Variation of the Lorentzian Hodge map and the linearized Maxwell equation

Differentiating $g_{ϵ}^{ac}{g}_{ϵcb}={\delta }_b^a$ at $ϵ=0$ gives ${\dot{g}}^{ab}=-h^{ab}$. In a positively oriented chart, ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g_{ϵ}}=\sqrt{\mathrm{\mid }\det (g_{ϵab})\mathrm{\mid }}d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. Since $\det (g_{ϵ})=\det (g)\det (I+ϵg^{-1}h)+O({ϵ}^2)$ and the linear term of $\det (I+ϵA)$ is $ϵ\mathrm{tr}A$, differentiation gives ${\dot{\mathrm{v}\mathrm{o}\mathrm{l}}}_g=\frac{1}{2}({\mathrm{tr}}_{g}h){\mathrm{v}\mathrm{o}\mathrm{l}}_g$.

Evaluating the defining identity in [F1] on increasing coordinate wedges gives the component formula $({\star }_g\omega {)}_{b_1\dots b_{4-k}}=\frac{1}{k!}({\mathrm{v}\mathrm{o}\mathrm{l}}_g{)}_{a_1\dots a_k{b}_1\dots b_{4-k}}{\omega }^{a_1\dots a_k}$, with the same orientation sign as the displayed two-form case. Thus the component Hodge map is the oriented volume tensor times the $k$ inverse-metric factors that raise the indices of ${\omega }_{ϵ}$. Differentiating the volume contributes $\frac{1}{2}({\mathrm{tr}}_{g}h)\omega$ inside the Hodge map by step 1.1; differentiating each of the $k$ inverse metrics contributes $-h_{a_j}{}^b$ in its corresponding covariant slot by step 1.1; differentiating ${\omega }_{ϵ}$ contributes $\tau$. Summing these $k+2$ terms yields $\dot{\star }{\omega }_{ϵ}={\star }_g(\tau +\frac{1}{2}({\mathrm{tr}}_{g}h)\omega -h^{\mathrm{♯}}\text{\hspace{0.17em}⁣}\cdot \omega )$, including $k=0$ where the slot sum is empty.

Substituting $\omega =\mathcal{F}$ and $\tau =f$ gives the displayed variation. The exterior derivative is independent of $ϵ$ and linear, so differentiating $d({\star }_{g_{ϵ}}{\mathcal{F}}_{ϵ})=0$ gives $d\delta (\star \mathcal{F})=0$. The closure equation differentiates to $df=0$ in the linearized definition.