the equation of continuity in tensor form


ρ=Σaeaδ(rra)
or
de=ρdV

dxide=ρdxidV

dxide=ρdxidtdVdt

dxide=ρvidΩc

dxide=jidΩc

where ji=(ρc,ρvi)
now, consider the term ecAkdxk from the action term
ecAkdxk=1cρdVAkdxk

=1cρAkdxkdtdVdt

=1c2jkAkdΩ

substituting this in the action equation
S=bamcdsbaecAμdxμ116πcbadΩFμνFμν

S=mcds1c2jkAkdΩ116πcdΩFμνFμν

δS=mcdvμδxμ1c2δjkAkdΩ1c2jkδAkdΩ116πcdΩδ(FμνFμν)

δ(FμνFμν)=2FμνδFμν

=2Fμνδ(dμAνdνAμ)

=2Fμν(2δdμAν)

=4FμνdμδAν

=4dμ(FμνδAν)4dμFμνδAν

δS=mcdvμδxμ1c2δjkAkdΩ1c2jkδAkdΩ116πc4(dμ(FμνδAν)dμFμνδAν)dΩ

considering the terms with a variation in Ak
δS=1c2jkδAkdΩ+14πcdμFμνδAνdΩ

since δS=0
1c2jkδAkdΩ+14πcdμFμνδAνdΩ=0

4πcjkδAkdΩ+dμFμνδAνdΩ=0

(4πcjνδAνdΩ+dμFμνδAνdΩ)=0

δAνdΩ(4πcjν+dμFμν)=0

4πcjν+dνFνμ=0

dνFνμ=4πcjν

dνFμν=4πcjν

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