T^{a}_{a} = F^{a c} F_{a c} - \frac{1}{4} δ_{a}^{a} F_{c d} F^{c d} = 0.

\nabla_{[a} F_{b c]} = 0

with

F^{a c}

gives

F^{a c} \nabla_{a} F_{b c} = \frac{1}{4} \nabla_{b} (F_{c d} F^{c d}),

which cancels the derivative of the trace term. Hence

\nabla_{a} T^{a}_{b} = 0, T^{a}_{a} = 0 .

For

J^{a} = T^{ab} V_{b}

T^{ab}

now imply

\nabla_{a} J^{a} = T^{ab} \nabla_{a} V_{b} = \frac{1}{2} T^{ab} (\nabla_{a} V_{b} + \nabla_{b} V_{a}) = \frac{1}{2} T^{ab} (L_{V} g)_{ab} .

ii

Solution

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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