Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 2 / b / ii

At the chosen event use the orthonormal frame in spacetime from the hint. The electromagnetic energy density is

ρ = T_{00} = \frac{1}{2} (∣ E ∣^{2} + ∣ B ∣^{2}),

S = E \times B

. Depending on the index convention, the required contraction is

ρ - w S_{1}

ρ + w S_{1}

. In either case,

T_{ab} V^{a} W^{b} \geq ρ - ∣ S ∣ \geq \frac{1}{2} (∣ E ∣^{2} + ∣ B ∣^{2}) - ∣ E ∣∣ B ∣ = \frac{1}{2} (∣ E ∣ - ∣ B ∣)^{2} \geq 0.

Rescaling

V

and rotating the spatial frame covers arbitrary future timelike

V

and future causal

W

. Thus the Maxwell field satisfies the dominant energy condition.

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