The asymptotic behavior of Lorentz-violating photon fields

Abstract

In this work, we derive the Newman-Penrose formalism of Maxwell's equations using two approaches: differential forms and intrinsic derivatives. Denoting $(k_{AF}{)}^{\hat{I}}¼$ as $k^{\hat{I}}¼$, with $k^{\hat{I}}¼=(k^t,k^r,0,0)$ 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 $r\gg 1/k^2$, higher powers of $k^2$ dominate over lower powers, as the latter decay more rapidly with increasing $r$. Although the Coulomb mode $\ddot{I}{•}_1\sim \mathcal{O}(\ln r/r^2)$ deviates from the LI expectation $\mathcal{O}(r^{-2})$ due to LV corrections, the leading outgoing radiation mode remains unaffected, i.e., $\ddot{I}{•}_2\sim \mathcal{O}(r^{-1})$. Given the constraint $|k_{AF}|\le 10^{-44}$GeV \cite{CMBLV-N}, the three complex scalars $\ddot{I}{•}_a$ ($a=0,1,2$) still obey the peeling theorem: $\ddot{I}{•}_a\sim \mathcal{O}(r^{(a-3)}),a=0,1,2$ for large, finite distances.

21 pages