Write down the components of the position four-vector $x_{\mu}$. Hence find the components of the four-momentum $p_{\mu}=M U_{\mu}$ of a particle of mass $M$, where $U_{\mu}=d x_{\mu} / d \tau$, with $\tau$ being the proper time.

The particle, viewed in a frame in which it is initially at rest, disintegrates leaving a particle of mass $m$ moving with constant velocity together with other remnants which have a total three-momentum $\mathbf{p}$ and energy $E$. Show that

$$m=\sqrt{\left(M-\frac{E}{c^{2}}\right)^{2}-\frac{|\mathbf{p}|^{2}}{c^{2}}}$$