The four-velocity $U_{\mu}$ of a particle of rest mass $m$ is defined by $U_{\mu}=d x_{\mu} / d \tau$, where $\tau$ is the proper time (the time as measured in the particle's rest frame). Derive the expression for each of the four components of $U_{\mu}$ in terms of the components of the three-velocity and the speed of light, $c$.

Show that $U \cdot U=c^{2}$ for an appropriately defined scalar product.

The four-momentum, $p_{\mu}=m U_{\mu}$, of a particle of rest mass $m$ defines a relativistic generalisation of energy and momentum. Show that the standard non-relativistic expressions for the momentum and kinetic energy of a particle with mass $m$ travelling with velocity $v$ are obtained in the limit $v / c \ll 1$. Show also how the concept of a rest energy equal to $m c^{2}$ emerges.