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comment How to prove this auxiliary lemma to Hawking's singularity theorem?
I forgot to include that O'Neil considers only P-Jacobi fields to define focality. There are just Jacobi fields above with the added condition that the 'near geodesics' too be normal to $S$. It can be a little tricky to put things perfectly in order, but your point is clear and I understand it now. I am very gratefull.
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2
comment How to prove this auxiliary lemma to Hawking's singularity theorem?
A point $q=\gamma(r)$ is focal for $S$, where $\gamma(0)\in S$ if there is a Jacobi field along $\gamma$ for which $J(0)$ is parallel to $S$ and $J(r)=0$.
Jul
2
comment How to prove this auxiliary lemma to Hawking's singularity theorem?
Well, well, I am starting to see something. So essentially what you are saying is: [Tentative of propostion In some sensible conditions (which?) the longest curve from an hypersurface to a point cannot be normal to that surface if there is a focal point among the normal curve.] I will try looking in those books.
Jul
2
comment How to prove this auxiliary lemma to Hawking's singularity theorem?
I taught about that but it is still not clear to me how this could solve the question. We CHOSE the maximal curve, if the presence of a focal point generated a lunger curve, we would have chosen the longer one... so are you implying that the presence of a focal point forbids the sup to be a maximum? From the hypothesis we know that $J^-(q)\cap S$ is compact and globally hyperbolic, hence the function $x\mapsto \tau(x,q)$ takes a maximum on this set (compactedness) and there is a normal curve of desired lenght (hyperbolicity). How does the presence of a focal point disrupt this?
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