Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / b

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Solution

The symmetric stress-energy tensor obtained by metric variation is

T^{μν} = - F^{μ ρ} F^{ν}_{ρ} + \frac{1}{4} η^{μν} F^{ρ σ} F_{ρ σ} + m^{2} (A^{μ} A^{ν} - \frac{1}{2} η^{μν} A^{ρ} A_{ρ}) .

(1)

It is conserved on shell. With signature

(+ - - -)

its energy density is

T^{00} = \frac{1}{2} (E^{2} + B^{2}) + \frac{m ^{2}}{2} {(A^{0})^{2} + A^{2}} \geq 0.

(2)

The canonical stress-energy tensor differs from this symmetric tensor by the divergence of

F^{μ ρ} A^{ν}

plus terms proportional to the field equation. Thus its integrated energy agrees after discarding the corresponding total spatial derivative, and positivity holds on shell.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / b

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