The Lorentz factor γ measures how strongly relativistic effects — time dilation and length contraction — apply at a given speed. This free calculator solves for γ from the speed, or the speed from γ, in any unit including fractions of c.
The Lorentz factor is found from the speed with γ = 1/sqrt(1 − (v/c)²), where β = v/c is the speed written as a fraction of the speed of light. To use it, enter the speed and — if you like — pick the unit c so you can type a fraction of light speed directly, such as 0.8. The calculator then computes β, squares it, subtracts from 1, takes the square root, and inverts to give γ.
As the speed v approaches c, the quantity 1 − β² shrinks toward zero and γ grows without limit. Note that v must be less than c: matter cannot reach or exceed the speed of light, so β always stays below 1. At low speeds γ is barely above 1, which is why the effects only become important close to light speed. The Lorentz factor is the same γ that appears in the time dilation calculator and in the energy relation of the E = mc² calculator.
Take a particle moving at v = 0.8c, so β = 0.8. Then 1 − β² = 1 − 0.64 = 0.36, the square root of which is 0.6, giving γ = 1/0.6 ≈ 1.667. At this speed clocks run about 1.667 times slower and lengths contract to roughly 60% of their rest value. Working backwards from a target factor, γ = 2 corresponds to v = c·sqrt(1 − 1/4) = c·sqrt(0.75) ≈ 0.866c — you need over 86% of light speed just to double the factor. For the definitions of these relativity terms, see the physics glossary.
The Lorentz factor appears throughout special relativity. It governs time dilation Δt = γΔt0, length contraction L = L0/γ, relativistic momentum p = γmv and total energy E = γmc². It is essential for the timing corrections that keep GPS accurate, for the design of particle accelerators (where γ reaches the thousands at the LHC), and it explains why fast-moving cosmic-ray muons survive long enough to reach the ground before decaying.
The Lorentz factor γ is a dimensionless number that is 1 at rest and grows without bound as speed approaches the speed of light. It scales relativistic time, length, momentum and energy: moving clocks slow by γ, moving lengths contract by γ, and relativistic energy is γ times the rest energy.
If v equals or exceeds c, then 1 − v^2/c^2 becomes zero or negative, and γ becomes infinite or imaginary. The speed of light c is the speed limit for matter, so v must always be less than c and β = v/c must be less than 1.
Beta is the speed written as a fraction of the speed of light: β = v/c. It ranges from 0 at rest up to (but never reaching) 1 at light speed. In terms of β the Lorentz factor is simply γ = 1/sqrt(1 − β^2).
At everyday speeds γ is only negligibly above 1. A jet flying at 300 m/s has γ ≈ 1.0000000000005, an unmeasurably tiny effect. You need roughly 10% of the speed of light before γ visibly exceeds 1, which is why relativity is invisible in daily life.
Moving clocks are slowed by the Lorentz factor, Δt = γΔt0, and a moving object’s total energy is E = γmc^2. When γ = 1 (at rest) this reduces to the famous rest energy E = mc^2, so the Lorentz factor is what connects motion to the mass-energy relation.