A particle of rest mass $m$ and four-momentum $P=(E / c, \mathbf{p})$ is detected by an observer with four-velocity $U$, satisfying $U \cdot U=c^{2}$, where the product of two four-vectors $P=\left(p^{0}, \mathbf{p}\right)$ and $Q=\left(q^{0}, \mathbf{q}\right)$ is $P \cdot Q=p^{0} q^{0}-\mathbf{p} \cdot \mathbf{q}$.

Show that the speed of the detected particle in the observer's rest frame is

$$v=c \sqrt{1-\frac{P \cdot P c^{2}}{(P \cdot U)^{2}}}$$