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  • Open Access

Motion of extended fluid bodies in the Newtonian limit of f(R) gravity

Bofeng Wu*

Xiao Zhang†

  • Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, China

  • *Contact author: wubofeng@gxu.edu.cn
  • †Contact author: xzhang@amss.ac.cn
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Phys. Rev. D 114, 044067 – Published 21 August, 2026

DOI: https://doi.org/10.1103/rlxh-zm39

Abstract

In the Newtonian limit of f(R) gravity, for an isolated self-gravitating system consisting of N extended fluid bodies, the interbody dynamics are studied by applying the symmetric and trace-free formalism in terms of irreducible Cartesian tensors. The multipole expansion of each body’s center-of-mass acceleration is derived, and the expansion comprises the Coulomb-type part and the Yukawa-type part, where the former, identical to that in general relativity, is encoded by the products of the mass multipole moments of the body with those of other bodies, and the latter, as the modification introduced by f(R) gravity, is encoded by the products of the scalar multipole moments of the body with those of other bodies. Due to the finite range of the massive scalar mode, the scalar multipole terms in the Yukawa part do not decay systematically with increasing order. As an essential component of the system’s orbital dynamics, the multipole expansion for the total gravitational potential energy is provided, and the expression for the total conserved energy in terms of the mass and scalar multipole moments of the bodies is offered. To investigate the system’s spin dynamics, the equation of motion for each body’s spin angular momentum is further deduced and presented in the form of multipole expansion. These findings constitute the main content of the coarse-grained description of interbody dynamics for the system within the framework of the Newtonian limit of f(R) gravity. As a by-product, for a two-body system, the effective one-body equation governing the relative motion between the two bodies and the total energy of this system are achieved.

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Phys. Rev. D 114, 044067– Published 21 August, 2026
  • Received 5 January 2026
  • Revised 20 July 2026
  • Accepted 31 July 2026
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