**Theorem**
Let $\beta\colon [0,1] \to M$ be a null geodesic. If $\beta(t\_0)$ is conjugate to $\beta(0)$ along $\beta$ for some $t\_0\in (0,1)$, then there is a timelike curve from $\beta(0)$ to $\beta(1)$.


This classical theorem can be found in


* Hawking and Ellis, The large scale structure of spacetime, Prop. 4.5.12,
* O'Neill, Semi-Riemannian geometry, Prop 10.48,
* Beem, Ehrlich and Easley, Global Lorentzian geometry, Theor. 10.72
* Kriele, Spacetime, Lemma 4.6.15


and the proofs are more or less refinements over Hawking and Ellis'. I am following the notation by Beem et al. I do not understand a step in the proof.


They construct a variation $\alpha\colon [0,1]\times (-\epsilon,\epsilon)\to M$, $(t,s)\mapsto \alpha(t,s)$, $\alpha(\cdot,0)=\beta$, $\alpha(0,\cdot)=\beta(0)$ and $\alpha(t\_2,\cdot)=\beta(t\_2)$ for some $t\_0<t\_2\le 1$. Denoting $T=\alpha\_\* \partial/\partial t$, $V=\alpha\_\*\partial/\partial s$, in all these references it is claimed that if


* $\frac{d }{d s} g(T,T)\vert\_{s=0}=0$, $t\in [0,t\_2]$
* $\frac{d^2 }{d s^2} g(T,T)\vert\_{s=0}\le 0$, $t\in [0,t\_2]$ with strict inequality in $(0,t\_2)$,


then for sufficiently small $s\ne 0$ the varied curves are timelike.


I assume that they are using here Taylor's theorem together with a continuity and compactness argument.
However, it seems to me that this argument does not work as it stand, at most one can conclude that for every $\delta>0$ there is some $b$, $0<b<\epsilon$, such that the varied curves $\alpha\_s$, $s\in (0, b]$ are timelike on the time interval $[\delta,t\_2-\delta]$. Indeed the third derivative could be diverging for $t\to 0,t\_2$. Observe that trying to bound the second derivative by a negative constant $-c$, $c>0$, as done e.g. by Beem et al. does not solve this problem as long as this bound holds only over $\beta$ (that is for $s=0$) (by the way the second derivative would not be continuous at $t=0,t\_2$ since there $\frac{d^2 }{d s^2} g(T,T)=0$).


Am I missing something? Incidentally, do you have completely different proofs, perhaps more topological? 



Conjugate or focusing points on null geodesics imply chronality
It seems that your main argument against the correctness of the proof in Beem, Ehrlich and Easley, Global Lorentzian geometry, Theor. 10.72, is that $\frac{d^2}{ds^2}g(T,T)|\_{s=0}=0$ at $t=0$ and $t=1$. Anyway, I don't see why $\frac{d^2}{ds^2}g(T,T)|\_{s=0}=0$ should be zero there. In my opinion the proof by Beem, Ehrlich and Easley is correct (and also the one in Hawking and Ellis). Please observe that finding a negative bound on the second derivative along $\beta$ in $(0,1)$ is enough, by continuity, to have negative second derivative on a small rectangle $ [0,1]\times (-\epsilon,\epsilon)$. 


