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

Bofeng Wu, Xiao Zhang

TL;DR

This work develops a systematic coarse-grained description of inter-body dynamics for an isolated system of extended fluid bodies in the Newtonian limit of $f(R)$ gravity. By decomposing the external gravitational potential into Coulomb-type GR-like and Yukawa-type $f(R)$-modifications, it derives the center-of-mass equations, the total energy and momentum, spin evolution, and a two-body effective equation all in terms of mass multipole moments $\hat{M}^A_I$ and scalar multipole moments $\hat{Q}^A_I$. A key finding is that, unlike GR, the scalar monopole and higher-order scalar multipoles generally influence orbital and spin dynamics, making body size and shape dynamically relevant through Yukawa couplings and preventing a straightforward multipole truncation. The results lay a foundational framework for celestial mechanics in $f(R)$ gravity and enable quantitative tests against solar-system and binary-system data to constrain the model parameter $m_{ ext{s}}$ and the form of $f(R)$.

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.

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

TL;DR

This work develops a systematic coarse-grained description of inter-body dynamics for an isolated system of extended fluid bodies in the Newtonian limit of gravity. By decomposing the external gravitational potential into Coulomb-type GR-like and Yukawa-type -modifications, it derives the center-of-mass equations, the total energy and momentum, spin evolution, and a two-body effective equation all in terms of mass multipole moments and scalar multipole moments . A key finding is that, unlike GR, the scalar monopole and higher-order scalar multipoles generally influence orbital and spin dynamics, making body size and shape dynamically relevant through Yukawa couplings and preventing a straightforward multipole truncation. The results lay a foundational framework for celestial mechanics in gravity and enable quantitative tests against solar-system and binary-system data to constrain the model parameter and the form of .

Abstract

In the Newtonian limit of gravity, for an isolated self-gravitating system consisting of 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 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 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.
Paper Structure (15 sections, 162 equations)

This paper contains 15 sections, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notation and related formulas in the STF formalism
  4. A brief review on $f(R)$ gravity
  5. Equations of motion for extended fluid bodies in an isolated self-gravitating system
  6. Fundamental variables in the system
  7. The internal and external potentials
  8. Multipole analysis on $1/|\boldsymbol{x}-\boldsymbol{x}'|$ and $\text{e}^{-m_{\text{s}}|\boldsymbol{x}-\boldsymbol{x}'|}/|\boldsymbol{x}-\boldsymbol{x}'|$
  9. Equations of motion for bodies in the system
  10. Conserved quantities and spin dynamics in the isolated self-gravitating system
  11. Conserved quantities of the system
  12. Spin dynamics in the system
  13. Orbital motion of the two-body system
  14. Summary and discussions
  15. The weak-field and slow-motion approximation of $f(R)$ gravity