On homothetic Killing vectors in stationary axisymmetric vacuum spacetimes

TL;DR

This paper investigates homothetic Killing vectors in stationary axisymmetric vacuum spacetimes, deriving their general forms and conformal factors, especially focusing on cases with zero twist and specific metric component relations.

Contribution

It provides a comprehensive analysis of homothetic Killing vectors in SAV spacetimes, including explicit forms and conditions, extending previous understanding of spacetime symmetries.

Findings

Component along z is constant

Radial component is either constant or proportional to g_{ρρ}

Explicit forms of vectors and conformal factors derived

Abstract

In this paper we consider homothetic Killing vectors in the class of stationary axisymmetric vacuum (SAV) spacetimes, where the components of the vectors are functions of the time and radial coordinates. In this case the component of any homothetic Killing vector along the $z$ direction must be constant. Firstly, it is shown that either the component along the radial direction is constant or we have the proportionality $g_{\phi\phi}\propto g_{\rho\rho}$, where $g_{\phi\phi}>0$. In both cases, complete analyses are carried out and the general forms of the homothetic Killing vectors are determined. The associated conformal factors are also obtained. The case of vanishing twist in the metric, i.e., $\omega= 0$ is considered and the complete forms of the homothetic Killing vectors are determined, as well as the associated conformal factors.