# Poisson type relativistic perfect fluid spheres

## TL;DR

This paper derives and analyzes static, spherically symmetric perfect fluid solutions to Einstein's equations using Newtonian potential-density pairs, illustrating with examples like harmonic oscillator, Plummer, and dark matter halo models, and studying particle orbits.

## Contribution

It introduces a method to generate relativistic perfect fluid spheres from Newtonian potentials and examines their stability and orbital dynamics.

## Key findings

- Models satisfy all energy conditions.
- Stable circular orbits are possible around these structures.
- Examples include harmonic oscillator, Plummer, and dark matter halo models.

## Abstract

Static spherically symmetric solutions of the Einstein's field equations in isotropic coordinates representing perfect fluid matter distributions from Newtonian potential-density pairs are investigated. The approach is illustrated with three simple examples based on the potential-density pairs corresponding to a harmonic oscillator (homogeneous sphere), the well-known Plummer model and a massive spherical dark matter halo model with a logarithmic potential. Moreover, the geodesic circular motion of test particles around such structures is studied. The stability of the orbits against radial perturbations is also analyzed using an extension of the Rayleigh criteria of stability of a fluid in rest in a gravitational field. The models considered satisfy all the energy conditions.