A factor describing how time, length, and relativistic mass change for an object moving at high speed.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.
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.
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.
The Lorentz factor is essential for GPS satellite corrections, particle accelerator design (e.g., at CERN), and calculating the energy of cosmic rays.
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.
| K | variable | kinetic energy — kinetic energy joule |
| γ | variable | Lorentz factor — lorentz factor |
| m | variable | rest mass — mass kilogram |
| c | variable | speed of light — speed of light metre per second |
| c | constant | The speed of light in vacuum, a fundamental constant of nature. — speed of light = 299792458 metre per second |
| L | variable | Luminosity (total power emitted) — luminosity watt |
| R | variable | Radius of the emitting region — radius metre |
| n | variable | Number density of particles — number density per cubic metre |
| γ | variable | Lorentz factor of the particles — lorentz factor |
| m | variable | Particle rest mass — mass kilogram |
| c | constant | Speed of light in vacuum, the maximum speed for any particle. — speed of light = 299792458 metre per second |
| Γ | variable | Lorentz factor — lorentz factor |
| β | variable | speed in units of c — velocity |
| \nu_c | variable | Critical synchrotron frequency — frequency hertz |
| \gamma | variable | Lorentz factor of electron — lorentz factor |
| e | variable | Elementary charge — elementary charge coulomb |
| B | variable | Magnetic field strength — magnetic flux density tesla |
| m_e | variable | Electron rest mass — electron mass kilogram |
| c | variable | Speed of light in vacuum — speed of light metre per second |
| \delta | variable | Doppler factor — doppler factor |
| \Gamma | variable | Lorentz factor — lorentz factor |
| \beta | variable | velocity in units of c — velocity |
| \theta | variable | angle between source velocity and line of sight — angle radian |
| E | variable | Kinetic energy of the charged particle — energy joule |
| x | variable | Path length in the material — length metre |
| \rho | variable | Density of the absorbing material — density kilogram per cubic metre |
| z | variable | Charge number of the incident particle — charge number |
| A | variable | Atomic mass number of the absorber — atomic mass number |
| \beta | variable | Speed of the particle relative to the speed of light — velocity ratio |
| \gamma | variable | Lorentz factor of the particle — lorentz factor |
| v | variable | Speed of the particle — velocity metre per second |
| W_{\text{max}} | variable | Maximum energy transfer in a single collision — energy joule |
| I | variable | Mean excitation potential of the absorber — energy joule |
| \delta | variable | Density correction factor — density correction |
| C | variable | Shell correction factor — shell correction |
| Z | variable | Atomic number of the absorber — atomic number |
| N_A | constant | Number of constituent particles per mole — avogadro constant = 6.02214076e23 per mole |
| r_e | constant | Radius of a classical electron — classical electron radius = 2.8179403262e-15 metre |
| m_e | constant | Rest mass of the electron — electron mass = 9.1093837015e-31 kilogram |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| -\frac{d}{dx} | operator | Negative derivative with respect to path length — negative derivative |
| \ln | operator | Natural logarithm — natural logarithm |

