Classical Electrodynamics · Problem 12.16

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · proca-stress-energy-tensor

Problem

(a) Starting with the Proca Lagrangian density (12.91) and following the same procedure as for the electromagnetic fields, show that the symmetric stress-energy-momentum tensor for the Proca fields is Θαβ=14π[gαγFγλFλβ+14gαβFλνFλν+μ2(AαAβ−12gαβAλAλ)]\Theta^{\alpha\beta} = \frac{1}{4\pi}\left[g^{\alpha\gamma}F_{\gamma\lambda}F^{\lambda\beta} + \frac{1}{4}g^{\alpha\beta}F_{\lambda\nu}F^{\lambda\nu} + \mu^2\left(A^\alpha A^\beta - \frac{1}{2}g^{\alpha\beta}A_\lambda A^\lambda\right)\right] (b) For these fields in interaction with the external source JβJ^\beta, as in (12.91), show that the differential conservation laws take the same form as for the electromagnetic fields, namely, ∂αΘαβ=JλFλβc\partial_\alpha \Theta^{\alpha\beta} = \frac{J_\lambda F^{\lambda\beta}}{c} (c) Show explicitly that the time-time and space-time components of Θαβ\Theta^{\alpha\beta} are Θ00=18π[E2+B2+μ2(A0A0+A⋅A)]Θi0=14π[(E×B)i+μ2AiA0]\begin{aligned} \Theta^{00} &= \frac{1}{8\pi}\left[E^2 + B^2 + \mu^2(A^0 A^0 + \mathbf{A}\cdot\mathbf{A})\right]\\ \Theta^{i0} &= \frac{1}{4\pi}\left[(\mathbf{E}\times\mathbf{B})_i + \mu^2 A^i A^0\right] \end{aligned}

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