Stationary and Axisymmetric Perfect Fluids with one Conformal Killing Vector

TL;DR

This paper investigates stationary, axisymmetric perfect-fluid solutions with conformal symmetry in Einstein's equations, identifying the structure of solutions and correcting previous misconceptions about their rotation properties.

Contribution

It classifies solutions based on two Lie algebras, shows they depend on one arbitrary function, and clarifies the nature of their conformal Killing vectors as homothetic.

Findings

Solutions depend on one arbitrary function of one coordinate.

The conformal Killing vector is necessarily homothetic.

Provides a comprehensive table of all possible solutions for the considered Lie algebras.

Abstract

We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in Mars and Senovilla, 1994, and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in the paper quated above concerning the differentially rotating character of one of the solutions presented