Non-spherical sources of Schwarzschild space-time

While it is known that any spherical fluid distribution may only source the spherically symmetric Schwarzschild space-time, the inverse is not true. Thus, in this manuscript, we find exact axially symmetric and static fluid (interior) solutions to Einstein equations, which match smoothly on the boundary surface to the Schwarzschild (exterior) space-time, even though the fluid distribution is not endowed with spherical symmetry. The solutions are obtained by using the general approach outlined in Hernandez-Pastora et al. (Class Quantum Gravity 33:235005, 2016), and satisfy the usual requirements imposed to any physically admissible interior solution. A discussion about the physical and geometric properties of the source is presented. The relativistic multipole moments (RMM) are explicitly calculated in terms of the physical variables, allowing to prove that spherical sources can only match to the Schwarzschild space-time. The complexity of the source is evaluated through the complexity factors. It is shown that there is only one independent complexity factor, as in the spherically symmetric case.