Static metric definition

I know that a stationary metric is one which possesses a timelike Killing Vector (so I have time invariance). I read that a static metric is one which has a timelike Killing vector (so it is stationary) AND this Killing vector is orthogonal to a family of hypersurface (of dimension n-1 is n is the dimension of my manifold). I don't understand the "physical" meaning behind this definition ? Is there a way to explain this in some more intuitive way ? Thanks

Answer

I think this a way of understanding the situation that is both intuitive and correct. The more experienced GRers will no doubt shout at me if this is wrong or misleading.

If we can write the metric so that the timelike Killing vector is everywhere orthogonal to the spacelike surfaces then there will be no cross terms between the time and spatial elements in the metric. If this isn't the case the metric will have always cross terms. So for example the stationary but non-static, Kerr metric necessarily has terms like $dtd\phi$. The presence of the cross terms means the geometry is no longer invariant under a change of time direction i.e. $t \rightarrow -t$.