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Let [itex](U,V,\theta, \phi)[/itex] be Kruskal coordinates on the Kruskal manifold, where [tex]-UV=\left(\frac{r}{2m}-1\right)e^{r/2m},\hspace{1cm} t=2m\ln\left(\frac{-V}{U}\right)[/tex] and [itex]\theta[/itex] and [itex]\phi[/itex] are the usual polar angles. The metric is [tex]ds^2=\frac{-32m^3}{r}e^{\frac{-r}{2m}}dUdV+r^2d\Omega^2[/tex]. The vector [tex]\xi=-U\partial_U+V\partial_V[/tex] is a Killing vector.

I need to express [itex]\xi[/itex] in exterior Schwarzschild coordinates, however I'm not sure how to go about doing this. I guess I need to transform the basis vectors [itex]\partial_U[/itex] and [itex]\partial_V[/itex] into basis vectors in the Schwarzschild coordinates, but can't see how to, as U and V are defined implicitly.

Any help would be much appreciated!

I need to express [itex]\xi[/itex] in exterior Schwarzschild coordinates, however I'm not sure how to go about doing this. I guess I need to transform the basis vectors [itex]\partial_U[/itex] and [itex]\partial_V[/itex] into basis vectors in the Schwarzschild coordinates, but can't see how to, as U and V are defined implicitly.

Any help would be much appreciated!

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