Lorentz Contraction Calculator | Relativistic Length Tool

Calculator Inputs

Calculation Mode

Proper Length (L₀)

Proper Length Unit

Contracted Length (L)

Contracted Length Unit

Velocity Value

Velocity Unit

Output Length Unit

Decimal Precision

Graph Maximum β

Graph Points

Use fraction of c for direct relativistic input, or choose a standard speed unit for conversion.

Example Data Table

Proper Length (m)	β = v/c	Lorentz Factor γ	Contracted Length (m)	Percent Contraction
10.00	0.10	1.005038	9.949874	0.501%
10.00	0.50	1.154701	8.660254	13.397%
10.00	0.80	1.666667	6.000000	40.000%
10.00	0.95	3.202563	3.122499	68.775%
10.00	0.99	7.088812	1.410674	85.893%

Formula Used

Primary Lorentz Contraction Equation
L = L₀ √(1 - v²/c²)

Using β = v/c
L = L₀ √(1 - β²)

Lorentz Factor
γ = 1 / √(1 - β²)

Equivalent Form
L = L₀ / γ

Solving for Velocity from Lengths
β = √(1 - (L/L₀)²), then v = βc

Here, L₀ is the proper length measured in the object’s rest frame, L is the shorter length measured by an observer seeing the object move, v is relative speed, c is the speed of light, and γ is the Lorentz factor. Contraction becomes noticeable only at relativistic speeds.

How to Use This Calculator

Select the calculation mode that matches your task.
Enter the known proper length, contracted length, or speed values.
Choose matching units for every field you use.
Set the desired output unit and decimal precision.
Adjust the graph range if you want a wider or tighter relativistic plot.
Press Submit to show the result beneath the header and above the form.
Use the CSV or PDF buttons to export the computed table.

Frequently Asked Questions

1) What is Lorentz contraction?

Lorentz contraction is the shortening of an object’s measured length along its direction of motion when observed from another inertial frame. The object keeps its proper length in its own rest frame.

2) When does the effect become noticeable?

The effect is tiny at everyday speeds. It becomes significant only when velocity is a sizable fraction of light speed, where the factor √(1 - β²) drops well below one.

3) What is proper length?

Proper length is the length measured in the frame where the object is at rest. It is the longest length associated with that object for straight-line motion along the measured axis.

4) Can this calculator solve for speed?

Yes. Choose the velocity mode, enter proper and contracted lengths, and the tool computes β, physical speed, gamma, contraction ratio, and percentage contraction automatically.

5) Why must the speed stay below light speed?

Special relativity requires massive objects to move slower than light. At β = 1, the denominator in the gamma expression reaches zero, so the model becomes nonphysical for matter.

6) Does contraction happen in every direction?

No. The contraction applies only along the direction of relative motion. Dimensions perpendicular to the motion remain unchanged in standard special relativity.

7) What does the graph show?

The graph plots contracted length against β for the solved proper length, or gamma against β in gamma mode. It also marks the currently solved operating point.

8) Can I use different length units?

Yes. The calculator converts among meters, kilometers, centimeters, millimeters, micrometers, nanometers, feet, and inches before solving and then reports the answer in your chosen output unit.