Length Contraction Calculator
Enter the rest length of an object and its velocity relative to you, and this calculator applies Einstein's length contraction formula to show the observed (contracted) length, the Lorentz factor, and the percentage by which the object shrinks along its direction of motion. Switch velocity units between fraction of c, km/s and m/s. Use the scenario presets to explore real-world cases from the ISS to cosmic-ray muons.
Formula
Worked example
A spaceship at rest measures 100 m long. It passes you at 0.9c (90% of the speed of light). Beta = 0.9, so gamma = 1 / sqrt(1 - 0.81) = 1 / sqrt(0.19) = 1 / 0.4359 = 2.294. The observed length is L = 100 / 2.294 = 43.59 m, a contraction of 56.41%.
What is length contraction?
Length contraction (also called Lorentz contraction) is a consequence of Einstein's special theory of relativity. When an object moves relative to an observer, the observer measures the object's length along the direction of motion to be shorter than it appears in its own rest frame. The "rest length" or "proper length" is the longest an object can ever be measured - it is the length measured when both the object and the measuring device are at rest with respect to each other. An observer who sees the object moving past them at high speed measures a contracted, shorter length. Crucially, the contraction only acts along the direction of motion; dimensions perpendicular to travel are unchanged.
The Lorentz contraction formula
The formula is L = L0 / gamma, where L0 is the rest (proper) length, L is the observed (contracted) length, and gamma is the Lorentz factor: gamma = 1 / sqrt(1 - v^2/c^2). Here v is the relative velocity between the object and the observer, and c is the speed of light (299,792,458 m/s). Because v must be less than c, the denominator is always a real positive number less than 1, so gamma is always greater than or equal to 1, and L is always less than or equal to L0. At everyday speeds (aircraft, satellites) the contraction is immeasurably small. As v approaches c, gamma grows without bound and L shrinks toward zero.
Why does length contraction happen?
Length contraction is not a physical "squeezing" of the object - the object does not change internally. It is a geometric consequence of the way space and time are intertwined in spacetime. Different observers in relative motion disagree about which events are simultaneous, and measuring the length of a moving object requires noting the positions of both its ends at the same time. Because observers in different frames disagree about simultaneity, they necessarily disagree about length. Both measurements are equally valid; there is no privileged frame of reference.
Real-world evidence: cosmic-ray muons
Muons created by cosmic rays in the upper atmosphere (about 10 km up) travel at roughly 0.9997c. Their half-life is only 2.2 microseconds, so classically they should decay after traveling about 660 m - far less than the 10 km to the surface. Yet detectors at sea level record them in large numbers. From our viewpoint on the ground, their clocks run slow (time dilation) and they survive the journey. From the muon's viewpoint, the atmosphere is length-contracted to only about 700 m / gamma = about 700 / 40 = 17 m, so they traverse it easily before decaying. Both explanations predict the same observable result, confirming special relativity.
Length contraction at common relativistic velocities
| Velocity (beta) | Example | Lorentz factor (gamma) | Observed length (100 m rest) | Contraction |
|---|---|---|---|---|
| 0.0000256c (7.66 km/s) | ISS | 1.0000000003 | ~100.000 m | < 0.0001% |
| 0.5c | Hypothetical spacecraft | 1.1547 | 86.60 m | 13.40% |
| 0.8c | Hypothetical spacecraft | 1.6667 | 60.00 m | 40.00% |
| 0.9c | Hypothetical spacecraft | 2.2942 | 43.59 m | 56.41% |
| 0.99c | Hypothetical spacecraft | 7.0888 | 14.11 m | 85.89% |
| 0.9997c | Cosmic-ray muon | 40.83 | 2.45 m | 97.55% |
| 0.9999991c | LHC proton | 2,358 | 0.042 m | 99.96% |
How much an object contracts along its direction of motion at representative speeds. Gamma (gamma) is the Lorentz factor; a 100 m spacecraft is used to illustrate the observed length.
Frequently asked questions
Does length contraction mean the object is physically compressed?
No. The object's internal structure is unchanged; its atoms remain the same distance apart as measured in its own rest frame. Length contraction is a measurement effect that arises from the geometry of spacetime. Two observers moving relative to each other simply do not agree on which events are simultaneous, so when each measures both ends of the object "at the same time," they get different numbers - and both are correct in their own inertial frame.
In which direction does the contraction occur?
Only along the axis of relative motion. If a spacecraft flies past you from left to right, you measure it as shorter from nose to tail, but its width and height appear unchanged. This asymmetry is built into the Lorentz transformation equations and has been confirmed by every experimental test of special relativity.
What happens to length at exactly the speed of light?
As v approaches c, gamma grows without bound and the observed length approaches zero. Reaching exactly c would require infinite energy and is impossible for any object with mass. Only massless particles (photons) travel at c, and the length-contraction formula does not apply to them in the same way.
Is length contraction the same as the barn-pole paradox?
The barn-pole paradox (or ladder paradox) is a thought experiment that uses length contraction as its starting point. A pole longer than a barn (at rest) is carried at relativistic speed and, in the barn's frame, it fits inside because it is contracted. In the pole's frame the barn is contracted and the pole does not fit - yet no contradiction arises because the two frames disagree about which events are simultaneous. So yes, the paradox is built on length contraction, but it illustrates the relativity of simultaneity rather than a real physical impossibility.
Can I use this calculator to find the velocity from two lengths?
Yes, by rearranging the formula: v = c * sqrt(1 - (L/L0)^2). If you know the rest length and the observed contracted length, you can solve for beta. Enter any two of the three quantities (L0, L, v) and solve for the third - this calculator solves for L given L0 and v. For the reverse calculation, compute beta = sqrt(1 - (L_obs/L0)^2) and enter that as the velocity.
Why is the Lorentz factor always at least 1?
Because v is always less than c, v^2/c^2 is always between 0 and 1 (exclusive), so 1 - v^2/c^2 is always between 0 and 1, and its square root is between 0 and 1. Dividing 1 by a number between 0 and 1 gives a number greater than or equal to 1. At v = 0 the Lorentz factor is exactly 1 and there is no contraction; it only exceeds 1 when the object is moving relative to the observer.