Hi, I'm trying to do this problem. It boils down to this. I have to show that for a Cauchy surface [tex]\Sigma[/tex] Maxwells equations imply that [tex]D_aE^a=-a\pi j_an^a[/tex] where D is the induced derivative operator on [tex]\Sigma[/tex] associated with [tex]h_{ab}=g_{ab}+n_an_b[/tex] with n the unit normal, and [tex]E_a=F_{ab}n^b[/tex], j four current. I have issue with the first term in the derivative
[tex]n^bD_aF^a_b=n^bh^a_ch_b^dh_a^e\nabla _eF^c_d[/tex].
Now [tex]n^bh^d_b = 0[/tex] so the whole thing equals zero. and the other term doesnt contain derivatives of F so the current part will not come out. The only thing I can think of is [tex]h^d_b ≠ h_b^d[/tex] however Wald doesnt elaborate on this distinction, and the things I've tried to distinguish them havent made a difference. Any help much appreciated!
[tex]n^bD_aF^a_b=n^bh^a_ch_b^dh_a^e\nabla _eF^c_d[/tex].
Now [tex]n^bh^d_b = 0[/tex] so the whole thing equals zero. and the other term doesnt contain derivatives of F so the current part will not come out. The only thing I can think of is [tex]h^d_b ≠ h_b^d[/tex] however Wald doesnt elaborate on this distinction, and the things I've tried to distinguish them havent made a difference. Any help much appreciated!
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