- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Spin and quadrupole sectors in nonrelativistic gravity

Utku Zorba*

  • *Contact author: utku.zorba@iuc.edu.tr
PDF

Phys. Rev. D 114, 084003 – Published 1 October, 2026

DOI: https://doi.org/10.1103/2czs-q2mm

Abstract

We study the large-c expansion of general relativity in Arnowitt-Deser-Misner (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 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 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.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (37)

  1. D. Van den Bleeken, Torsional Newton–Cartan gravity from the large c expansion of general relativity, Classical Quantum Gravity 34, 185004 (2017).
  2. D. Hansen, J. Hartong, and N. A. Obers, Action principle for Newtonian gravity, Phys. Rev. Lett. 122, 061106 (2019).
  3. D. Van den Bleeken, Torsional Newton-Cartan gravity and strong gravitational fields, arXiv:1903.10682.
  4. D. Hansen, J. Hartong, and N. A. Obers, Non-relativistic gravity and its coupling to matter, J. High Energy Phys. 06 (2020) 145.
  5. J. Hartong, N. A. Obers, and G. Oling, Review on non-relativistic gravity, Front. Phys. 11, 1116888 (2023).
  6. J. Hartong, E. Have, N. A. Obers, and I. Pikovski, A coupling prescription for post-Newtonian corrections in quantum mechanics, SciPost Phys. 16, 088 (2024).
  7. J. Hartong and J. Musaeus, Toward a covariant framework for post-Newtonian expansions for radiative sources, Phys. Rev. D 109, 124058 (2024).
  8. M. Elbistan, E. Hamamci, D. Van den Bleeken, and U. Zorba, A 3+1 formulation of the 1/c expansion of general relativity, J. High Energy Phys. 02 (2023) 108.
  9. B. Kol and M. Smolkin, Non-relativistic gravitation: From Newton to Einstein and back, Classical Quantum Gravity 25, 145011 (2008).
  10. B. Kol, M. Levi, and M. Smolkin, Comparing space+time decompositions in the post-Newtonian limit, Classical Quantum Gravity 28, 145021 (2011).
  11. B. Kol and M. Smolkin, Einstein’s action and the harmonic gauge in terms of Newtonian fields, Phys. Rev. D 85, 044029 (2012).
  12. A. Guerrieri and R. F. Sobreiro, Non-relativistic limit of gravity theories in the first order formalism, J. High Energy Phys. 03 (2021) 104.
  13. E. Ekiz, O. Kasikci, M. Ozkan, C. B. Senisik, and U. Zorba, Non-relativistic and ultra-relativistic scaling limits of multimetric gravity, J. High Energy Phys. 10 (2022) 151.
  14. E. Bal, E. Ekiz, E. O. Kahya, and U. Zorba, Stationary solutions in the small-c expansion of GR, arXiv:2604.23677.
  15. M. Elbistan, The 1/c expansion of general relativity in a 3+1 formulation revisited, Turk. J. Phys. 50, 37 (2026).
  16. D. Hansen, N. A. Obers, G. Oling, and B. T. Sogaard, Carroll expansion of general relativity, SciPost Phys. 13, 055 (2022).
  17. L. Donnay, A. Fiorucci, Y. Herfray, and R. Ruzziconi, Carrollian perspective on celestial holography, Phys. Rev. Lett. 129, 071602 (2022).
  18. R. Ruzziconi, Carrollian physics and holography, Phys. Rep. 1182, 1 (2026).
  19. M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89, 061901 (2014).
  20. F. J. Ernst, New formulation of the axially symmetric gravitational field problem, Phys. Rev. 167, 1175 (1968).
  21. F. J. Ernst, New formulation of the axially symmetric gravitational field problem. II, Phys. Rev. 168, 1415 (1968).
  22. V. S. Manko and I. D. Novikov, Generalizations of the Kerr and Kerr-Newman metrics possessing an arbitrary set of mass-multipole moments, Classical Quantum Gravity 9, 2477 (1992).
  23. F. Frutos-Alfaro and M. Soffel, On the post-linear quadrupole-quadrupole metric, Rev. Mate. Teor. Aplic. 24, 239 (2017).
  24. R. L. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gen. Relativ. Gravit. 40, 1997 (2008).
  25. M. Ergen, E. Hamamci, and D. Van den Bleeken, Oddity in nonrelativistic, strong gravity, Eur. Phys. J. C 80, 563 (2020).
  26. I. Kolář, D. Kubiznak, and P. Tadros, Rotating Carroll black holes: A no go theorem, Phys. Rev. D 112, L121504 (2025).
  27. J. B. Hartle, Slowly rotating relativistic stars. 1. Equations of structure, Astrophys. J. 150, 1005 (1967).
  28. J. B. Hartle and K. S. Thorne, Slowly rotating relativistic stars. II. Models for neutron stars and supermassive stars, Astrophys. J. 153, 807 (1968).
  29. J. Lense and H. Thirring, Über den einfluss der eigenrotation der zentralkörper auf die bewegung der planeten und monde nach der einsteinschen gravitationstheorie, Phys. Z. 19, 156 (1918).
  30. F. D. Ryan, Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments, Phys. Rev. D 52, 5707 (1995).
  31. M. Bauböck, E. Berti, D. Psaltis, and F. Özel, Relations between neutron-star parameters in the Hartle-Thorne approximation, Astrophys. J. 777, 68 (2013).
  32. R. P. Geroch, Multipole moments. II. Curved space, J. Math. Phys. (N.Y.) 11, 2580 (1970).
  33. R. O. Hansen, Multipole moments of stationary space-times, J. Math. Phys. (N.Y.) 15, 46 (1974).
  34. N. A. Collins and S. A. Hughes, Towards a formalism for mapping the space-times of massive compact objects: Bumpy black holes and their orbits, Phys. Rev. D 69, 124022 (2004).
  35. K. Glampedakis and S. Babak, Mapping spacetimes with LISA: Inspiral of a test-body in a quasi-Kerr field, Classical Quantum Gravity 23, 4167 (2006).
  36. J. R. Gair, C. Li, and I. Mandel, Observable properties of orbits in exact bumpy spacetimes, Phys. Rev. D 77, 024035 (2008).
  37. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).

Outline

Information

Phys. Rev. D 114, 084003– Published 1 October, 2026
  • Received 22 June 2026
  • Accepted 19 August 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation