# Approximate perfect fluid solutions with quadrupole moment

## TL;DR

This paper explores approximate solutions to Einstein's equations for static, axially symmetric perfect fluids with quadrupole moments, providing new classes of solutions under small deviations from spherical symmetry.

## Contribution

It introduces a specific line element suitable for such fields and derives linearized solutions for vacuum and perfect fluid cases, including matching conditions for physical relevance.

## Key findings

- Derived linearized solutions for small deviations from spherical symmetry.
- Presented classes of vacuum and perfect fluid solutions.
- Established matching conditions for physically meaningful spacetimes.

## Abstract

We investigate the interior Einstein's equations in the case of a static, axially symmetric, perfect fluid source. We present a particular line element that is specially suitable for the investigation of this type of interior gravitational fields. Assuming that the deviation from spherically symmetry is small, we linearize the corresponding field equations and find several classes of vacuum and perfect fluid solutions. We find physically meaninful spacetimes by imposing appropriate matching conditions.