About

The barn-pole paradox is a standard thought experiment in special relativity that exposes the failure of absolute simultaneity. A pole of rest length L_pole moves at relativistic velocity v through a barn of rest length L_barn. In the barn frame, Lorentz contraction shortens the pole to L_pole × √1 − β², so it fits inside. In the pole frame, the barn contracts instead, so the pole never fits. The resolution lies in relativity of simultaneity: the two door-closing events that are simultaneous in one frame are not simultaneous in the other. Incorrect resolution of this paradox in engineering contexts involving relativistic particle beams or synchrotron timing can propagate systematic measurement errors.

This simulator calculates contracted lengths, the Lorentz factor γ, and simultaneity offsets for both frames. It renders a real-time animation of the pole traversing the barn and a Minkowski spacetime diagram showing worldlines of the barn doors and pole endpoints. The tool assumes flat Minkowski spacetime with no gravitational effects. Results diverge from reality at β > 0.99999 due to floating-point precision limits.

Formulas

The Lorentz factor governs all relativistic effects in this scenario:

γ = 1√1 − β²

where β = vc is the velocity as a fraction of the speed of light c = 299,792,458 m/s.

Length contraction in the direction of motion:

L = L₀γ = L₀ √1 − β²

Relativity of simultaneity. Two events separated by spatial distance Δx that are simultaneous (Δt = 0) in the barn frame have a time difference in the pole frame:

Δt′ = −γ v Δxc²

where L₀ = rest length of the object, L = contracted length observed from the other frame, Δx = spatial separation of the two door-closing events (equal to barn rest length L_barn), and Δt′ = time difference between door closings as measured in the pole frame. A negative value means the far door closes before the near door in the pole frame.

Reference Data

β (v/c)	γ (Lorentz Factor)	Contraction Ratio 1/γ	10m pole contracted	Time dilation factor	Simultaneity offset per 10m
0.10	1.005	0.995	9.95m	1.005×	3.34ns
0.20	1.021	0.980	9.80m	1.021×	6.67ns
0.30	1.048	0.954	9.54m	1.048×	10.0ns
0.40	1.091	0.917	9.17m	1.091×	13.3ns
0.50	1.155	0.866	8.66m	1.155×	16.7ns
0.60	1.250	0.800	8.00m	1.250×	20.0ns
0.70	1.400	0.714	7.14m	1.400×	23.3ns
0.80	1.667	0.600	6.00m	1.667×	26.7ns
0.85	1.898	0.527	5.27m	1.898×	28.3ns
0.90	2.294	0.436	4.36m	2.294×	30.0ns
0.92	2.552	0.392	3.92m	2.552×	30.7ns
0.95	3.203	0.312	3.12m	3.203×	31.7ns
0.97	4.113	0.243	2.43m	4.113×	32.3ns
0.99	7.089	0.141	1.41m	7.089×	33.0ns
0.999	22.37	0.0447	0.447m	22.37×	33.3ns
0.9999	70.71	0.0141	0.141m	70.71×	33.33ns

Frequently Asked Questions

In the barn's rest frame, the moving pole undergoes Lorentz contraction and becomes shorter than the barn. Both doors can be simultaneously closed while the pole is inside. In the pole's rest frame, the barn is the moving object and contracts instead, so the pole is longer than the barn at all times. The resolution is that the two door-closing events are not simultaneous in the pole frame. The front door closes and opens before the rear door closes, so the pole passes through without contradiction. No physical inconsistency arises because simultaneity is frame-dependent.

No. Lorentz contraction is not a material stress or mechanical compression. It is a geometric consequence of how spacetime intervals project onto different inertial frames. The pole experiences no internal forces from contraction. If you placed strain gauges on the pole, they would read zero. The effect is real in the sense that measurements of length genuinely differ between frames, but no structural deformation occurs.

In special relativity, perfectly rigid bodies do not exist. Information about a door closing propagates through the material at the speed of sound, which is always less than the speed of light. If you attempt to trap the pole by closing both doors, the pole's far end does not "know" the door has closed until a compression wave reaches it. The pole would crumple or break rather than instantaneously stop. The paradox is resolved before material properties matter, purely through simultaneity, but rigid body assumptions add a separate layer of physical impossibility.

Lorentz contraction scales as the square root of (1 − β²). At β = 0.1 (10% of light speed), contraction is only 0.5%. At β = 0.5, the object shortens to 86.6% of its rest length. The effect becomes dramatic above β = 0.9, where contraction exceeds 56%. For the barn-pole paradox to produce a clear "fit" scenario where a 10 m pole fits in a 5 m barn, you need β ≥ 0.866, corresponding to γ = 2.

On a Minkowski diagram, the barn doors are vertical worldlines (stationary in the barn frame) and the pole endpoints are tilted worldlines moving at angle arctan(β). The simultaneity line in the barn frame is horizontal. In the pole frame, the simultaneity line tilts by arctan(β), intersecting the door worldlines at different times. The diagram shows geometrically that "both doors closed at the same time" maps to two distinct times in the other frame. The pole's worldstrip passes through the barn's worldstrip without overlap in both frames when you account for the tilted simultaneity.

Yes. Relativistic heavy ion collisions at facilities like RHIC and LHC involve gold or lead nuclei traveling at β > 0.999. A gold nucleus with a rest diameter of about 14 fm contracts to under 1 fm in the lab frame. Detector timing and geometry must account for this contraction. The barn-pole logic applies directly: the nucleus "fits" inside spatial regions smaller than its rest size from the lab perspective, while in the nucleus rest frame, the detector apparatus is contracted instead.