Why a timelike geodesic maximizes path length? I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is this statement easy to prove? I mean, does it follow straightly from the locally minimizing property of geodesics in Riemannian manifolds? If yes, please explain it to me, otherwise suggest me some reference where I can found the proof.
It seems that those minimizing/maximizing properties depend upon the geodesics' causal structure (timelike, spacelike or null-like) and, in general, nothing can be said about those geodesic's properties without knowing its causal structure. Is this correct?
 A: First we sketch a proof that a timelike geodesic is a maximum of proper time. (We exclude saddle points for now.) Let $\gamma$ be a curve satisfying the geodesic equation, i.e. it is an extremum of proper time defined by $\tau[\gamma]:=\int\sqrt{-\langle\dot\gamma,\dot\gamma\rangle}\,\mathrm{d}t$. It is fairly simple to show that there always exists a curve $\mu$ for which $\tau[\mu]<\tau[\gamma]$, implying $\gamma$ is not a minimum. Construct along $\gamma$ a "tube" which is arbitrarily wide. Let $\mu$ be a curve which has the same start and end points as $\gamma$. Let $\mu$ be confined to the tube along $\gamma$. Now wind $\mu$ along the tube so that it is almost null, i.e. the curve's tangent approaches the null cone at every point on the tube. Thus we have constructed a curve with $\tau[\mu]$ arbitrarily close to zero, which can be made less than $\tau[\gamma]$. 
This implies that a geodesic is not a minimum, but cannot determine that a timelike geodesic is not a saddle. However, this is not entirely true either. Here we quote Theorem 9.9.3 in [1]$^1$.
Let $\gamma$ be a smooth timelike curve connecting two points $p,q$. Then the necessary and sufficient condition that $\gamma$ locally maximize the proper time between $p$ and $q$ over smooth one parameter variations is that $\gamma$ be a geodesic with no point conjugate to $p$ between $p$ and $q$.
So a timelike geodesic is not necessarily a maximum of proper time. The study of geodesics does tie in to causal structure, Refs. [1] and [2] are highly recommended for this purpose.
Two standard references on causal structure are:
[1] R.M. Wald, General Relativity (1984).
[2] S.W. Hawking & G.F.R. Ellis, The large scale structure of space-time (1973).

$^1$This is in turn quoted from Proposition 4.5.8 in [2], but I prefer [1]'s wording. Note that the full proof is found in [2].
A: Think of the twin paradox, in which one twin travels out into space in a spaceship at high speed and the returns, while the other twin remains stationary on Earth. In the end, the twin traveling in a spaceship will have aged less because of time dilation. Arguably, the twin who remains on Earth travels along a geodesic, and he will have aged more than his twin (who deviated from the geodesic), i.e., his timelike path will have a greater proper time (as the “length” of a timelike path is its proper time). This is exactly the path maximization principle in general relativity.
