Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section B / B6 / a / i

The upper index identifies $P^{\alpha }$ as a contravariant Lorentz four-vector, obeying the rank-one contravariant tensor transformation law. Its components transform with the coordinate Jacobian,It is not invariant component by component. Its contracted square $P_{\alpha }{P}^{\alpha }=m^2{c}^2$, for metric signature $(+,-,-,-)$, is a Lorentz invariant. This distinguishes a contravariant transformation law from an invariant scalar.