Maxwell's equations in curved spacetime.

[eok20]

Maxwell's equations in curved spacetime can be written as
$$\nabla^a F_{ab} = -4\pi j_b, \nabla_{[a} F_{bc] = 0$$ or as $$d*F = 4\pi*j, dF = 0$$, 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.

 

[arkajad]

Any help would GREATLY be appreciated.

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

Are you done with problem 1? Completely? Sure?

 

[eok20]

Are you done with problem 1? Completely? Sure?

Yes, I've done problem 1 (showing that $$\nabla^b j_b = 0$$) as well as problem 2a (showing that ** = +- 1). I am confident in both of my solutions to these-- does one of these help?

 

[arkajad]

Check here:

http://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry

Are you familiar with the formula after "The divergence of an antisymmetric tensor field of type (2,0) simplifies to"

Can you show that

$$\nabla^a F_{ab}=g_{bc}\nabla_a F^{ac}$$