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 equations acquire a structure, in which the leading lapse and spatial metric are determined by a closed subsystem, the NLO rotational field satisfies a linear equation on that background, and the NNLO fields satisfy a linear system sourced by the NLO data. The stationary sector exhibits both strong-gravity and weak-field 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 O(J) O ( J ) at NLO and up to O(J³) O ( J 3 ) at NNLO. In the weak-field 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 =0,2 ℓ = 0 , 2 sector by including higher multipoles up to =4 ℓ = 4 , where ℓ 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 motivated by relativistic compact-object geometries.