Equilibrium hydrostatic equation and Newtonian limit of the singular f(R) gravity

TL;DR

This paper derives the hydrostatic equilibrium equation for spherical stars in f(R) gravity and investigates its Newtonian limit, demonstrating consistency with classical gravity for specific f(R) models.

Contribution

It provides a general derivation of the hydrostatic equilibrium equation in f(R) gravity using the Palatini formalism and explores the Newtonian limit for particular f(R) functions.

Findings

Hydrostatic equilibrium equation derived for f(R) gravity.

Newtonian limit established for specific f(R) models.

Consistent classical gravity limit demonstrated.

Abstract

We derive the equilibrium hydrostatic equation of a spherical star for any gravitational Lagrangian density of the form $L=\sqrt{-g}f(R)$. The Palatini variational principle for the Helmholtz Lagrangian in the Einstein gauge is used to obtain the field equations in this gauge. The equilibrium hydrostatic equation is obtained and is used to study the Newtonian limit for $f(R)=R-\frac{a^{2}}{3R}$. The same procedure is carried out for the more generally case $f(R)=R-\frac{1}{n+2}\frac{a^{n+1}}{R^{n}}$ giving a good Newtonian limit.