# Four-momentum: example

Required math: calculus

Required physics: special relativity

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 3; Problem 3.3.

As a simple example of the calculation of four-momentum, suppose we have a neutral pion ${\pi^{0}}$ moving at a speed ${v=\frac{3}{5}}$ at an angle of ${38.7^{\circ}}$ in the first quadrant. The rest mass of a ${\pi^{0}}$ is ${m=135}$ MeV.

The four-momentum is defined as

$\displaystyle \mathbf{p}=\gamma m\left[1,v_{x},v_{y},v_{z}\right] \ \ \ \ \ (1)$

where ${\gamma=1/\sqrt{1-v^{2}}}$.

The angle is chosen because ${\cos38.7^{\circ}=\frac{4}{5}}$ and ${\sin38.7^{\circ}=\frac{3}{5}}$, so

 $\displaystyle v_{x}$ $\displaystyle =$ $\displaystyle \frac{12}{25}\ \ \ \ \ (2)$ $\displaystyle v_{y}$ $\displaystyle =$ $\displaystyle \frac{9}{25}\ \ \ \ \ (3)$ $\displaystyle \gamma$ $\displaystyle =$ $\displaystyle \frac{5}{4} \ \ \ \ \ (4)$

Thus

$\displaystyle \mathbf{p}=\left[168.75,81,60.75,0\right] \ \ \ \ \ (5)$