Barn–pole paradox

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.

Change the lengths or β to update directly. Choose the frame, then type an exact time in ns or use the slider. Its range is the displayed time window; it does not round typed values. The A/B buttons pause at those fixed door events, and t=0 returns to the center alignment. The highlighted time slice shows what is simultaneous in this frame. Changing frame pauses and selects its time zero; it does not preserve a universal now. Invalid time clears the scene and slice but retains worldlines and reference tables; invalid physical inputs clear the comparison.

Play continues from the selected time and restarts from the lower limit at the end. Traversing the whole window takes about five screen seconds; each frame contributes at most 50 ms so lag is not caught up. Pause stops at the current slice. Options contains examples and a step of one fortieth of the half-range. Reset restores L₀=10 m, B=8 m, β=0.8, the barn frame and time zero. Reduced motion or a hidden tool/page pauses playback; all manual controls remain available.

Only raw lengths, β and the chosen frame are saved, including incomplete or invalid entries. Navigation, reload and language changes rebuild fresh diagrams at time zero without playback. Time, results, plots and open menus are not saved. Within a view, exact typed time survives resize and physical parameter changes;it may need correction if the new time window is smaller. Small nonzero times use scientific notation.

Sources: Illinois PHYS 419: consequences of special relativity and UNSW Einstein Light: Lorentz transformations.