New definition of complexity for self-gravitating fluid distributions: The spherically symmetric, static case

TL;DR

This paper introduces a new definition of complexity for static, spherically symmetric self-gravitating systems in general relativity, based on a complexity factor derived from the Riemann tensor, which measures deviations from homogeneous, isotropic configurations.

Contribution

It proposes a novel complexity factor for self-gravitating fluids, linking it to the active gravitational mass and balancing effects of density inhomogeneity and pressure anisotropy.

Findings

Defined the zero complexity condition for specific fluid configurations.

Derived exact solutions satisfying the zero complexity criterion.

Discussed potential applications to compact object structure and evolution.

Abstract

We put forward a new definition of complexity, for static and spherically symmetric self--gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self--gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.