 Solution

ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2021/section-b/b6/a/i/solution

The upper index identifies

P^{α}

as a contravariant Lorentz four-vector, obeying the rank-one contravariant tensor transformation law. Its components transform with the coordinate Jacobian,

P^{' α} = \frac{\partial x ^{' α}}{\partial x ^{β}} P^{β} .

It is not invariant component by component. Its contracted square

P_{α} P^{α} = m^{2} c^{2}

(+, -, -, -)

, is a Lorentz invariant. This distinguishes a contravariant transformation law from an invariant scalar.

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