γ = 1/√(1−v²/c²) with 1/γ, βγ, and √(1−β²) for any subluminal speed — the single number behind all of special relativity.
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Key facts
Category
Science & Education
Input types
select, number
Output type
text
Sample coverage
4
API ready
Yes
Overview
The Lorentz Factor Calculator computes the relativistic factor γ (gamma) along with 1/γ, βγ, and √(1−β²) for any subluminal velocity. Whether you specify velocity as a fraction of the speed of light (β = v/c) or in meters per second (m/s), this tool instantly provides the core numerical values needed for special relativity calculations including time dilation, length contraction, and relativistic momentum.
When to use
Analyzing relativistic kinematics or particle accelerator beam parameters.
Solving physics problem sets involving time dilation, length contraction, or relativistic momentum.
Evaluating how close a subluminal speed is to the speed of light and understanding the corresponding relativistic divergence.
How it works
1Choose your input mode: speed as a fraction of light speed (β) or speed in meters per second (m/s).
2Enter the numeric velocity value below the speed of light (c ≈ 2.99792e8 m/s).
3The tool computes the Lorentz factor using γ = 1/√(1−v²/c²) and derives complementary values: 1/γ, βγ (normalized momentum p/mc), and √(1−β²).
Use cases
Physics education: demonstrating how γ diverges non-linearly as β approaches 1.0.
Accelerator physics: calculating beam momentum parameters (βγ) and relativistic time stretch for unstable particles.
Spacecraft concept analysis: finding time dilation factors and contraction scales for high-fraction c travel scenarios.
Examples
1. Standard High-Energy Beam Calculation
Physics Student
Background
Working on a modern physics assignment analyzing muons traveling through a detection tube at 99% the speed of light.
Problem
Calculate the time dilation factor and momentum factor for particles moving at β = 0.99.
How to use
Set speed input mode to 'Fraction of c (β)' and enter 0.99.
speedMode: "beta", beta: 0.99
Outcome
Obtains γ = 7.088812, 1/γ = 0.141067, and βγ = 7.017924, showing that moving clocks run at ~14.1% of their rest rate.
2. Ultra-Relativistic Particle Acceleration
Accelerator Engineer
Background
Evaluating beam parameters where particles are accelerated to 0.999c.
Problem
Determine the rapid scaling in γ and normalized momentum when adding an extra nine to the beam velocity.
How to use
Select 'Fraction of c (β)' and enter 0.999.
FAQ
What is the Lorentz factor (γ)?
The Lorentz factor, denoted by γ (gamma), is a quantity defined as γ = 1/√(1−v²/c²) that determines the scale of time dilation, length contraction, and relativistic mass-energy effects in special relativity.
What does the parameter β (beta) represent?
β is the normalized speed expressed as a fraction of the speed of light in a vacuum, β = v/c, ranging between 0 and 1 for subluminal motion.
Why is the product βγ significant?
βγ corresponds to the relativistic momentum scale p/(mc), which particle physicists frequently quote to describe a beam particle's momentum relative to its rest mass.
What is the physical meaning of 1/γ?
The reciprocal 1/γ equals √(1−β²) and represents the exact contraction ratio for lengths in the direction of motion as well as the rate at which moving clocks tick relative to stationary clocks.
Can this calculator accept speeds equal to or greater than c?
No. The Lorentz factor is defined strictly for subluminal speeds (v < c). At v = c, the denominator becomes zero, resulting in an undefined mathematical singularity.