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)

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