# Anisotropic geodesic fluid in non-comoving spherical coordinates

## TL;DR

This paper analyzes a spherically symmetric geodesic fluid model with anisotropic pressure, providing analytical solutions and exploring potential applications in astrophysics such as galactic halos and dark energy stars.

## Contribution

It presents a self-contained set of Einstein's equations for anisotropic geodesic fluids in non-comoving coordinates and solves them analytically except for a key nonlinear ODE.

## Key findings

- Analytical solutions for the Einstein equations in this model.
- Reformulation of the key ODE as a Lienard and Abel differential equation.
- Discussion of potential astrophysical applications and open mathematical problems.

## Abstract

We start with a recently introduced spherically symmetric geodesic fluid model (arXiv: 1601.07030) whose energy-momentum tensor in the comoving frame is dust-like with nontrivial energy flux. In the non-comoving energy frame (vanishing energy flux) the same EMT contains besides dust only radial pressure. We present Einstein's equations together with the matter equations in static spherically symmetric coordinates. These equations are self-contained (four equations for four unknowns). We solve them analytically except for a resulting nonlinear ordinary differential equation (ODE) for the gravitational potential. This ODE can be rewritten as a Lienard differential equation which, however, may be transformed into a rational Abel differential equation of the first kind. Finally we list some open mathematical problems and outline possible physical applications (galactic halos, dark energy stars) and related open problems.