The asymptotic behavior of Lorentz-violating photon fields

In this work, we derive the Newman-Penrose formalism of Maxwell's equations using two approaches: differential forms and intrinsic derivatives. Denoting $(k_{AF})^\mu$ as $k^\mu$, with $k^\mu=(k^t,k^r,0,0)$ in spherically symmetric spacetimes, we show that the expansion in $r^{-1}$ fails to produce consistent, closed solutions due to the inability to separate Lorentz-violating (LV) phase factors, as the Lorentz-invariant (LI) null tetrad does not adapt to the LV wavefront. Moreover, with exact formal solutions, we demonstrate that the expansion is nonperturbative in the LV parameter $k^2\equiv k^t-k^r$. For $r\gg1/k^2$, higher powers of $k^2$ dominate over lower powers, as the latter decay more rapidly with increasing $r$. Although the Coulomb mode $\phi_1\sim\mathcal{O}(\ln{r}/r^2)$ deviates from the LI expectation $\mathcal{O}(r^{-2})$ due to LV corrections, the leading outgoing radiation mode remains unaffected, i.e., $\phi_2\sim\mathcal{O}(r^{-1})$. Given the constraint $|k_{AF}|\le10^{-44}$GeV \cite{CMBLV-N}, the three complex scalars $\phi_a$ ($a=0,1,2$) still obey the peeling theorem: $\phi_a\sim\mathcal{O}(r^{(a-3)}),~a=0,1,2$ for large, finite distances.