It is known that pure electrodynamics in curved space-time is invariant under Weyl transformations
gμν→Ω(x)gμν,Fμν→Ω−1(x)Fμν(1)
Now let's assume that there is external current Jμ. Now an action becomes
S=∫d4x√−g(−14FμνFαβgμνgαβ−JμAνgμν)(2)
Is it true that it isn't Weyl invariant? I.e., is it possible to construct the explicit form of the "conformal" transformation of Jμ, which preserves Weyl invariance in some generalized sense?
The question appears because of Maxwell equation in FRLW space-time in presence of
Jμ=(0,σE)(3)
This equation takes the form
B″+σB′+k2B=0,(4)
which is completely the same as corresponding equation in flat space-time, which seems to mean that conformal invariance is preserved even in presence of current Jμ (3). This current isn't external, since it is expressed in terms of E, but I can't get an appropriate transformation for A, so that
∫d4x√−gJμAνgμν→∫d4xσ√−ga−2(E⋅A)
becomes invariant.