Maxwell's equations in curved spacetime can be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\nabla^a F_{ab} = -4\pi j_b, \nabla_{[a} F_{bc] = 0[/tex] or as [tex] d*F = 4\pi*j, dF = 0[/tex], where F is a two-form, j is a one-form and * is the Hodge star. How do you show that these two sets of equations are equivalent (basically, that the first from each set are equivalent since I see how the second ones are equivalent)? I tried working it out in local coordinates but it got really messy e.g. derivatives of the determinant of the metric and I was unable to get anywhere. I also tried to work it out in terms of an orthonormal frame (tetrad) but that didn't work since I don't know if the Christoffel symbols for the connection have a nice form in that basis.

Any help would GREATLY be appreciated.

Note this is problem 2 in chapter 4 of Wald's text.

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# Maxwell's equations in curved spacetime.

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