Lorentz Factor Calculator Formula

Understand the math behind the lorentz factor calculator. Each variable explained with a worked example.

Use the Lorentz Factor Calculator

Formulas Used

Lorentz Factor (gamma)

gamma = 1 / sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2))

Beta (v/c)

beta = velocity / 2.998e8

1 - beta^2

one_minus_beta2 = 1 - pow(velocity, 2) / pow(2.998e8, 2)

Variables

VariableDescriptionDefault
velocityVelocity(m/s)269800000

How It Works

The Lorentz Factor

Gamma is the cornerstone of special relativity, appearing in time dilation, length contraction, and relativistic energy.

Formula

gamma = 1 / sqrt(1 - beta^2)

where beta = v/c.

  • gamma = 1 at rest
  • gamma = 1.15 at 0.5c
  • gamma = 7.09 at 0.99c
  • gamma approaches infinity as v approaches c

    • Worked Example

    Object at 0.9c.

    velocity = 269800000
    1. 01beta = v/c = 2.698e8 / 2.998e8 = 0.8999
    2. 02beta^2 = 0.8099
    3. 031 - beta^2 = 0.1901
    4. 04gamma = 1 / sqrt(0.1901) = 1 / 0.4360 = 2.294

    Frequently Asked Questions

    Why is gamma always at least 1?

    Because v can never exceed c, the denominator sqrt(1-v^2/c^2) is always between 0 and 1, making gamma at least 1.

    What is the physical meaning of gamma?

    It tells you the factor by which time dilates, length contracts, and inertia increases. A gamma of 2 means clocks run half as fast and lengths shrink by half.

    What gamma do particle accelerators achieve?

    The LHC accelerates protons to gamma of about 7500, meaning they travel at 99.999999% of light speed.

    Ready to run the numbers?

    Open Lorentz Factor Calculator