Post-Newtonian orbital mechanics around a black hole in modified gravity

¹ Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
² Institute of Physics and Astronomy, University of Potsdam, 28, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany
³ Advanced Concepts Team, European Space Agency, TEC-SF, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴ Deutsche Zentrum für Astrophysik DZA, Postplatz 1, 02826 Görlitz, Germany
⁵ Gravitation and Astroparticle Physics Amsterdam (GRAPPA), University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 February 2026
Accepted: 15 May 2026

Abstract

Context. Scalar-tensor-vector gravity, also known as modified gravity (MOG), has emerged as an alternative to General Relativity (GR). It aims to explain astrophysical phenomena without invoking dark matter. The theory introduces a dynamic scalar field, a vector field, and modifications to the gravitational constant. The S-stars orbiting the supermassive black hole at the Galactic centre provide a unique opportunity to test the predictions of MOG because the orbital measurements are highly precise.

Aims. We investigate the perturbations in the orbits of S-stars under MOG, focusing on the effects on orbital elements, observables such as right ascension, declination, and radial velocity, and the potential degeneracy with dark matter scenarios.

Methods. We numerically integrated the first post-Newtonian equations of motion for S-stars within the MOG framework, considering contributions from the space-time geometry and the fifth force. We analysed the time evolution of orbital elements and projected the orbits onto the plane of the sky to assess deviations from GR. Furthermore, we compared the MOG-induced effects with those expected from a dark matter distribution.

Results. We found that MOG significantly alters the orbital precession, particularly for higher values of the MOG parameter α. For sufficiently large α or long observational baselines, the deviations in the observables can reach amplitudes comparable to current observational precision. Furthermore, we demonstrate that MOG effects can mimic those of a dark matter distribution, particularly in the argument of pericentre, and we reveal an unexplored connection between MOG and GR with electromagnetism.

Conclusions. The effects of MOG on stellar orbits are distinct from those predicted by GR and can be tested with precise astrometric and spectroscopic measurements of the S-stars. However, a potential degeneracy with dark matter signatures necessitates careful interpretation of observational data. Further observations over longer periods are required to conclusively distinguish between these scenarios.