# I need help with a proof in Hawking & Ellis [closed]

Here's a proof in Hawking and Ellis (1973) of proposition 6.4.6:

The definition of "strong causality" used in the book is that for every point $$p$$ and every neighborhood $$U$$ of $$p$$, there exists a neighborhood $$O$$ of $$p$$ contained in $$U$$ which no non-spacelike curve intersects more than once.

The conditions (a) to (c) of proposition 6.4.5 are given below:

a) The null energy condition holds on the spacetime $$M$$, so that $$R_{ab}K^aK^b\ge0$$ for all null vectors $$K^a$$.

b) The null generic condition holds, so that every null geodesic contains a point where $$K_{[a}R_{b]cd[e}K_{f]}K^cK^d$$ is non-zero ($$K^a$$ is the tangent vector to the null geodesic).

c) the chronology condition holds on $$M$$, ie. there are no closed timelike curves in $$M$$.

In proposition 6.4.5, the authors proved that if conditions (a) to (c) hold, then the causality condition holds on $$M$$, ie. there are no closed non-spacelike curves in $$M$$.

I have some questions about the proof of proposition 6.4.6:

1: Why can we assume that for each $$V_n$$, there exists a non-spacelike curve that passes through $$V_n$$, leaves $$\mathscr U$$, and then reenters $$V_n$$? By the definition of strong causality, can't we only assume that the non-spacelike curve will leave $$V_n$$ (and not necessarily $$\mathscr U$$)?

2: I don't quite see why you can "join up some $$\lambda_n$$ to give a closed non-spacelike curve" if any two points of $$\lambda$$ have a timelike separation. I would appreciate if anyone could explain more fully how to prove this.