Orthogonal splitting of the Riemann curvature tensor and its implications in modeling compact stellar structures

TL;DR

This paper explores the decomposition of the Riemann tensor to understand the complexity of static, charged, anisotropic stellar models within a modified gravity framework, revealing conditions for minimal complexity.

Contribution

It introduces a new analysis of complexity in $f(R,L_{m},\mathcal{T})$ gravity using orthogonal Riemann tensor decomposition and derives conditions for vanishing complexity in stellar models.

Findings

Compact objects with anisotropic matter and density inhomogeneity have maximum complexity.

Spherical matter distributions can have minimal or zero complexity in $f(R,L_{m},\mathcal{T})$ gravity.

New relations between mass, curvature, and conformal tensors are established.

Abstract

Although the interpretation of complexity in extended theories of gravity is available in the literature, its illustration in $f(R,L_{m},\mathcal{T})$ theory is still ambiguous. The orthogonal decomposition of the Riemann tensor results in the emergence of complexity factor as recently proposed by Herrera [1]. We initiate the analysis by contemplating the interior spacetime as a static spherical anisotropic composition under the presence of charge. The modified field equations are derived along with the establishment of association between the curvature and conformal tensors that have significant relevance in evaluating complexity of the system. Furthermore, the generalized expressions for two different masses are calculated, and their link with conformal tensor is also analyzed. Moreover, we develop a particular relation between predetermined quantities and evaluate the complexity in terms of a certain scalar $Y_{TF}$. Several interior solutions admitting vanishing complexity are also determined. Interestingly, compact objects having anisotropic matter configuration along with the energy density inhomogeneity possess maximum complexity. It is concluded that the spherical distribution of matter might not manifest complexity or admitting minimal value of this factor in the framework of $f(R,L_{m},\mathcal{T})$ theory due to the appearance of dark source terms.