Barn Pole Paradox
Explore the barn-pole paradox (ladder paradox) in special relativity. Compute length contraction, Lorentz factor γ, and door closing times in both frames with charts.
About This Calculator
The Barn Pole Paradox Calculator explores a classic thought experiment in Einstein's special relativity — also known as the ladder paradox. A pole longer than a barn is accelerated to relativistic speed directed straight through the barn's two open doors. From the barn's frame of reference, the pole undergoes length contraction and briefly fits entirely inside the barn, allowing both doors to close simultaneously. From the pole's frame, the barn rushes toward it at relativistic speed and contracts even further, making the pole too long to ever fit completely. Yet the doors still close without hitting the pole — how is this possible?
The paradox is resolved by the relativity of simultaneity: the two events of the doors closing are simultaneous only in the barn's frame. In the pole's frame, they occur at different times — the back door closes and opens while the front of the pole is inside, then the front door closes after the back end enters. This calculator computes the full set of relativistic parameters: the Lorentz factor γ = 1/√(1−β²), the contracted lengths in each frame, and the precise door closing times for both reference frames using Lorentz transformations. It uses the speed of light c = 299,792,458 m/s and assumes no acceleration — both the barn and pole are inertial frames moving at constant velocity.
The calculator outputs the key times in the barn frame (when the back enters, when the front exits, and the earliest moment both doors can close) and the corresponding times in the pole frame (where the front and back door closings occur at different instants, demonstrating the relativity of simultaneity). The interactive event timeline shows exactly why the paradox resolves.
Regional Context
Global: Special relativity is a universal physical theory, independent of location. The speed of light c = 299,792,458 m/s is a defined physical constant used worldwide in India, the United States, and the United Kingdom. Physics students and researchers globally study the barn-pole paradox as a cornerstone thought experiment for understanding simultaneity and Lorentz transformations. This calculator uses SI units throughout and is suitable for undergraduate physics courses and independent study worldwide.
Frequently Asked Questions
What is the barn-pole paradox?
The barn-pole paradox (also known as the ladder paradox) is a thought experiment in special relativity. A pole longer than a barn moves at relativistic speed toward the barn. In the barn's frame, length contraction makes the pole fit inside momentarily. In the pole's frame, the barn is contracted making the pole too long to fit. The paradox is resolved by the relativity of simultaneity — the two doors do not close simultaneously in the pole's frame.
How is the barn-pole paradox resolved?
The barn-pole paradox is resolved through the relativity of simultaneity. While both barn doors close at the same time in the barn's frame of reference, they close at different times in the pole's frame. In the pole's frame, the back door closes first, trapping the front of the pole, then opens to let it out. Later, the front door closes behind the back end of the pole. The pole is never entirely inside the barn at any single moment in the pole's frame.
What is length contraction in special relativity?
Length contraction is a relativistic effect where objects moving at a significant fraction of the speed of light appear shorter along their direction of motion to a stationary observer. The contracted length is given by L = L₀/γ where L₀ is the rest length and γ = 1/√(1-v²/c²) is the Lorentz factor. This effect is reciprocal — observers in different inertial frames each see the other's objects as contracted.
What is the Lorentz factor?
The Lorentz factor γ (gamma) is a fundamental quantity in special relativity defined as γ = 1/√(1-β²) where β = v/c is the speed ratio. At rest γ = 1. At 90% of light speed (β=0.9), γ ≈ 2.29. As v approaches c, γ approaches infinity. The Lorentz factor determines the magnitude of length contraction, time dilation, and relativistic mass increase.
Does the pole really fit in the barn?
From the barn's frame of reference, yes — the length-contracted pole fits entirely inside the barn for a brief moment, allowing both doors to close simultaneously. From the pole's frame of reference, no — the barn is even more contracted and the pole never fully fits inside. However, the doors still close without hitting the pole because they do not close simultaneously in the pole's frame.
What is the ladder paradox?
The ladder paradox is another name for the barn-pole paradox. A ladder longer than a garage is moved at relativistic speed toward it. To a stationary observer, the ladder contracts and fits inside. To an observer moving with the ladder, the garage contracts and the ladder does not fit. The resolution is the same — the front and back doors of the garage do not close simultaneously in the ladder's reference frame.
What is the relativity of simultaneity?
The relativity of simultaneity is the principle that two events that are simultaneous in one inertial frame of reference are not necessarily simultaneous in another frame moving relative to the first. This is a cornerstone of Einstein's special relativity and is the key to resolving the barn-pole paradox. In the barn frame the doors close at the same time; in the pole frame they close at different times.
Is the barn-pole paradox experimentally testable?
While the exact barn-pole scenario cannot be tested with macroscopic objects (we cannot accelerate a pole to relativistic speeds), the underlying physics — length contraction and relativity of simultaneity — is confirmed daily in particle accelerators and through cosmic ray muon observations. Muons created in the upper atmosphere reach Earth's surface because their lifetime is time-dilated from our perspective, and the atmosphere is length-contracted from theirs.