The past set of a stationary observer in Kerr space-time

rurouni_sohji

Hi,

I'm studying the general relativity with the book by Woodhouse, and I have a question about the past set in Kerr metric.

On page 161, there's a question asking that

: Show that there exists [itex]r_{0}[/itex] such that the past set [itex]I^{-}(\omega)[/itex] is the whole of the region [itex] r\geq r_{0}[/itex], where [itex]\omega[/itex] is the world line of any stationary observer.

The hint for it on page 197 says that

1. Define [itex]g_{ab}\geq g'_{ab}[/itex] by that for every vector which is timelike with repect to [itex]g'_{ab}[/itex], it's also time like with respect to [itex]g_{ab}[/itex]. (OK)

2. Lemma1. Prove that this is a partial ordering. (Trivial)

3. Lemma2. Show that if [itex]g_{ab}\geq g'_{ab}[/itex], then [itex]I^{-}(\omega)\supseteq I'^{-}(\omega)[/itex]
(I can't prove it. Is a geodesic in [itex]g'_{ab}[/itex] also a geodesic in [itex]g_{ab}[/itex]? I don't think so...)

4. For the Kerr metric [itex]g_{ab}[/itex], taking [itex]g'_{ab}=g_{ab} - k t_{a} t_{b}[/itex] (where [itex]t_{a} = (1,0,0,0)[/itex] in B-L coordinate), construct a flat space-time metric on [itex]r \geq r_{0}[/itex] with [itex]g_{ab}\geq g'_{ab}[/itex].
(I can't again. Does 'a flat space' mean 'an approximate flat space'? If so, the Kerr space-time itself becomes flat when r is large... What's wrong with my idea?)


Finally assuming 1~4 above, I still cannot answer the original problem... What's the relation with them? Please anybody going to help me?