What is the Lorentz factor?

The Lorentz factor (symbol gamma, γ) is a dimensionless number that appears throughout special relativity. It equals 1 / sqrt(1 - v^2/c^2). At rest (v = 0), gamma = 1. As velocity approaches c, gamma approaches infinity.

What does beta (β) represent?

Beta is the velocity as a fraction of the speed of light: β = v/c. It ranges from 0 (at rest) to values approaching 1 (near the speed of light). Beta is dimensionless and is the natural unit for relativistic speeds.

How does time dilation work?

A clock moving at velocity v runs slow by the Lorentz factor. If gamma = 7, one second on the moving clock corresponds to 7 seconds for a stationary observer. The moving clock measures proper time, which is always shorter.

What is length contraction?

An object moving at velocity v appears contracted along its direction of motion by the factor 1/gamma. At gamma = 7 the object appears seven times shorter along the direction of travel compared to its rest length.

Why can nothing with mass reach the speed of light?

As v approaches c, gamma approaches infinity. The kinetic energy required (gamma - 1)mc^2 also approaches infinity, so infinite energy would be needed to accelerate any massive object to exactly c. This is a fundamental result of special relativity.

What is the speed of light used in this calculator?

The exact SI-defined value: c = 299,792,458 m/s (approximately 299,792 km/s).

Lorentz Factor Calculator

Enter a velocity (as a fraction of c, m/s, or km/s) to compute the Lorentz factor gamma, time dilation, and length contraction.

Enter Velocity

Velocity

Unit

Reference Table — Lorentz Factor at Common Speeds

Velocity (v/c)	v (km/s)	Lorentz Factor (γ)	Time Dilation	Context

Formulas

β = v / c where c = 299,792,458 m/s

γ = 1 / √(1 − β²)

Time dilation: Δt′ = γ Δt — a stationary observer sees the moving clock run slow by factor γ.

Length contraction: L′ = L / γ — an object in motion is contracted along its direction of travel by factor 1/γ.

Summary

The Lorentz factor (gamma, γ) is the central quantity in special relativity. It describes how time, length, and relativistic mass change for an object moving at velocity v relative to an observer. The formula is γ = 1 / √(1 − v²/c²), where c = 299,792,458 m/s. At everyday speeds gamma is essentially 1 — the correction is negligible. As v approaches c, gamma grows without bound, which is why massive objects cannot reach the speed of light. This calculator accepts velocity in three common units and returns gamma, beta (v/c), time dilation factor, and length contraction factor.

How it works

Select your velocity unit: fraction of c (beta), meters per second, or kilometers per second.
Enter the velocity value in the input field.
Click Calculate. The tool computes beta = v/c, then gamma = 1 / sqrt(1 - beta^2).
Results show gamma, beta, time dilation factor (clocks on the moving object tick 1/gamma times slower), and length contraction factor (objects are shortened by factor 1/gamma along the direction of motion).
A warning appears if the entered velocity equals or exceeds c, which is physically impossible for massive objects.

Use cases

Check the Lorentz factor for cosmic ray protons traveling at 0.999c.
Estimate time dilation for hypothetical near-light-speed spacecraft travel.
Verify textbook special relativity problems for particle physics or astrophysics courses.
Compute length contraction for a relativistic particle in an accelerator beam.
Explore how gamma scales across the range of 0 to 0.9999c using the reference table.

Frequently Asked Questions

Last updated: 2026-06-15 · Reviewed by Nham Vu