Final part of proof of timelike Killing w/ Frobenius --> static

[ergospherical]

If $V$ is timelike Killing with Frobenius condition $V_{[\alpha} \nabla_{\mu} V_{\nu]} = 0$ then you can derive the equation:$$\nabla_{\mu} (|V|^2 V_{\nu}) - \nabla_{\nu} (|V|^2 V_{\mu}) = 0$$which has the solution$$V_{\alpha} = \partial_{\alpha} \phi \quad \mathrm{where} \quad \phi = x^0 + f(x^i)$$The final part of the proof is to show that you can use this function to transform the coordinates $\bar{x} = \bar{x}(x)$ such that $\bar{g}_{0i} = 0$ and $\partial_0 \bar{g}_{\mu \nu} = 0$. I don't see it?

 

[cianfa72]

The final part of the proof is to show that you can use this function to transform the coordinates $\bar{x} = \bar{x}(x)$ such that $\bar{g}_{0i} = 0$ and $\partial_0 \bar{g}_{\mu \nu} = 0$. I don't see it?

I don't know the details of such proof, however the result is that the timelike congruence at rest in such a chart with coordinates $\bar x$ (i.e. the set of timelike worldlines described by fixed spacelike coordinates in this chart) turns out to be hypersurface orthogonal.