# Static trace free Einstein equations and stellar distributions

## TL;DR

This paper develops models of static spherical perfect fluid distributions within trace-free Einstein gravity, revealing new solutions and corrections to previous models, with implications for compact objects and isothermal spheres.

## Contribution

It introduces novel solutions for static fluid spheres in trace-free Einstein gravity, including corrections to prior models and analysis of specific metrics like Finch--Skea.

## Key findings

- Incompressible fluid does not produce Schwarzschild interior metric in trace-free Einstein gravity.
- A complete isothermal fluid sphere model with inverse square law pressure and density is obtained.
- The solutions encompass standard Einstein results as special cases and admit compact objects.

## Abstract

We construct models of static spherical distributions of perfect fluid in trace--free Einstein gravity theory. The equations governing the gravitational field are equivalent to the standard Einstein's equations however, their presentation is manifestly different which motivates the question whether new information would emerge due to the nonlinearity of the field equations. The incompressible fluid assumption does not lead to the well known Schwarzschild interior metric of Einstein gravity and a term denoting the presence of a cosmological constant is present on account of the integration process. The Schwarzschild interior is regained as a special case of a richer geometry. On the other hand, when the Schwarzschild geometry is prescribed, a constant density fluid emerges consistent with the standard equations. A complete model of an isothermal fluid sphere with pressure and density obeying the inverse square law is obtained. Corrections to the model previously presented in the literature by Saslaw {\it {et al}} are exhibited. The isothermal ansatz does not yield a constant gravitational potential in general but both potentials are position dependent. Conversely, it is shown that assuming a constant $g_{rr}$ gravitational potential does not yield an isothermal fluid in general as is the case in standard general relativity. The results of the standard Einstein equations are special cases of the models reported here. Noteworthy is the fact that whereas the previously reported isothermal solution was only of cosmological interest, the solution reported herein admit compact objects by virtue of the fact that a pressure-free hypersurface exists. Finally we analyze the consequences of selecting the Finch--Skea metric as the seed solution.