The asymptotic behavior of Lorentz-violating photon fields

TL;DR

This paper analyzes the asymptotic behavior of photon fields in Lorentz-violating theories, showing that LV effects dominate at large distances and that the peeling theorem still holds under current experimental constraints.

Contribution

It derives the Newman-Penrose formalism for Maxwell's equations with Lorentz violation and demonstrates the nonperturbative nature of LV effects on photon field asymptotics.

Findings

LV effects dominate at large distances for higher powers of k^2

The Coulomb mode deviates from LI expectation due to LV corrections

The peeling theorem remains valid under current LV constraints

Abstract

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.