Lorentz Factor Calculator
Enter a speed in metres per second to get the Lorentz factor γ — the heart of special relativity — plus that speed as a percentage of the speed of light.
Gamma and light-speed share at once
Enter a velocity and the calculator returns the Lorentz factor γ = 1 / √(1 − (v/c)²) together with the speed as a percentage of light speed.
Stay below light speed
γ is undefined at and beyond the speed of light (299,792,458 m/s), so the velocity must be less than c for a result.
What is the Lorentz factor?
The number behind special relativity
This Lorentz factor calculator turns a single measurement — the speed of an object in metres per second — into gamma (γ), the dimensionless number that governs how time, length, and mass change with motion in Einstein's special relativity. Gamma is 1 when an object is at rest and climbs ever more steeply as the speed approaches that of light, never quite reaching infinity. At everyday speeds γ is so close to 1 that relativity is invisible, but for particles in accelerators, cosmic rays, and GPS satellites the factor matters. Alongside γ the calculator shows the speed as a percentage of the speed of light so you can see how close to c you are.
Enter a speed in metres per second to get the Lorentz factor and the percentage of light speed instantly.
The Lorentz factor is one divided by the square root of one minus the squared ratio of the speed to the speed of light.
γ = 1 / √(1 − (v/c)²)Here c is the speed of light in vacuum, exactly 299,792,458 m/s. The ratio v/c is always between 0 and 1 for a real object, so 1 − (v/c)² is a positive number less than one, its square root is less than one, and dividing one by it gives a factor of one or more. The closer v gets to c, the smaller the denominator and the larger γ becomes.
Suppose an object travels at 149,896,229 m/s — exactly half the speed of light.
Form the speed ratio
149,896,229 ÷ 299,792,458 = 0.5 — half the speed of light.
Square it and subtract from one
1 − 0.5² = 1 − 0.25 = 0.75.
Square-root and invert
1 ÷ √0.75 ≈ 1 ÷ 0.8660 ≈ 1.1547 — the Lorentz factor. The speed is 50 % of light speed.
The Lorentz factor answers a single question: by how much do relativistic effects multiply at this speed? A γ of 1.1547, from the half-light-speed example, means a moving clock runs about 13 % slower, lengths shrink by the same fraction along the direction of travel, and relativistic mass is 15 % higher than the rest mass. The growth is sharply non-linear. At 10 % of light speed γ is just 1.005, a fraction of a percent; at 90 % it is 2.29; at 99 % it leaps to about 7.09. That steep climb is why crossing the last sliver toward c is impossible: each increment of speed costs disproportionately more energy, and at exactly c the factor would be infinite. The percentage-of-light-speed figure puts your input in context — it is the v/c ratio expressed as a percent, the same quantity that drives γ.
The formula is exact within special relativity, but a couple of points are worth keeping in mind.
Below light speed, special relativity only
γ is only defined for speeds below c (299,792,458 m/s); at or above light speed the calculator returns no result because the factor would be infinite or imaginary. The figure describes special relativity in flat spacetime and does not include gravity (general relativity) or rotation. Enter the speed in metres per second — divide km/h by 3.6, or multiply a fraction of light speed by 299,792,458 first.