In this note I will present three interesting (and useful) identities of the Lorentz factor (which most people call the gamma factor).

\(\frac{d\gamma}{dv} = \frac{{\gamma}^{3}v}{{c}^{2}}\)

\({\gamma}^{2} - 1 = {(\beta \gamma)}^{2}\)

\(\frac{d(\gamma v)}{dv} = {\gamma}^{3}\)

Before we begin, I would like to make it clear that \(\beta = \frac{v}{c}\) and \(\gamma = \frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\).

**Identity 1**

\[\frac{d\gamma}{dv} = \frac{-1}{2}{\left(1-{\frac{{v}^{2}}{{c}^{2}}}\right)}^{-3/2}\frac{2v}{{c}^{2}}\]

\[\frac{d\gamma}{dv} = \frac{{\gamma}^{3}v}{{c}^{2}}\]

**Identity 2**

\[{\gamma}^{2} - 1 = {\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}^{-1} -1\]

\[{\gamma}^{2} - 1 = {\left(1-\frac{{v}^{2}}{{c}^{2}}\right)}^{-1} -1\]

\[{\gamma}^{2} - 1 = \frac{{c}^{2}}{{c}^{2} -{v}^{2}} - 1\]

\[{\gamma}^{2} - 1 = \frac{{c}^{2}}{{c}^{2} -{v}^{2}} - \frac{{c}^{2} -{v}^{2}}{{c}^{2} -{v}^{2}}\]

\[{\gamma}^{2} - 1 = \frac{{v}^{2}}{{c}^{2}}\frac{1}{1 - \frac{{v}^{2}}{{c}^{2}}}\]

\[{\gamma}^{2} - 1 = {(\beta \gamma)}^{2}\]

**Identity 3**

\[\frac{d(\gamma v)}{dv} = \frac{d\gamma}{dv}v+\gamma \]

\[\frac{d(\gamma v)}{dv} = \frac{{\gamma}^{3}{v}^{2}}{{c}^{2}} + \gamma \]

\[\frac{d(\gamma v)}{dv} = \gamma \left(\frac{{\gamma}^{2}{v}^{2}}{{c}^{2}} + 1\right)\]

\[\frac{d(\gamma v)}{dv} = \gamma \left(\frac{{v}^{2}}{{c}^{2} - {v}^{2}} + \frac{{c}^{2} -{v}^{2}}{{c}^{2} - {v}^{2}}\right)\]

\[\frac{d(\gamma v)}{dv} = \gamma \left(\frac{{c}^{2}}{{c}^{2} - {v}^{2}} \right)\]

\[\frac{d(\gamma v)}{dv} = \gamma ({\gamma}^{2})\]

\[\frac{d(\gamma v)}{dv} = {\gamma}^{3}\]

Check out my other notes at Proof, Disproof, and Derivation

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestYou need to fix up the LaTeX for the last line. Apart from that, awesome.

Log in to reply

What's wrong with it? I can't see it. By last line do you mean the very last line?

Log in to reply

Yeah, there was something wrong but it's fixed now.

Log in to reply

Log in to reply

Log in to reply

Seems pretty straightforward, does these formulas have an importance in physics ? and please answer me in English LOL (I do not know much physics).

Log in to reply

Yes! If you read my derivation of E=mc^2 then identity 3 is used. They often occur in relativistic dynamics. Pretty much whenever calculus becomes important in relativity, these identities will be convenient.

Log in to reply