# Complexity factors for axially symmetric static sources

## TL;DR

This paper extends the concept of complexity from spherically symmetric to axially symmetric static fluid sources, defining three complexity factors based on structure scalars, and explores conditions under which these factors vanish, indicating simple configurations.

## Contribution

It introduces three new complexity factors for axially symmetric static sources, generalizing previous spherically symmetric results and analyzing their implications for fluid configurations.

## Key findings

- Three complexity factors vanish for simple spheroid configurations.
- Exact solutions show complexity factors can vanish in more general cases.
- Complexity is not solely determined by symmetry, but also by energy density and pressure anisotropy.

## Abstract

A previously found definition of complexity for spherically symmetric fluid distributions [1], is extended to axially symmetric static sources. In this case there are three different complexity factors, defined in terms of three structure scalars obtained from the orthogonal splitting of the Riemann tensor. All these three factors vanish, for what we consider the simplest fluid distribution, i.e a fluid spheroid with isotropic pressure and homogeneous energy density. However, as in the spherically symmetric case, they can also vanish for a variety of configurations, provided the energy density inhomogeneity terms cancel the pressure anisotropic ones in the expressions for the complexity factors. Some exact analytical solutions of this type are found and analyzed. At the light of the obtained results, some conclusions about the correlation (the lack of it) between symmetry and complexity, are put forward.