Definitions of relativistic multipole moments in Newtonian gravity of massive objects


The gravitational field of a limited mass distribution is specified as a solution to the Poisson equation in the Newtonian approximation. One can investigate some limiting cases of this metric in order to understand its physical meaning and structure. In this paper, we give a short introduction to essential concepts of multipole moments in Newtonian gravitation to motivate the definition for the relativistic definitions. In the static Newtonian case, we can get a complete description of the gravitational field outside a massive object by means of the multipole moments and these multipole moments are relatively easy to obtain by an expansion in terms of spherical harmonics. In the relativistic case, however, the situation is much more difficult. There are several relativistic definitions of coordinate independent multipole moments and they can be compared to the Newtonian multipole moments. The explicit calculation of multipole moments is really quite tedious and laborious. We present some recurrence formulas which simplify the calculations. We will present some examples for calculating multipole moments of the static q-metric. We will use the Geroch-Hansen method because the calculations in this case are straightforward and the method is coordinate-independent.In addition, the Elers definition of the Newtonian limit is presented and it is used to determine the multipolemoments in the Newtonian approximation of this metric.


1 T. Bäckdahl, and Magnus Herberthson, Classical and Quantum Gravity 22.9: 1607 (2005).

2 T. Bäckdahl, Classical and Quantum Gravity 26.17, 175021 (2009).

3 R. Geroch, Journal of Mathematical Physics, 11.2, 437-449 (1970).

4 P. Krtouš, et al, Journal of High Energy Physics, 2007.02, 004 (2007).

5 R.O. Hansen, Journal of Mathematical Physics, 15.1, 46-52 (1974).

6 R. Beig, and S. Walter, On the multipole expansion for stationary space-times, Proc. of the Royal Society of London. A. Mathematical and Physical Sciences, 376.1765, 333-341 (1981).

7 R.P. Kerr, , and A. Schild, General Relativity and Gravitation 41.10, 2485-2499 (2009).

8 H. Quevedo, Intern. J. of Modern Physics D 20.10, 1779-1787 (2011).

9 H. Quevedo, S. Toktarbay, and Ye. Aimuratov, arXiv preprint arXiv:1310.5339 (2013).

10 F.L. Dubeibe, F.D. Lora-Clavijo, and Guillermo A. González, Physics Letters A 381.6, 563-567 (2017).

11 Frutos-Alfaro, Francisco, Hernando Quevedo, and Pedro A. Sanchez, Royal Society open science 5.5, 170826 (2018).

12 Frutos-Alfaro, Francisco, and Michael Soffel, Royal Society open science 5.7, 180640 (2018).
How to Cite
ABISHEV, M. et al. Definitions of relativistic multipole moments in Newtonian gravity of massive objects. Recent Contributions to Physics (Rec.Contr.Phys.), [S.l.], v. 72, n. 1, p. 11-18, mar. 2020. ISSN 2663-2276. Available at: <>. Date accessed: 03 june 2020. doi:
Theoretical Physics. Nuclear and Elementary Particle Physics. Astrophysics

Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.