We study the large-$c$ expansion of general relativity in Arnowitt-Deser-Misner (ADM) variables. Using a unified even $\omega$-expansion, the ADM formulation gives a common starting point for Galilean and Carrollian limits. We focus on the Galilean branch and derive the ADM action and field equations up to next-to-next-to-leading order (NNLO). We then construct stationary vacuum solutions of the Galilean ADM equations up to NNLO in weak and strong branches. In the weak branch, we find next-to-leading order (NLO) Kerr-type, Hartle-Thorne-type and mixed-type solutions. The NLO weak equations also allow a simple extension to higher mass multipoles. At NNLO, the weak Kerr-type and extended Hartle-Thorne-type sectors solve the equations separately, but their naive sum is not a solution. The nonlinear NNLO equations generate mixed $J^{2}Q$ source terms, which require additional corrections to the NNLO lapse and NNLO spatial tensor field. This gives a mixed weak-branch Galilean solution of the truncated NNLO equations in the ADM gauge, whose full relativistic completion is not fixed by the truncated data alone. In the strong branch, Kerr-type data solve the equations through NNLO while the strong Hartle-Thorne-type data solve the NLO equations. We also explain how the ADM data can be reconstructed order by order into approximate spacetime metrics. These reconstructed metrics should not be interpreted by themselves as unique full relativistic completions. Since these metrics include spin, quadrupole and mixed spin–quadrupole effects, they may be useful for studying the spacetime around rotating compact objects such as black holes and neutron stars.

Close

PrevNext

View figure in article

• Alternative gravity theories