KILLING VECTOR FIELDS OF A SPACETIME

Abstract

We studied the geodesics of a spacetime with the pseudo-Riemannian metric:

$$d{s}^2=\frac{1}{x_4{x}_4}\{{\displaystyle \sum }_{b,c=1}^3\left({\delta }_{bc}-\frac{a{x}_b{x}_c}{1+a{r}^2}\right)d{x}_{b}d{x}_c-\frac{1}{1+a{x}_4{x}_4}d{x}_{4}d{x}_4\}$$

on $R^3\times R_{+}$, where $r^2={\displaystyle \sum }_{b=1}^3{x}_b{x}_b$ and $a=\text{ constant}$, which are plane quadratic curves (in [12]). In this paper, we shall determine all the Killing vector fields of this spacetime and choose special pairs out of them with interesting properties for the case .