Complexity factors for axially symmetric static sources

Abstract

A previously found definition of complexity for spherically symmetric fluid distributions [L. Herrera, Phys. Rev. D 97, 044010 (2018).] 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. In light of the obtained results, some conclusions about the correlation (the lack of it) between symmetry and complexity are put forward.