General Relativity: An Introduction for Physicists · Problem 6.9

Electromagnetism — Hobson, Efstathiou & Lasenby

Physics · maxwell-equations-arbitrary-coordinates

Problem

In an arbitrary coordinate system, the second Maxwell equation reads ∇σFμν+∇νFσμ+∇μFνσ=0.\nabla_\sigma F_{\mu\nu} + \nabla_\nu F_{\sigma\mu} + \nabla_\mu F_{\nu\sigma} = 0. Show that this can be written as ∂σFμν+∂νFσμ+∂μFνσ=0,\partial_\sigma F_{\mu\nu} + \partial_\nu F_{\sigma\mu} + \partial_\mu F_{\nu\sigma} = 0, and hence show that ∇[σFμν]=0\nabla_{[\sigma} F_{\mu\nu]} = 0.

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