LorentzLen

Length Contraction Calculator

Objects shrink in the direction of motion — see by how much at any speed.

Proper length (m)

Velocity (fraction of c)

Contracted length

43.589 m

Lorentz factor (γ): 2.294

What you entered

Proper length: 100 mVelocity: 0.9c

1
v/c (velocity as fraction of speed of light)
0.9= 0.9c
2
v² / c²
0.9²= 0.81
3
Lorentz factor: γ = 1 / √(1 − v²/c²)
1 / √(1 − 0.81)= 2.2942
4
Contracted length: L = L₀ / γ
100 / 2.2942= 43.589 m

Result

Contracted length: 43.589 m

At 90.0% the speed of light, a 100 m object contracts to 43.589 m along the direction of motion — an observer moving with the object still measures 100 m.

Length contraction at relativistic speeds

A moving object's length along the direction of travel is its proper length divided by the Lorentz factor.

Proper length	Speed (fraction of c)	Lorentz factor	Contracted length
1	0.1	1.0050	0.9950
1	0.5	1.1547	0.8660
10	0.8	1.6667	6.0000
1	0.9	2.2942	0.4359
100	0.99	7.0888	14.1067

Contraction happens only along the direction of motion - the object's other two dimensions are unaffected, so a fast-moving sphere is measured as a flattened ellipsoid rather than a smaller sphere. The effect is exactly reciprocal to time dilation, and the two together are why muons created high in the atmosphere reach the ground: in their frame the distance is contracted, while in ours their lifetime is dilated. Both descriptions give the same answer, which is the consistency relativity requires.

Lorentz contraction explained

A moving object's length in the direction of motion is contracted: L = L₀/γ, where L₀ is the proper length (measured at rest) and γ is the Lorentz factor. At 86.6% c, γ = 2, so the object appears half its rest length. This isn't an optical illusion — it's a real consequence of spacetime geometry.

Observing length contraction

Length contraction is negligible at everyday speeds. Even the ISS at 7.7 km/s has γ ≈ 1.0000000003. But for particles in accelerators moving at 99.999% c, contraction is dramatic — a 1-meter rod would measure just 4.5 mm. Muons from cosmic rays survive to reach Earth partly because of their contracted travel distance.

Frequently asked questions

A 100-meter spacecraft travels at 0.8c — how long does it appear to a stationary observer?

γ at 0.8c = 1/√(1 − 0.64) = 1/√0.36 ≈ 1.667. Contracted length = 100 / 1.667 ≈ 60 meters. The spacecraft appears 40% shorter in its direction of motion to the outside observer.

How is length contraction different from relativistic mass?

They're two sides of the same coin — both stem from the Lorentz factor γ. Length contraction shrinks distances (L = L₀/γ), while relativistic mass grows inertia (m = γm₀). Both become significant only at speeds above about 10% of light speed. The relativistic mass calculator shows the mass side.

Does the object actually get shorter, or does it just appear shorter?

It's a real physical effect in the observer's reference frame — not an optical illusion. In the spacecraft's own frame, its length is unchanged. This is because space and time are relative to the observer's motion, a core principle of special relativity. Both frames are equally valid.

At what speed does length contraction become noticeable?

At 10% of c (about 30,000 km/s), contraction is only 0.5% — barely measurable. At 50% c, it's about 13%. At 90% c, objects shrink to 44% of their rest length. At 99.99% c, just 1.4%. The effect follows a curve — mostly negligible until speeds get very close to c, then dramatic.