Spin and Quadrupole Sectors in Nonrelativistic Gravity

TL;DR

This paper analyzes the large-$c$ expansion of general relativity in ADM variables, deriving solutions up to NNLO for Galilean limits, including rotating compact objects with spin and quadrupole effects.

Contribution

It introduces a unified even $oldsymbol{ extomega}$-expansion in ADM variables, deriving new solutions for weak and strong Galilean branches up to NNLO.

Findings

Derived stationary vacuum solutions including Kerr-type and Hartle-Thorne-type.

Extended solutions to higher mass multipoles in the weak branch.

Constructed approximate spacetime metrics with spin and quadrupole effects.

Abstract

We study the large-$c$ expansion of general relativity in 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 NNLO. We then construct stationary vacuum solutions in weak and strong branches. In the weak branch, we find 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 $J^2Q$ source terms, which require additional corrections to the NNLO lapse and NNLO spatial tensor field. This gives a mixed weak-branch Galilean solution in the ADM gauge. 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 into approximate spacetime metrics. 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.