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

TL;DR

This paper explores the small-$c$ expansion of general relativity in ADM variables, deriving exact stationary solutions at NNLO that include rotating and multipolar geometries, relevant to astrophysical objects.

Contribution

It provides explicit stationary solutions at NNLO in the small-$c$ expansion, including rotating and multipolar geometries, enhancing the understanding of stationary backgrounds in GR.

Findings

Derived exact vacuum solutions at NLO and NNLO for strong- and weak-gravity branches.

Constructed rotating solutions from small-$c$ expansion of Kerr and C-metric geometries.

Extended solutions to include higher multipoles up to $ ext{l}=4$.

Abstract

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.