quantity

lorentz factor

A factor describing how time, length, and relativistic mass change for an object moving at high speed.
9 related articles · 6 formulas

Description

The Lorentz factor, denoted γ (gamma), is the mathematical heart of special relativity. It quantifies how much time slows down, lengths contract, and relativistic mass increases for an object moving relative to an observer. At everyday speeds, γ is essentially 1, meaning relativistic effects are negligible; as speed approaches the speed of light, γ grows without bound, making these effects dramatic. It appears in nearly every relativistic equation, from the energy-momentum relation E = γmc² to the transformation of electric and magnetic fields.

History

The factor was introduced by Hendrik Lorentz in 1904 as part of his theory of electrons, and later incorporated into Einstein's special relativity in 1905. Lorentz derived it to explain the null result of the Michelson–Morley experiment, though Einstein's derivation from postulates was more fundamental.

How it works

The factor is defined as γ = 1 / √(1 − v²/c²), where v is the object's speed and c is the speed of light. When v is small compared to c, the denominator is close to 1, so γ ≈ 1. As v approaches c, the denominator shrinks toward zero, so γ skyrockets. This single number scales time intervals (time dilation), lengths (length contraction), and momentum in a consistent way, ensuring that the speed of light remains constant for all observers.

In practice

The Lorentz factor is essential for GPS satellite corrections, particle accelerator design (e.g., at CERN), and calculating the energy of cosmic rays.

Fun fact

At 99.999999% of the speed of light, the Lorentz factor is about 7,000, meaning time for a particle runs 7,000 times slower than for a stationary observer.

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Formulas

$$K > (\gamma - 1) m c^2$$
Particle kinetic energy exceeds rest mass energy threshold.
What is in the formula
Kvariablekinetic energy — kinetic energy joule
γvariableLorentz factor — lorentz factor
mvariablerest mass — mass kilogram
cvariablespeed of light — speed of light metre per second
cconstantThe speed of light in vacuum, a fundamental constant of nature. — speed of light = 299792458 metre per second
$$L = 4\pi R^2 n \gamma m c^3$$
Limits luminosity from kinetic energy flux.
What is in the formula
LvariableLuminosity (total power emitted) — luminosity watt
RvariableRadius of the emitting region — radius metre
nvariableNumber density of particles — number density per cubic metre
γvariableLorentz factor of the particles — lorentz factor
mvariableParticle rest mass — mass kilogram
cconstantSpeed of light in vacuum, the maximum speed for any particle. — speed of light = 299792458 metre per second
$$\Gamma = \frac{1}{\sqrt{1 - \beta^2}}$$
Factor relating time dilation and length contraction in relativity.
What is in the formula
ΓvariableLorentz factor — lorentz factor
βvariablespeed in units of c — velocity
$$\nu_c = \frac{3}{2}\gamma^2 \frac{e B}{2\pi m_e c}$$
Frequency of radiation from relativistic electrons in magnetic field.
What is in the formula
\nu_cvariableCritical synchrotron frequency — frequency hertz
\gammavariableLorentz factor of electron — lorentz factor
evariableElementary charge — elementary charge coulomb
BvariableMagnetic field strength — magnetic flux density tesla
m_evariableElectron rest mass — electron mass kilogram
cvariableSpeed of light in vacuum — speed of light metre per second
$$\delta = \frac{1}{\Gamma (1 - \beta \cos\theta)}$$
Relativistic Doppler factor for a moving source.
What is in the formula
\deltavariableDoppler factor — doppler factor
\GammavariableLorentz factor — lorentz factor
\betavariablevelocity in units of c — velocity
\thetavariableangle between source velocity and line of sight — angle radian
$$-\frac{dE}{dx} = \frac{4\pi N_A r_e^2 m_e c^2 \rho z^2}{A \beta^2} \left[ \ln\left(\frac{2m_e \gamma^2 v^2 W_{\text{max}}}{I^2}\right) - 2\beta^2 - \delta - 2\frac{C}{Z} \right]$$
Energy loss of charged particles due to ionization in matter.
What is in the formula
EvariableKinetic energy of the charged particle — energy joule
xvariablePath length in the material — length metre
\rhovariableDensity of the absorbing material — density kilogram per cubic metre
zvariableCharge number of the incident particle — charge number
AvariableAtomic mass number of the absorber — atomic mass number
\betavariableSpeed of the particle relative to the speed of light — velocity ratio
\gammavariableLorentz factor of the particle — lorentz factor
vvariableSpeed of the particle — velocity metre per second
W_{\text{max}}variableMaximum energy transfer in a single collision — energy joule
IvariableMean excitation potential of the absorber — energy joule
\deltavariableDensity correction factor — density correction
CvariableShell correction factor — shell correction
ZvariableAtomic number of the absorber — atomic number
N_AconstantNumber of constituent particles per mole — avogadro constant = 6.02214076e23 per mole
r_econstantRadius of a classical electron — classical electron radius = 2.8179403262e-15 metre
m_econstantRest mass of the electron — electron mass = 9.1093837015e-31 kilogram
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
-\frac{d}{dx}operatorNegative derivative with respect to path length — negative derivative
\lnoperatorNatural logarithm — natural logarithm

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