Question

How to prove this auxiliary lemma to Hawking's singularity theorem?

Answer 1

Lemma

Let S be a closed achronal spatial hypersurface in spacetime M. Let q∈D+(S), there is a geodesic γ from S to q which has lenght τ(S,q), is normal to S and does not have focal ponts of S before q. (γ is time-like aside for the trivial case q∈S).

Notations

D(S) is the Cauchy development of S D(S)=D+(S)∪D−(S), where, for example, D+(S)={q∈M: every past pointing inextensible causal curve from q intersects S}

τ(S,q)=sup{lenght(α):α is a causal curve from a point in S to q}

Now I know/I can prove that

-D(S) is open and globally hyperbolic

-There is a geodesic of the desired lenght normal to S

Answer 2

Hey there :- Follow the above attachment