A region far from any charge contains a static electromagnetic field with…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch The Einstein field equations and the Newtonian limit

Problem

A region far from any charge contains a static electromagnetic field with Jμ=0J^\mu=0 (source-free) and proper energy density ρEM=4.0×1010 J m−3\rho_{\rm EM}=4.0\times10^{10}\,\mathrm{J\,m^{-3}} measured in the local rest frame, with stress-energy tensor TEMμνT^{\mu\nu}_{\rm EM} satisfying Tμμ=0T^\mu{}_\mu=0 identically. Take Λ=0\Lambda=0. ((a)) Using Tμμ=0T^\mu{}_\mu=0, show that the Newtonian-limit field equation for this source is ∇2Φ=8πGρEM/c2\nabla^2\Phi=8\pi G\rho_{\rm EM}/c^2, twice the coefficient that the same energy density would give as pressureless dust. Evaluate ∇2Φ\nabla^2\Phi numerically, and state the value that dust of the same energy density would give for comparison. ((b)) Explain why ∇μTEMμν=0\nabla_\mu T^{\mu\nu}_{\rm EM}=0 in this source-free region must follow from the field equations rather than being assumed about the electromagnetic field, and name the identity responsible. ((c)) State the number of constraint equations and evolution equations among the ten field equations sourced by this field, and explain why this count is exactly the same as for the dust case even though TEMμνT^{\mu\nu}_{\rm EM} has a completely different structure.

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