Barn-Pole Paradox Calculator
Explore special relativity through length contraction and simultaneity
Paradox Parameters
Relativistic velocity (β = v/c = 0.900)
Length of pole in its rest frame
Length of barn in its rest frame
Choose reference frame for detailed analysis
Relativistic Parameters
Rest Frame Lengths
Contracted Lengths
Barn Frame of Reference
Event Timeline
Analysis
Paradox Resolution
Key Insight: The resolution lies in the relativity of simultaneity.
• Barn Frame: The pole contracts and fits in the barn. Both doors can close simultaneously.
• Pole Frame: The barn contracts, but more importantly, the door closings are NOT simultaneous. The pole is never entirely contained.
Both perspectives are correct! Events that are simultaneous in one frame are not simultaneous in another moving frame.
Classic Example
Setup
Pole length: 12 meters (rest frame)
Barn length: 6 meters (rest frame)
Velocity: 0.9c (270,000 km/s)
Lorentz factor: γ = 1/√(1-0.9²) = 2.294
Barn Frame Analysis
Contracted pole length: 12m / 2.294 = 5.23m
Result: Pole fits in 6m barn!
Both doors can close simultaneously at t = 20.81 ns
Pole Frame Analysis
Contracted barn length: 6m / 2.294 = 2.62m
Result: 12m pole never fits in 2.62m barn!
Door closings are not simultaneous: Δt = 41.33 ns
Relativity Concepts
Length Contraction
L' = L/γ
Objects contract in direction of motion
Simultaneity
t' = γ(t - vx/c²)
Events simultaneous in one frame may not be in another
Lorentz Factor
γ = 1/√(1-β²)
Measures relativistic effects strength
Key Insights
No absolute reference frame exists
Both perspectives are physically valid
Simultaneity depends on reference frame
Space and time are interwoven
Causality is preserved in all frames
Understanding the Barn-Pole Paradox
The Paradox Setup
A pole longer than a barn moves at relativistic speed through the barn. The question: can the pole fit inside the barn with both doors closed simultaneously?
Two Contradictory Views
- •Barn frame: Pole contracts, fits inside barn
- •Pole frame: Barn contracts, pole never fits
- •Both views seem logically consistent
- •Yet they lead to opposite conclusions
The Resolution
Relativity of Simultaneity
The key insight is that events that are simultaneous in one reference frame are not necessarily simultaneous in another frame moving relative to it.
Resolution: In the barn frame, doors close simultaneously. In the pole frame, they close at different times, so the pole is never completely enclosed.
Mathematical Foundation
The paradox is resolved using the Lorentz transformation, particularly the transformation for time coordinates between different reference frames.
t' = γ(t - vx/c²)
Time transformation between reference frames
Physical Interpretation
The paradox demonstrates fundamental properties of spacetime in special relativity. It shows that space and time are not separate, absolute entities but are interwoven in a four-dimensional spacetime continuum.
Lesson: There is no universal "now" - simultaneity is relative to the observer's state of motion.