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Maxwell's equations in curved spacetime.

  1. Aug 27, 2010 #1
    Maxwell's equations in curved spacetime can be written as
    [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.
  2. jcsd
  3. Aug 27, 2010 #2
    Are you done with problem 1? Completely? Sure?
  4. Aug 27, 2010 #3
    Yes, I've done problem 1 (showing that [tex]\nabla^b j_b = 0[/tex]) as well as problem 2a (showing that ** = +- 1). I am confident in both of my solutions to these-- does one of these help?
  5. Aug 28, 2010 #4
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