Generating new perfect-fluid solutions from known ones

TL;DR

This paper explores a method to generate new perfect-fluid solutions in Einstein's gravity using an $SL(2,R)$ transformation, specifically for barotropic fluids with $ ho+3p=0$, and applies it to known solutions to unify and extend their forms.

Contribution

It demonstrates that an $SL(2,R)$ transformation can produce new perfect-fluid solutions only for the specific equation of state $ ho+3p=0$, and provides a unified form for known solutions.

Findings

Transformation generates new solutions for $ ho+3p=0$

Known solutions can be expressed in a unified form

Derivation of Petrov type D solutions included

Abstract

Stationary perfect-fluid configurations of Einstein's theory of gravity are studied. It is assumed that the 4-velocity of the fluid is parallel to the stationary Killing field, and also that the norm and the twist potential of the stationary Killing field are functionally independent. It has been pointed out earlier by one of us (I.R.) that for these perfect-fluid geometries some of the basic field equations are invariant under an $SL(2,R)$ transformation. Here it is shown that this transformation can be applied to generate possibly new perfect-fluid solutions from existing known ones only for the case of barotropic equation of state $\rho+3p=0$. In order to study the effect of this transformation, its application to known perfect-fluid solutions is presented. In this way, different previously known solutions could be written in a singe compact form. A new derivation of all Petrov type D stationary axisymmetric rigidly rotating perfect-fluid solutions with an equation of state $\rho+3p=constant$ is given in an appendix.