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Lorentz factor (γ) — the relativistic scaling of time, length and mass

The Lorentz factor γ = 1/√(1−(v/c)²) quantifies how time, distance and energy scale at speeds near light. It appears in Lorentz transformations, time dilation, length contraction and particle physics.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 3, 2026

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Overview

The Lorentz factor, conventionally denoted by the Greek letter γ, is a dimensionless quantity that measures how measurements of time, length and related quantities change for observers in relative motion at speeds approaching the speed of light. It arises in Einstein's special theory of relativity and in the Lorentz transformations that relate space and time coordinates between inertial frames. The factor depends only on the speed v of the moving object and the universal constant c, the speed of light.

Definition and basic properties

Mathematically the Lorentz factor is written as strongly as a function of the velocity v: γ = 1 / sqrt(1 − (v/c)²). If one introduces β = v/c, the same expression becomes γ = 1/√(1−β²). At v = 0 the factor equals 1, and it increases without bound as v approaches c, tending to infinity in the limit v → c. For small speeds (β ≪ 1) γ is very close to 1; to leading order γ ≈ 1 + ½β².

$\gamma ={\frac {1}{\sqrt {1-({\frac {v}{c}})^{2}}}}$

Physical consequences

The Lorentz factor governs several well-known relativistic effects:

Time dilation: Clocks moving at speed v relative to an observer tick more slowly by the factor γ. A proper time interval Δτ measured by a clock moving with the object is related to the coordinate time Δt measured in the laboratory by Δt = γ Δτ.
Length contraction: Lengths measured along the direction of motion are reduced by 1/γ: L = L0/γ, where L0 is the proper length in the object's rest frame.
Energy and inertia: In relativistic dynamics the energy and momentum of particles depend on γ. Older literature used the term "relativistic mass" equal to γ times the rest mass, but modern practice favors keeping the invariant rest mass and attributing γ to energy and momentum formulas instead.

Historical context

The factor is named after Hendrik Lorentz, who, together with other late‑19th and early‑20th century physicists, developed transformations that left Maxwell's equations invariant and foreshadowed modern relativity. Henri Poincaré contributed mathematical refinements, and Albert Einstein incorporated the same structure into his 1905 formulation of special relativity, where γ appears naturally in the relation between moving frames. See also discussions of special relativity for the broader theoretical setting.

Examples and applications

The Lorentz factor is central in many practical and experimental contexts. Particle accelerators routinely accelerate particles to speeds where γ is large, affecting their lifetimes and trajectories. Atmospheric muons produced by cosmic rays reach the ground in greater numbers than classical estimates predict because their lifetimes are extended by γ as seen from the Earth frame. Global navigation satellite systems require relativistic corrections to clock rates, including effects proportional to γ, to maintain positional accuracy. For clear discussion of limiting regimes and common examples, see standard textbooks and reviews on relativistic kinematics and experimental tests of relativity relativistic speeds.

$\beta$

Notable facts and distinctions

Because γ grows without bound as speed approaches c, no massive object can reach or exceed the speed of light without requiring infinite energy according to special relativity. Numerical values are sometimes quoted for illustration: for example, γ = 2 corresponds to v ≈ 0.866c. The Lorentz factor appears throughout relativistic formulas (time intervals, spatial coordinates, momentum and energy) and is a compact way to express how classical intuitions about space and time must be modified at high velocities.

For further reading on derivations, experimental verifications and advanced uses, consult introductory texts on special relativity and resources that explain Lorentz transformations in more detail.

Lorentz factor for accelerations

The time derivative of γ $\gamma$ is interesting to formulate the relativistic form of Newton's second law $\vec F = m \vec a$ for accelerations in the direction of motion, since the relativistically correct relation is ${\vec {F}}={\tfrac {\mathrm {d} }{\mathrm {d} t}}{\vec {p}}$ over momentum. It holds: ${\vec p}=\gamma m{\vec v}$ .

It follows directly:

${\vec {F}}=\left({\frac {\mathrm {d} }{\mathrm {d} t}}\gamma \right)m{\vec {v}}+\gamma m{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}+\gamma {\vec {v}}{\frac {\mathrm {d} }{\mathrm {d} t}}m$

and one obtains for the time derivative of the Lorentz factor:

${\frac {\mathrm {d} }{\mathrm {d} t}}\gamma =\gamma ^{3}{\frac {\vec {v}}{c}}\cdot {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\vec {v}}{c}}$

and thus for the correct relationship between force and acceleration:

${\vec {F}}=m\gamma ^{3}\left({\frac {\vec {v}}{c}}\cdot {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\vec {v}}{c}}\right){\vec {v}}+\gamma m{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}+\gamma {\vec {v}}{\frac {\mathrm {d} }{\mathrm {d} t}}m.$

Lorentz factor as a function of momentum p

The Lorentz factor can also be expressed as:

$\gamma ={\sqrt {1+\left({\frac {\vec {p}}{mc}}\right)^{2}}}$

with

the relativistic momentum ${\vec {p}}$ of the object under consideration
its mass

This notation is mainly found in theoretical physics.

The equivalence can be proven by an equation with the "normal" Lorentz factor, which yields the relativistic momentum.

${\vec {p}}^{2}=\underbrace {(\gamma ^{2}-1)c^{2}} _{\left({\frac {1}{1-\beta ^{2}}}-{\frac {1-\beta ^{2}}{1-\beta ^{2}}}\right)c^{2}=\beta ^{2}c^{2}\gamma ^{2}}m^{2}=\gamma ^{2}v^{2}m^{2}$