Lorentz Factor (γ) Calculator
Calculate the Lorentz factor from velocity.
Find time dilation, length contraction, and relativistic mass 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.