Lorentz Factor Calculator

Created by Krishna Nelaturu
Last updated: Jul 01, 2022

Do you want to calculate the gamma factor of a moving object? Look no further than this Lorentz factor calculator! Input the object's speed, and you can find out the relativistic gamma.

If you're curious about the Lorentz factor, how to calculate it, and why it is essential, we promise you that this article is a must-read.

If you're exclusively interested in the Lorentz force, we direct you to our Lorentz Force Calculator.

What is Lorentz factor and why is it important?

Thanks to Einstein's theory of special relativity, we know that an object at rest and the same object in motion have different physical properties. The Lorentz factor (γ) is a number that represents this change in the physical properties of the object in motion, based on its speed.

The Lorentz factor (also called Lorentz term) is essential to determining the following transformations and physical properties of a moving object:

  • Lorentz transforms
  • Gravitational time dilation
  • Length contraction
  • Relativistic mass
  • Relativistic momentum
  • Relativistic kinetic energy

How do you calculate the gamma factor?

We can express the Lorentz factor or relativistic gamma by:

γ=11u2c2=cc2u2\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} = \frac{c}{\sqrt{c^2-u^2}}

where:

  • γ\gamma is the Lorentz factor (or special relativity gamma);
  • cc is the speed of light in vaccum, equal to 299792458 m/s or 186000 miles/s; and
  • uu is the object's speed.

We can rewrite this to get a more common form of the relativistic gamma:

γ=11u2c2=11β2\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}} = \frac{1}{\sqrt{1-\beta^2}}

Here, we introduce β\beta as the ratio of the object's speed to light speed:

β=uc=11γ2\beta = \frac{u}{c} = \sqrt{1 - \frac{1}{\gamma^2}}

For example, consider a particle moving at 10% of the speed of light. To calculate its special relativity gamma:

  1. Determine the particle's β\beta:
β=0.1cc=0.1\kern{4em} \small\beta = \frac{0.1c}{c} = 0.1
  1. Calculate the Lorentz factorγ\gamma:
γ=11β2=110.12γ=1.005\kern{4em} \small \begin{align*} \gamma &= \frac{1}{\sqrt{1-\beta^2}}\\ &= \frac{1}{\sqrt{1-0.1^2}}\\ \gamma&= 1.005 \end{align*}

Using this gamma, you can figure out the particle's relativistic mass, momentum, and kinetic energy.

Now, do you understand what the Lorentz factor is? It is a simple concept, but if you ever feel the calculation is giddying, our calculator is here to help.

How to use this Lorentz factor calculator

This Lorentz factor calculator is simple to use:

  • Enter the speed of the object u in the units you prefer, and the calculator will determine its β and the Lorentz factor γ.
  • Input the object's β, and the calculator will find its speed u and its relativistic gamma γ.
  • Give the object's Lorentz factor γ, and this calculator will show you its speed u and β.
Krishna Nelaturu
Speed of the object (u)
ft/s
Beta (β)
Gamma (γ)
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