Length Contraction Calculator (Special Relativity)

Length Contraction: Special Relativity

Einstein’s Special Theory of Relativity (1905) predicts that an object moving relative to an observer appears shorter along the direction of motion. This is called Lorentz-FitzGerald contraction (or simply length contraction).

The Formula L = L₀ / γ = L₀ × √(1 − v²/c²)

Where:

• L = contracted length (measured by stationary observer watching the object pass)
• L₀ = proper length (measured in the object’s own rest frame)
• v = relative velocity
• c = speed of light = 2.998 × 10⁸ m/s
• γ = Lorentz factor = 1 / √(1 − v²/c²)

Key Properties

• Length contraction only occurs along the direction of motion: transverse dimensions are unchanged.
• The object does not physically compress: observers in the moving frame measure the normal proper length. Only a stationary observer sees the contracted length.
• At everyday speeds (airplanes, satellites), contraction is unmeasurably small. At v = 0.866c (√3/2 × c), γ = 2 and the object appears half its proper length. At v = 0.9999c, γ ≈ 70.7 and the object appears 1.4% of its proper length.

The Barn-Pole Paradox A classic thought experiment: a pole of proper length 10 m passes through a barn of proper length 8 m. At sufficient speed, from the barn’s frame the pole appears contracted to 8 m and fits inside. From the pole’s frame, the barn appears contracted and the pole never fits. Both are correct simultaneously — simultaneity of events differs between frames.

Einstein’s Train Thought Experiment Einstein imagined a very fast train passing an embankment. Light flashes at the train’s nose and tail appear simultaneous to a passenger on the train but NOT simultaneous to a stationary observer. This illustrates that simultaneity — and therefore measurement of length — is relative to the observer’s frame.