Lorentz Factor Calculator

The Lorentz factor (γ) is a dimensionless quantity from Einstein's special relativity that describes how much time, length, and relativistic mass change for an object moving at a given speed relative to an observer. At everyday speeds γ ≈ 1, meaning relativistic effects are negligible. As velocity approaches the speed of light, γ grows dramatically, causing significant time dilation, length contraction, and mass increase.

The Lorentz factor is calculated using the formula γ = 1 / √(1 − v²/c²), where v is the object's velocity and c is the speed of light (~299,792,458 m/s). You can also write it as γ = 1 / √(1 − β²), where β = v/c is the ratio of the object's speed to the speed of light.

Beta (β) is simply the ratio of an object's velocity (v) to the speed of light (c), so β = v/c. It ranges from 0 (at rest) to just under 1 (approaching the speed of light). Beta is a convenient dimensionless measure of relativistic speed and is used directly in the Lorentz factor formula.

As an object's speed approaches the speed of light (β → 1), the Lorentz factor γ approaches infinity. This means infinite energy would be required to accelerate a massive object to the speed of light, which is why it is physically impossible for objects with mass to reach c.

Time dilation is the phenomenon where time passes more slowly for a moving observer relative to a stationary one. The relationship is Δt′ = γ · Δt, where Δt is the proper time (measured in the moving frame) and Δt′ is the dilated time (measured by the stationary observer). A higher γ means a greater difference in elapsed time between the two frames.

Length contraction means that an object moving at relativistic speeds appears shorter along its direction of motion to a stationary observer. The contracted length L = L₀ / γ, where L₀ is the proper (rest) length. The contraction factor is 1/γ — the same Lorentz factor that describes time dilation and mass increase.

Relativistic effects become noticeable (γ > 1.01) at roughly 10% of the speed of light (~30,000 km/s). At 50% of c, γ ≈ 1.155; at 90% of c, γ ≈ 2.294; and at 99% of c, γ ≈ 7.089. For everyday speeds — even spacecraft — γ is so close to 1 that relativistic corrections are immeasurably small.

Yes. In relativistic mechanics, the relativistic momentum and energy of a moving object increase by a factor of γ compared to classical predictions. This is sometimes described as relativistic mass m = γ · m₀, where m₀ is the rest mass. This is why particle accelerators require exponentially more energy to push particles closer to the speed of light.