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

TL;DR

This paper analyzes the motion of extended fluid bodies in the Newtonian limit of $f(R)$ gravity, deriving multipole expansions for acceleration, potential energy, and spin dynamics, revealing modifications to General Relativity.

Contribution

It introduces a multipole expansion framework for inter-body dynamics in $f(R)$ gravity's Newtonian limit, including scalar multipole effects and energy expressions, extending prior GR results.

Findings

Derived multipole expansion for center-of-mass acceleration including Yukawa terms.

Provided expressions for gravitational potential energy and total conserved energy.

Formulated the effective one-body equation for two-body systems.

Abstract

In the Newtonian limit of $f(R)$ gravity, for an isolated self-gravitating system consisting of $N$ extended fluid bodies, the inter-body 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. 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 inter-body 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.