Lorentz Factor Calculator
The Lorentz factor (symbol: gamma) is the cornerstone of Einstein's special relativity. It quantifies how dramatically time, length, and relativistic momentum differ between a moving object and a stationary observer. At the speeds of everyday life, gamma is so close to 1 that relativistic effects are immeasurable. But push a particle to 90% of the speed of light and gamma reaches 2.29, meaning a moving clock ticks at less than half the rate of a stationary one. At 99% of c, gamma climbs to 7.09. At 99.9% of c it exceeds 22. This calculator accepts velocity as a fraction of c (beta), in metres per second, or in kilometres per second, then returns the exact Lorentz factor along with the resulting time dilation and length contraction. The speed of light used is the 2019 SI-defined exact value: c = 299,792,458 m/s. Results are suitable for physics coursework, exam preparation, and quick engineering sanity checks.
Enter a velocity below to calculate gamma.
Formula
The Lorentz factor is defined as:
gamma = 1 / sqrt(1 - beta^2) where beta = v / c
Here v is the speed of the moving object and c = 299,792,458 m/s is the exact SI-defined speed of light. Beta is the dimensionless velocity ratio. When beta = 0 (at rest), gamma = 1 and there are no relativistic effects. As beta approaches 1, gamma grows without limit.
Time dilation
t_moving = t_rest / gamma
A clock moving at velocity v ticks more slowly. An interval of 1 second measured by a stationary observer corresponds to only 1/gamma seconds on the moving clock (proper time). Equivalently, a stationary observer watching a moving clock sees it run gamma times slower.
Length contraction
L_moving = L_rest / gamma
An object moving at velocity v appears contracted along its direction of motion by factor 1/gamma. A rod of rest length 1 metre appears only 1/gamma metres long to a stationary observer.
Worked example
A spacecraft travels at v = 0.8c (80% of the speed of light). Beta = 0.8.
gamma = 1 / sqrt(1 - 0.8^2) = 1 / sqrt(1 - 0.64) = 1 / sqrt(0.36) = 1 / 0.6 = 1.666667
Time dilation: 1 second on the spacecraft equals 1.666667 seconds for an Earth observer. Length contraction: a 1-metre rod on the spacecraft appears 0.6 metres long to an Earth observer.
Reference table: gamma at common velocities
The table below shows how the Lorentz factor rises as velocity approaches c. Values are computed from the exact formula.
| Velocity (fraction of c) | Beta (v/c) | Gamma | Time dilation factor | Length contraction factor |
|---|---|---|---|---|
| 0.1c (10%) | 0.1 | 1.005038 | 1 s = 1.005038 s | 1 m = 0.994987 m |
| 0.5c (50%) | 0.5 | 1.154701 | 1 s = 1.154701 s | 1 m = 0.866025 m |
| 0.9c (90%) | 0.9 | 2.294157 | 1 s = 2.294157 s | 1 m = 0.435890 m |
| 0.99c (99%) | 0.99 | 7.088812 | 1 s = 7.088812 s | 1 m = 0.141067 m |
| 0.999c (99.9%) | 0.999 | 22.366272 | 1 s = 22.366272 s | 1 m = 0.044699 m |
Frequently asked questions
What does the Lorentz factor (gamma) actually mean?
The Lorentz factor gamma tells you by how much time, length, and relativistic mass change for an object moving at velocity v relative to a stationary observer. At everyday speeds gamma is effectively 1 (no noticeable effect). As v approaches c, gamma rises steeply toward infinity, meaning time slows dramatically and lengths contract to nearly zero.
Why can nothing with mass reach the speed of light?
As an object with mass accelerates toward c, the Lorentz factor grows without bound. Kinetic energy is proportional to (gamma - 1) * m * c^2, so reaching c would require infinite energy. The math shows gamma becomes undefined at v = c for any massive object, making it a physical impossibility rather than merely a practical engineering challenge.
How does the Lorentz factor affect GPS satellites?
GPS satellites orbit at about 14,000 km/h (roughly 0.0000131c). Special relativity causes their clocks to tick slightly slower than ground clocks by about 7 microseconds per day. General relativity (gravitational time dilation) adds a further 45 microseconds per day in the opposite direction. Both corrections are applied continuously, and without them GPS position errors would accumulate at roughly 10 km per day.
What is the twin paradox and how does gamma explain it?
In the twin paradox, one twin travels at high speed to a distant star and returns while the other stays on Earth. The traveling twin's clock runs slower by factor gamma throughout the journey, so on return that twin is measurably younger. This is not actually a paradox: the asymmetry arises because the traveling twin must accelerate and decelerate (changing inertial frames), which breaks the symmetry of the situation.
Where does the Lorentz factor appear in particle accelerators?
Particle accelerators like the LHC push protons to gamma values exceeding 7,000. At those energies, a proton's relativistic momentum and energy are about 7,000 times higher than at rest. Engineers must account for gamma when designing magnetic focusing systems, calculating beam energies, and predicting synchrotron radiation losses. Muons produced in accelerators also live far longer than their 2.2 microsecond rest lifetime precisely because of their high gamma.
Sources
- NIST CODATA 2018: Speed of light in vacuum, c = 299,792,458 m/s (exact, SI-defined). physics.nist.gov/cgi-bin/cuu/Value?c
- Einstein, A. (1905). "Zur Elektrodynamik bewegter Korper" (On the Electrodynamics of Moving Bodies). Annalen der Physik, 17, 891. Referenced via NIST Physics resources at physics.nist.gov.
- NIST Special Publication 330 (2019 Edition): The International System of Units (SI). www.nist.gov/pml/special-publication-330
Reviewed by the CalculatorHub team, edited by James Graham. Last reviewed 14 June 2026.