Barn-Pole Paradox Calculator — Special Relativity Length Contraction

Formula

How this is calculated

The barn-pole paradox (also called the ladder paradox) is a thought experiment in special relativity. A long pole moves at relativistic speed v through a barn. In the barn's rest frame, the pole is Lorentz-contracted to L_pole = L₀_pole / γ — so if β = v/c is large enough, the contracted pole fits inside the barn simultaneously. But in the pole's rest frame, it is the barn that is contracted to L_barn = L₀_barn / γ, and the pole remains at its full rest length — apparently too long to fit.

Both descriptions are correct within their respective reference frames. The apparent paradox is resolved by the relativity of simultaneity: the two events — "front door closes on the pole's rear" and "rear door opens for the pole's front" — are simultaneous in the barn frame but are NOT simultaneous in the pole frame. In the pole frame, the rear door opens before the front door closes, so the pole is never fully enclosed. No physical contradiction exists; only the notion of "at the same time" differs between frames.

The simultaneity shift Δt = γ · β · L₀_barn / c gives the time difference in the pole frame between the two door events. For a 8 m barn at β = 0.8 (γ = 5/3), this shift is about 44 ns — too small to perceive but experimentally consistent with time dilation measured in particle accelerators and GPS corrections.