A Conformal Mapping and Isothermal Perfect Fluid Model

TL;DR

This paper introduces a new static, inhomogeneous isothermal perfect fluid model based on a conformal spacetime with zero gravitational force but non-zero curvature, expanding solutions for cosmological and bounded fluid spheres.

Contribution

It presents a novel conformal metric approach leading to a unique three-parameter family of static, inhomogeneous isothermal perfect fluid solutions in general relativity.

Findings

Density and pressure decay as R^{-2} in the radial coordinate.

The model can describe both unbounded cosmological and bounded fluid sphere solutions.

The solution is characterized by a conformal metric with specific curvature properties.

Abstract

Instead of conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as ``minimally'' curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three parameter family of perfect fluid solution which is static and inhomogeneous. The density and pressure fall off in the curvature radial coordinates as $R^{-2}, $ for unbounded cosmological model with a barotropic equation of state. This is the characteristic of isothermal fluid. We thus have an ansatz for isothermal perfect fluid model. The solution can also represent bounded fluid spheres.