Relativity Paradox Calculator (barn–pole and bug–rivet, length contraction)

Understanding the relativity paradox calculator — concept and principle

This calculator works through two classic puzzles of special relativity, the barn–pole (ladder) paradox and the bug–rivet paradox, using length contraction and the relativity of simultaneity. It shows the contracted lengths in each frame, whether the pole fits, and the timing gap that resolves the apparent contradiction. It is used in physics courses to build intuition about relativity.

How it is calculated

γ = 1 ÷ √(1 − v²/c²)

Moving length seen from the barn = L_pole ÷ γ; barn seen from the pole = L_barn ÷ γ

Door-closing gap in the pole frame: Δt′ = γ v L_barn ÷ c² (from the Lorentz transformation t′ = γ(t − vx/c²))

• Speed is entered as a percentage of c; lengths in metres.

The principle

Each observer sees the other's objects contracted along the direction of motion. The farmer sees a short pole that fits inside the barn with both doors shut at once. The runner sees a short barn far too small for the pole. Both are right, because "at the same time" depends on the frame: the two door events that are simultaneous for the farmer happen at different times for the runner. In the runner's frame the far door shuts and opens before the near door closes, so the pole is never fully enclosed. Physical outcomes, like whether something breaks, are the same in every frame.

Worked example

Default inputs: barn–pole, 86.6% of c, pole 10 m, barn 6 m.

1. γ = 1 ÷ √(1 − 0.866²) = 1.9998.
2. The barn sees the pole as 10 ÷ 2 = 5 m: it fits, yes — doors shut together.
3. The pole sees the barn as 6 ÷ 2 = 3 m: it does not fit, no — barn is shorter here.
4. Door-closing gap in the pole frame = 1.9998 × 0.866 × 6 ÷ c = 34.661 ns.
5. The bug–rivet rows apply only to the other paradox, so they show —.

How to read the result

The timing gap is the key: it shows that the two doors do not close simultaneously in the runner's frame. At everyday speeds γ ≈ 1 and both effects vanish.

Common mistakes

• Assuming simultaneity is the same in every frame.
• Thinking the pole is physically squeezed.
• Applying the result to accelerating or stopping objects without considering how forces propagate.

Frequently asked questions

Does the pole really fit in the barn?

In the barn frame, yes; in the pole frame, no. Both descriptions are consistent.

What resolves the paradox?

The relativity of simultaneity.

What speed halves lengths?

About 86.6% of c, where γ = 2.

Is the bug squashed in the rivet paradox?

The outcome is the same in every frame; select the bug–rivet option to see the answer for your inputs.

Sources: OpenStax, University Physics Volume 3 (2016), chapter 5 key equations