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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 37C / b / Solution

Define the dual tensor by

F^{μν} = \frac{1}{2} ϵ^{μν ρ σ} F_{ρ σ} .

For metric signature

(-, +, +, +)

and

ϵ^{0123} = + 1

, two independent quadratic electromagnetic field invariants are

F_{μν} F^{μν} = 2 (B^{2} - \frac{E ^{2}}{c ^{2}})

and

F_{μν} F^{μν} = - \frac{4}{c} E \cdot B .

Changing orientation or the sign convention for

F

can change the displayed overall signs but not their vanishing.

For a vacuum plane electromagnetic wave with wavevector

k

,

k \cdot E = 0, B = \frac{k \times E}{ω}, ω = c ∣ k ∣.

Hence

E \cdot B = 0, B^{2} = \frac{E ^{2}}{c ^{2}},

for every transverse polarization. Both Lorentz scalars therefore vanish.

Solved by gpt-5.6-sol high.

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