Complexity Factor For Static Anisotropic Self-Gravitating Source in $f(R)$ Gravity

TL;DR

This paper extends Herrera's complexity definition for static self-gravitating fluids to $f(R)$ gravity, analyzing how modifications to General Relativity influence the complexity factor and providing exact solutions and applications to compact objects.

Contribution

It introduces the concept of complexity in $f(R)$ gravity, exploring its behavior and solutions, and compares it with the classical General Relativity case.

Findings

Homogeneous energy density and isotropic pressure reduce complexity.

Inhomogeneous and anisotropic fluids have maximum complexity.

Zero complexity can occur when effects of inhomogeneity and anisotropy cancel each other.

Abstract

In a recent paper, Herrera \cite{2} (L. Herrera: Phys. Rev. D97, 044010(2018)) have proposed a new definition of complexity for static self-gravitating fluid in General Relativity. In the present article, we implement this definition of complexity for static self-gravitating fluid to case of $f(R)$ gravity. Here, we found that in the frame of $f(R)$ gravity the definition of complexity proposed by Herrera, entirely based on the quantity known as complexity factor which appears in the orthogonal splitting of the curvature tensor. It has been observed that fluid spheres possessing homogenous energy density profile and isotropic pressure are capable to diminish their the complexity factor. We are interested to see the effects of $f(R)$ term on complexity factor of the self-gravitating object. The gravitating source with inhomogeneous energy density and anisotropic pressure have maximum value of complexity. Further, such fluids may have zero complexity factor if the effects of inhomogeneity in energy density and anisotropic pressure cancel the effects of each other in the presence of $f(R)$ dark source term. Also, we have found some interior exact solutions of modified $f(R)$ field equations satisfying complexity criterium and some applications of this newly concept to the study of structure of compact objects are discussed in detail. It is interesting to note that previous results about the complexity for static self-gravitating fluid in General Relativity can be recovered from our analysis if $f(R)=R$, which General Relativistic limit of $f(R)$ gravity. Some future research directions have been mentioned in the end of the summary.