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

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 $r_{0}$ such that the past set $I^{-}(\omega)$ is the whole of the region $r\geq r_{0}$, where $\omega$ is the world line of any stationary observer.

The hint for it on page 197 says that

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

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

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

4. For the Kerr metric $g_{ab}$, taking $g'_{ab}=g_{ab} - k t_{a} t_{b}$ (where $t_{a} = (1,0,0,0)$ in B-L coordinate), construct a flat space-time metric on $r \geq r_{0}$ with $g_{ab}\geq g'_{ab}$.
(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?
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