the pole. Since the length of the pole is (L) from the perspective of Team Pole, then the distance between these two members from Team Pole’s perspective must be (L) as well to be able to grab the pole simultaneously. This distance between these two members will not change from Team’s Pole perspective after grabbing the pole. That is why the length of the pole will remain (L) from the perspective of Team Pole after it is grabbed.

Team Barn’s Perspective of this scenario is quite different. The two members of their Team are a distance (L.gamma) apart, but the pole’s length is (L/gamma). One of these members holds the rear of the pole’s first (say at 7:00 PM on the rear clock), then the front of the pole continues to move until the other member holds it (at 7:00 PM on the front clock). That indeed makes sense because the clock at the pole’s rear is ahead of that at its front from Team Barn’s perspective.

Mathematical Details:

A) Before the pole is grabbed:

– Team Pole’s perspective: pole’s length is (L).

– Team Barn’s perspective: pole moves and Lorentz contracted. Pole’s length is (L/gamma).

B) After the pole is grabbed:

– Team Pole’s perspective: pole’s length is (L). The pole is stretched by a factor of gamma, but they measure its Lorentz contracted length so the gamma factors cancel.

– Team Barn’s perspective: the pole is stretched. Pole’s length becomes equal to the sum of the initial length before grabbing and the stretched length (L/gamma + v*vL*gamma/c^2) = L(1/gamma + (v/c)^2 gamma) = L.gamma.