Relativistic Energy-momentum Relation

Begin with the relativistic momentum and energy: $p = \frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}$

$E = \frac{m{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}.$ Derive the relativistic energy-momentum relation: ${E}^{2} = {(pc)}^{2} + {(m{c}^{2})}^{2}.$

Solution

There are some technical details we must note:

1) The momentum $p$ and velocity $v$ are vectors.

2) The gamma factor is usually written as $\gamma = \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}.$

However, it is more accurate to write $\gamma = \frac{1}{\sqrt{1-\frac{v\cdot v}{{c}^{2}}}}$ or $\gamma = \frac{1}{\sqrt{1-\frac{{\left|v\right|}^{2}}{{c}^{2}}}}.$

Part 1

We begin with the relativistic momentum and energy: $p = \frac{mv}{\sqrt{1-{v}^{2}/{c}^{2}}}$

$E = \frac{m{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}.$

With a little algebra we discover that

$\frac{p}{E} = \frac{v}{{c}^{2}}.$

Part 2

Now we square the relativistic energy and rearrange the equation

${E}^{2} = {E}^{2}\frac{{v}^{2}}{{c}^{2}} +{(m{c}^{2})}^{2}.$

Using the relation derived in Part 1, we find that

${E}^{2} = {(pc)}^{2} + {(m{c}^{2})}^{2}.$

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

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Comments

it can also be derived from Rayleigh jean's experimental equation

nibedan mukherjee - 6 years, 9 months ago

Really? I can't see how the Rayleigh-Jean experiment on radiation has anything to do with the gamma factor.

Steven Zheng - 6 years, 9 months ago

What is the simplest set of assumptions you need to arrive at your first equation?

Josh Silverman Staff - 6 years, 9 months ago

Good point. I was debating on whether I should include the relativistic momentum and energy. I will add them.

I was thinking of how to arrive at those expressions in the first place.

@Josh Silverman – Ah. That would be a different exercise. Consider the light-clock thought experiment.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$