Stationary solutions in the small-c expansion of GR
- 1Istanbul Technical University
- 2Istanbul University-Cerrahpaşa
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 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) at NLO and up to 𝒪(J3) 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 ℓ =0,2 sector by including higher multipoles up to 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.
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BibTeX
@article{bal2026stationary,
title = {{Stationary solutions in the small-c expansion of GR}},
author = {Enes Bal and Ertuğrul Ekiz and Emre Onur Kahya and Utku Zorba},
journal = {The European Physical Journal C},
year = {2026},
month = nov,
doi = {10.1140/epjc/s10052-026-16509-y},
url = {https://link.springer.com/article/10.1140/epjc/s10052-026-16509-y},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Stationary solutions in the small-c expansion of GR
AU - Enes Bal
AU - Ertuğrul Ekiz
AU - Emre Onur Kahya
AU - Utku Zorba
PY - 2026
DA - 2026/11/06
JO - The European Physical Journal C
DO - 10.1140/epjc/s10052-026-16509-y
UR - https://link.springer.com/article/10.1140/epjc/s10052-026-16509-y
AB - 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) at NLO and up to 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 ℓ =0,2 sector by including higher multipoles up to 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.
SN - 1434-6052
ER -