Lorentz Factor Calculator

The Lorentz factor, γ (gamma), is a dimensionless number that quantifies how much time dilation, length contraction, and relativistic mass/energy increase occur for an object moving at a given speed, according to Einstein's special relativity. It equals 1 at rest and grows without bound as speed approaches the speed of light.

γ = 1/√(1−β²), where β = v/c is the object's speed as a fraction of the speed of light c. This can also be written γ = 1/√(1−v²/c²) using velocity directly.

As v approaches c, β approaches 1, making 1−β² approach zero and γ diverge to infinity. For v equal to or greater than c, the term under the square root becomes zero or negative, which is mathematically undefined for real numbers, reflecting the physical fact that no massive object can reach or exceed the speed of light.

A moving clock runs slow by exactly the factor γ compared to a stationary observer: Δt = γΔt₀, where Δt₀ is the time measured in the moving object's own rest frame (proper time) and Δt is the (longer) time measured by the stationary observer.

An object moving at speed v appears contracted along its direction of motion by exactly the factor γ, as measured by a stationary observer: L = L₀/γ, where L₀ is the object's proper length (measured in its own rest frame).

Total relativistic energy is E = γmc², and relativistic momentum is p = γmv, both scaled up from their classical (non-relativistic) counterparts by the same factor γ. This is why particle accelerators report particle energies in terms of γ, it directly reflects how much the particle's energy and momentum exceed their rest-mass values.

Protons at the LHC's design energy of 6.5 TeV per beam reach a Lorentz factor of roughly γ ≈ 6,900 to 7,000, meaning their proper time runs about 7,000 times slower than laboratory time, a dramatic real-world confirmation of relativistic time dilation.

No, at everyday speeds (cars, planes, even orbital spacecraft), β is extremely small (typically less than 0.0001), so γ is indistinguishable from 1 to many decimal places. Relativistic effects only become significant when speeds reach a meaningful fraction of the speed of light, which is why classical (Newtonian) mechanics works so well for ordinary life.

Not quite, γ is the dimensionless scaling factor itself. The older concept of 'relativistic mass' was defined as m_rel = γm₀ (rest mass times γ), but this terminology has fallen out of favor in modern physics in favor of keeping mass as an invariant (rest mass) and applying γ directly to momentum and energy instead.

Yes, this calculator supports both: enter a velocity directly in metres per second, or switch modes to enter β = v/c directly (a number between 0 and 1) if you already know the speed as a fraction of light speed.