Lorentz Factor Calculator
Instantly compute the Lorentz gamma factor (γ) for any velocity, a key quantity in all of Special Relativity. Use it when calculating time dilation, length contraction, or relativistic momentum and energy.
About this calculator
The Lorentz factor γ (gamma) is the central quantity of Special Relativity, appearing in formulas for time dilation, length contraction, relativistic momentum, and relativistic energy. It is defined as γ = 1 / √(1 − v²/c²), where v is the object's speed and c = 299,792,458 m/s is the speed of light in a vacuum. At v = 0, γ = 1 and all relativistic effects vanish. As v → c, γ → ∞, meaning time dilation and length contraction become infinite and it would take infinite energy to accelerate a massive object to c. At 10% of the speed of light γ ≈ 1.005, almost negligible. At 90% of c, γ ≈ 2.29; at 99% of c, γ ≈ 7.09; at 99.9% of c, γ ≈ 22.4. Particle accelerators like the LHC routinely accelerate protons to γ values exceeding 7,000.
How to use
Suppose you want the Lorentz factor for a particle moving at v = 270,000,000 m/s. Step 1 — compute v²/c²: (2.7×10⁸)² / (2.998×10⁸)² = 7.29×10¹⁶ / 8.988×10¹⁶ ≈ 0.8112. Step 2 — subtract from 1: 1 − 0.8112 = 0.1888. Step 3 — take the square root: √0.1888 ≈ 0.4345. Step 4 — take the reciprocal: γ = 1 / 0.4345 ≈ 2.30. This particle's clock runs 2.30× slower and its length is compressed to 1/2.30 ≈ 43% of its rest length as seen by a stationary observer.
Frequently asked questions
What does a Lorentz factor of 2 mean physically?
A Lorentz factor of γ = 2 means the moving object is traveling at v = c × √(1 − 1/γ²) = c × √(1 − 0.25) = c × √0.75 ≈ 0.866c, about 86.6% of the speed of light. Physically it means a stationary observer sees the moving clock tick at half the normal rate (time dilation by a factor of 2), the object's length along its direction of motion is halved, and its relativistic momentum is twice the Newtonian momentum. Relativistic kinetic energy is (γ − 1)mc² = mc², equal to its rest energy.
How is the Lorentz factor used in relativistic energy and momentum calculations?
The relativistic momentum is p = γmv and the total relativistic energy is E = γmc², where m is the rest mass. When γ = 1 (low speeds), these reduce to the classical expressions. The famous mass–energy relation E = mc² applies to a particle at rest; a moving particle has total energy γmc². The kinetic energy is (γ − 1)mc². These formulas are essential in nuclear physics, particle accelerator design, and understanding cosmic ray interactions.
Why can nothing with mass reach or exceed the speed of light according to the Lorentz factor?
As a massive object accelerates, γ increases and so does its relativistic momentum p = γmv and energy E = γmc². To reach exactly v = c, γ would have to become infinite, requiring infinite energy — which is physically impossible for any finite energy source. This is a hard limit built into Special Relativity, not just a technological barrier. Massless particles like photons always travel at exactly c and have γ undefined in the classical sense; they are described differently using the energy–momentum relation E = pc.