Let $(A,V,g)$ be a Lorentzian affine space. Prove that:
- For no $P\in A$, it is verified that P < P (that is, there is no causal curve that start and end in P).
My problem is that I do not know how to proceed to show that any causal curve cannot start and end at the same point because I only arrive again and again at the definition of strongly causal but that is not what I should prove. Hence my question, if someone could give me an idea to solve it ...
- The chronological future of P, $I^+(P)$, in affine coordinates centered on P coincides with the future temporal cone T of V, that is, $I^+(P)$ $= P + T$.
My idea for this exercise is prove that if $\alpha: [a, b] \rightarrow A$ is a future time curve starting from P, then initially enters T and also whenever $\alpha(s)$ is in T, the function $s \rightarrow g (\alpha (s), \alpha (s))$ is decreasing.