Lorentz Factor (γ) Calculator
Calculate the Lorentz factor gamma (γ) from velocity.
Find time dilation, length contraction, relativistic mass, and kinetic energy at any fraction of the speed of light.
The Lorentz Factor γ (gamma) = 1 / √(1 − v²/c²) Where v = velocity, c = speed of light (299,792,458 m/s). The Lorentz factor is the central quantity of Einstein’s special relativity — it appears in every relativistic formula. At low speeds (v « c), γ ≈ 1 and all relativistic effects vanish. As v → c, γ → ∞, which is why no massive object can reach the speed of light.
Time Dilation Moving clocks run slower: t’ = γ × t₀ t₀ = proper time (time measured in the moving frame), t’ = dilated time (time measured by stationary observer). At 99% of c: γ ≈ 7.09 → moving clocks run 7 times slower. This was directly measured using muons created by cosmic rays at the top of the atmosphere. Muons have a half-life of 2.2 µs, yet many survive the journey to Earth’s surface — their internal “clocks” slow down.
Length Contraction Moving objects appear shorter along the direction of motion: L = L₀ / γ L₀ = proper length, L = contracted length observed by stationary frame. At 99% of c: a 1-meter rod appears to be only 14 cm long. Contraction only occurs along the direction of motion — transverse dimensions are unchanged.
Relativistic Momentum and Mass Relativistic momentum: p = γ × m × v “Relativistic mass” (historical concept): m_rel = γ × m₀ Modern physics prefers to say momentum increases, not mass — the rest mass m₀ is invariant. Even at high speeds, E = mc² uses rest mass m₀, not relativistic mass.
Relativistic Kinetic Energy KE = (γ − 1) × m₀ × c² At low speeds this approximates ½mv² (classical kinetic energy). Total energy: E_total = γ × m₀ × c² = rest energy + kinetic energy. At the Large Hadron Collider (LHC), protons are accelerated to γ ≈ 7,461 at full energy.
Practical Applications GPS satellites: orbit at v ≈ 3.87 km/s (γ ≈ 1.0000000083). Without correction, GPS would drift ~7 µs/day from time dilation alone — causing ~2 km positional error. Particle accelerators: protons at CERN reach 99.9999991% of c (γ ≈ 7461). Cosmic rays: some particles hit Earth with γ > 10¹¹ — energies of ~50 joules in a single subatomic particle (the “Oh-My-God particle” detected over Utah in 1991). Synchrotron radiation: relativistic electrons emit X-rays used in medical imaging and materials research.