Proving Killing Vector of Static Spacetime - David

[dman12]

Hello,

I am reading through some GR lecture notes and have come across the following:

"A spacetime is static if there exists a coordinate chart where:

∂₀g_μν = 0
g_0i = 0

This spacetime admits a Timelike Killing vector X that satisfies:

X_[α∇_βX_γ] = 0 "

How do I go about proving that this relation is true?

Thanks!

David

 

[PeterDonis]

How do I go about proving that this relation is true?

Have you tried expanding out the relation that $X$ has to satisfy? Do any terms drop out or cancel because of the properties that the coordinate chart satisfies?