Spin and Quadrupole Sectors in Nonrelativistic Gravity
Abstract
We study the large-c expansion of general relativity in ADM variables. Using a unified even ω-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 of the Galilean ADM equations up to NNLO 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 J2Q 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.
Versions
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BibTeX
@misc{zorba2026spin,
title = {{Spin and Quadrupole Sectors in Nonrelativistic Gravity}},
author = {Utku Zorba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.06410},
eprint = {2605.06410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06410},
}
RIS
TY - UNPB
TI - Spin and Quadrupole Sectors in Nonrelativistic Gravity
AU - Utku Zorba
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2605.06410
UR - https://arxiv.org/abs/2605.06410
AB - We study the large-c expansion of general relativity in ADM variables. Using a unified even ω-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 of the Galilean ADM equations up to NNLO 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 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.
ER -