Stationary solutions in the small-$c$ expansion of GR

We study the small-$c$ expansion of general relativity in ADM variables up to next-to-next-to-leading order (NNLO). We show that, in the stationary sector, this formulation renders the field equations more tractable for explicit solution building. The stationary sector exhibits both strong- and weak-gravity branches, whose structure becomes richer at NNLO. In the strong-gravity branch, we first obtain exact vacuum solutions of NLO Carroll gravity, including the Lense--Thirring and rotating C-metric backgrounds. At NNLO, we then construct the corresponding Lense--Thirring-type and C-metric-type exact vacuum geometries. These solutions also arise from the small-$c$ expansion of the Kerr and rotating C-metric geometries around the strong-gravity background, up to $\mathcal{O}(J)$ at NLO and up to $\mathcal{O}(J^3)$ at NNLO. In the weak-gravity branch, we find exact Hartle--Thorne-type solutions with an independent quadrupole moment, together with exact spin-squared corrections and a mixed quadrupolar-rotating solution. We further extend the $\ell=0,2$ sector by including higher multipoles up to $\ell=4$, where $\ell$ denotes the multipole index. These results show that the full NLO/NNLO theory admits a richer stationary vacuum sector than the magnetic Carroll truncation. More broadly, the ADM formulation provides a practical framework for constructing and analyzing stationary backgrounds in the small-$c$ expansion of general relativity, and may also offer a useful framework for organizing rotational and higher-multipole deformations relevant to compact astrophysical objects such as slowly rotating black holes and neutron stars.