Paradoxe de la grange

L₀ is the pole’s proper length and B is the barn’s proper length, each measured in its own rest frame. The relative speed is v = βc with c = 299,792,458 m/s exactly. Positive β means the pole moves right in the barn frame; negative β reverses the direction. At the common origin, x = ct = 0, the two centers coincide.

In the barn rest frame, its ends are at ±B/2 and the pole ends are βct ± L₀/(2γ). In the pole rest frame, its ends are at ±L₀/2 and the barn ends are −βct′ ± B/(2γ), where γ = 1/√(1 − β²). Lengths are measured between endpoints at the same time in the selected frame. Therefore, two frame-dependent containment statements need not describe the same pair of simultaneous events.

A and B always identify the left and right barn doors, not the leading and trailing ends. Their ideal signal events are fixed at (ct = 0, x = −B/2) and (ct = 0, x = +B/2) in the barn frame, whether or not the pole lies fully inside at that instant. These are event markers, not moving doors, forces, a collision model or a mechanism for trapping or stopping the pole.

The Lorentz transformation is x′ = γ(x − βct), ct′ = γ(ct − βx). Thus the door signals are simultaneous in the barn frame, but their pole-frame difference is Δct′ = ct′B − ct′A = −γβB. Negative β reverses their order; β = 0 is regular and both signals remain simultaneous. Their invariant separation is Δ(ct)² − Δx² = −B².

The diagram uses x and ct in meters with equal axis scale. Times in the table are nanoseconds: t(ns) = ct/c × 10⁹. Green solid lines are the pole endpoints; blue dashed lines are the barn endpoints. The dotted horizontal line selects one common time in the displayed frame. The containment result refers only to this current slice and includes endpoint equality within floating-point roundoff.

Run traverses the full displayed ct range in about five screen seconds; the motion is slowed for illustration. Step advances by one fortieth of the displayed half-range. The time-zero and A/B buttons pause at those coordinates. Changing frame stops playback and sets that frame’s clock to zero; it does not preserve a universal “now.” Input changes clear results. Reset retains inputs, and validated inputs and the selected frame are retained when navigating back.

Sources: Illinois PHYS419: consequences of special relativity and UNSW Einstein Light: Lorentz transformations.