Lorentz Factor Calculator
Compute the Lorentz factor γ = 1/√(1 − (v/c)²), the central quantity of special relativity that scales time dilation, length contraction, relativistic mass, momentum, and energy. Returns a dimensionless number ≥ 1.
Last updated: May 2026
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About this calculator
The Lorentz factor γ = 1/√(1 − (v/c)²) appears in nearly every formula of special relativity: dilated time t = γ·τ, contracted length L = L₀/γ, relativistic momentum p = γ·m·v, total energy E = γ·m·c², and relativistic mass m_rel = γ·m. It is named for Hendrik Lorentz, who derived it as a coordinate-transformation factor before Einstein's relativistic interpretation. The factor is 1 at v = 0 and grows monotonically with velocity: γ = 1.005 at v = 0.1c, γ = 1.155 at v = 0.5c, γ = 2.294 at v = 0.9c, γ = 7.089 at v = 0.99c, γ = 22.37 at v = 0.999c, and γ → ∞ as v → c. The formula assumes special relativity (flat spacetime, no gravity) and an inertial reference frame. Edge cases: at v = 0 the formula evaluates to 1 (no relativistic correction). For v ≥ c, (v/c)² ≥ 1 makes the square root imaginary or zero — massive particles cannot reach light speed because doing so would require infinite energy (E = γmc² diverges). The calculator hardcodes c = 299,792,458 m/s; for non-metric inputs (v in km/h or as a fraction of c) you would convert first. The Lorentz factor is also the time-dilation factor between two inertial frames moving with relative velocity v, and the same factor governs the relativistic Doppler effect for light.
How to use
Example 1 — particle-accelerator-like velocity. v = 0.99c = 2.969 × 10⁸ m/s. Step 1: v/c = 0.99. Step 2: (v/c)² = 0.9801. Step 3: 1 − 0.9801 = 0.0199. Step 4: √0.0199 ≈ 0.1411. Step 5: γ = 1 / 0.1411 ≈ 7.089. Verify: at 99% the speed of light, γ ≈ 7.09 — a clock moving at this speed ticks 7× slower than a stationary clock, and the particle's kinetic energy is (γ−1)·mc² ≈ 6.09·mc², about 6× its rest energy ✓. Example 2 — extreme ultra-relativistic. v = 0.9999c. Step 1: v/c = 0.9999. Step 2: (v/c)² = 0.99980001. Step 3: 1 − 0.99980001 ≈ 0.00019999. Step 4: √0.00019999 ≈ 0.01414. Step 5: γ ≈ 1 / 0.01414 ≈ 70.71. Verify: at 0.9999c, γ ≈ 70.7. The Large Hadron Collider accelerates protons to γ ≈ 7,000 (v ≈ c × √(1 − 1/γ²) ≈ 0.99999999c), corresponding to a kinetic energy of ~6.5 TeV per proton — roughly 7,000× the proton's rest-mass energy of 938 MeV ✓.
Frequently asked questions
Why does γ grow without bound as v approaches c?
Because the denominator √(1 − (v/c)²) approaches zero as v approaches c, and 1 divided by a number approaching zero diverges. Physically this reflects the impossibility of accelerating a massive object to light speed: relativistic energy E = γ·m·c² also diverges, meaning infinite energy would be required. As v approaches c, each additional unit of energy delivered to the object goes into increasing γ rather than increasing v meaningfully — the LHC, for example, has γ around 7000 for protons, but the protons travel at only about 30 cm/s slower than c. This is fundamentally different from classical kinematics where kinetic energy is ½mv², which grows quadratically with v with no upper bound on v itself. The asymptotic structure also explains why massless particles (photons) always travel at exactly c — they have no rest frame, no rest mass, and γ is undefined; instead, their energy and momentum are characterised by E = hν and p = h/λ, separate from any γ factor.
How does γ enter the formulas for time, length, momentum, and energy?
It scales each as: dilated time t = γ·τ (moving clocks tick slower); contracted length L = L₀/γ (moving objects appear shorter along the direction of motion); relativistic momentum p = γ·m·v (replacing classical p = m·v); total energy E = γ·m·c² (the famous E = mc² is the rest energy when v = 0, giving γ = 1, so total energy is γ·m·c² with kinetic energy KE = (γ − 1)·m·c²); relativistic mass m_rel = γ·m (an older convention some authors still use). Notice that γ ≥ 1 always: all four scaled quantities are larger than their non-relativistic counterparts (except length, which is smaller because it is divided by γ). The interplay between these is rich: at low v (γ ≈ 1) all formulas reduce to Newton; at high v (γ ≫ 1) the corrections become dramatic and dominate any phenomenon involving fast-moving objects. The full Lorentz transformation matrix itself contains γ in every entry, reflecting how γ governs the mixing of space and time coordinates between inertial frames. Pedagogically, learning relativity often centers on building intuition for how γ rescales each classical quantity as you cross into the relativistic regime.
What's the relationship between γ and rapidity, and why does the latter sometimes simplify problems?
Rapidity φ (sometimes ω or η) is defined by v/c = tanh(φ), or equivalently γ = cosh(φ) and γ·v/c = sinh(φ). Where velocities don't add linearly (relativistic velocity-addition: u' = (u + v)/(1 + uv/c²) is awkward), rapidities do: φ_total = φ₁ + φ₂, just like Galilean velocity addition. This is why rapidity is the natural angle parameter for Lorentz boosts, analogous to ordinary rotation angles for spatial rotations. In particle physics, the rapidity along the beam axis (often denoted y) is invariant under boosts along that axis, simplifying jet and decay-product analysis. For low velocities (φ ≪ 1), tanh(φ) ≈ φ, so v ≈ φ·c — rapidity coincides with classical velocity in the non-relativistic limit. For high velocities, rapidity grows without bound while velocity asymptotes to c, making rapidity a more 'natural' relativistic measure of how fast something is moving. In cosmology, the concept generalises to the Hubble flow and to recession-velocity descriptions in expanding spacetime.
What are the common mistakes when computing the Lorentz factor?
The biggest mistake is using non-SI units without conversion — v must be in the same units as c (m/s) before computing v/c. The second is forgetting that γ → 1 (not 0) as v → 0; at low velocities the factor is just barely above 1, and inexperienced students sometimes incorrectly write γ ≈ v/c. The third is using γ in non-inertial or curved-spacetime contexts where it doesn't directly apply; for accelerating objects use instantaneous rest frames or general-relativistic four-velocity, and for gravitational fields use general relativity's metric. People also confuse the Lorentz factor with the Doppler factor √((1−β)/(1+β)) (longitudinal) — they're related but distinct. Floating-point precision matters: at v ≥ 0.9999999c, double-precision arithmetic starts losing significant digits in the 1 − (v/c)² subtraction; for very high γ, use γ = 1/(2·√(1 − v/c)·√(1 + v/c)) and similar rewrites or use extended precision. Finally, applying γ to systems with non-uniform velocity (extended objects with different parts at different speeds) requires integration, not a single factor.
When should I not use this calculator?
Do not use it for accelerating reference frames — these require instantaneous-rest-frame analysis or general-relativistic four-velocity, not a single γ. It is not appropriate for problems involving gravity, where general relativity replaces special relativity; use Schwarzschild or other metric solutions. Do not use it for problems with v ≥ c, which is unphysical for massive objects; the formula gives NaN or zero. It is unsuitable for massless particles (photons, gluons), which have no rest frame and infinite γ; their kinematics use E = pc and frequency/wavelength directly. For very high γ (above ~10⁸), use extended-precision arithmetic to avoid catastrophic cancellation in the 1 − (v/c)² subtraction. Avoid it for cosmological problems involving the expansion of the universe; recession velocities of distant galaxies are not Lorentz boosts but coordinate effects from spacetime expansion. Finally, the calculator's hardcoded c works only for SI inputs (m/s); for other units (km/h, c-fractions, eV-velocity equivalents) convert first.