Length Contraction Calculator

Created by Luis Fernando
Last updated: Sep 21, 2022

If you're dealing with speeds comparable to the speed of light, this length contraction calculator is for you! Input the proper length and relative velocity, and the calculator will determine the Lorentz contraction immediately.

Still don't know how to calculate length contraction? In the following sections, we present the length contraction formula and the concept behind it.

🙋 Studying relativity, you'll probably deal with the widely-known E = mc² formula (mass-energy equivalence). To calculate this famous relationship, use our E = mc² calculator.

What is length contraction?

In the theory of relativity, length contraction is the phenomenon in which the length of an object measures shorter for an observer in a reference frame moving relative to it. The reference frame moving with the object (or in which such object is stationary) is known as the rest frame, while the object's length in this frame is known as the proper length or rest length (the equivalent to the proper time interval in time dilation).

🔎 Length contraction is also known as Lorentz contraction and Lorentz–FitzGerald contraction, as it was initially postulated by George FitzGerald (1889) and Hendrik Antoon Lorentz (1892) before Einstein's further refinement (1905).

So, the length measured in a reference frame moving relative to the rest frame will always be shorter than the proper length, and how much it contracts will depend on the relative velocity between the frames. The length contraction formula of the following section describes this dependence mathematically.

How to calculate length contraction

The equation to calculate the length contraction is:

$L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - v^2/c^2}$

, where:

• $L$ – Observed length (length measured by the observer);
• $L_0$ – Proper length (length of the object in its rest frame);
• $v$ – Relative velocity between the object and the observer;
• $c$ – Speed of light (299,792,458 m/s); and
• $\gamma$Lorentz factor, defined as $\gamma \equiv 1/\sqrt{1 - v^2/c^2}$.

Some important points about length contraction and the equation above:

• Lorentz contraction and the equation above apply for lengths measured in the direction parallel to the relative motion. There's no contraction in the directions perpendicular to the movement.
• Like relativistic kinetic energy or time dilation, the relative velocity must be comparable to the speed of light for length contraction to be noticeable. You won't notice it for a car moving at 70 mph, but if we could build a spaceship that travels at 200,000 km/s, it would decrease its length by 25% (you can verify this using the calculator).
• Apart from requiring very high speeds, another challenge to the observation of Lorentz contraction is the difficulty of measuring the length of objects moving at such high speeds.
Luis Fernando
Proper length (L₀)
m
Relative velocity (v)
km/s
Observed length (L)
m
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