Tachyons are hypothetical (and so-far undetected) particles that always travel faster than light.

(a) Show that all components of a tachyon's momentum four-vector are real if we assign the tachyon an imaginary mass, say \(m=i m_{0}\), where \(m_{0}\) is real.

(b) Then show that the invariant square of the momentum four-vector \(\eta_{\mu u} p^{\mu} p^{u}\) is necessarily positive. Thus the world of particles could be separated into three regimes: (i) Ordinary massive particles, with \(\eta_{\mu u} p^{\mu} p^{u}<0\); photons or other possible massless particles, with \(\eta_{\mu u} p^{\mu} p^{u}=0\); and tachyons, with \(\eta_{\mu u} p^{\mu} p^{u}>0\).