Some questions about geometry on pseudo-Riemannian manifolds

I will be using relativistic terminology for pseudo-Riemannian manifolds as in the book by Barret O' Neil.

• If one can show that the chronal future and the chronal past of some set are disjoint then why is this enough for showing that the set is achronal?

I mean one can have a set such that there is precisely one time-like curve intersecting it at just two distinct points. Then clearly this set is not achronal (because this one curve exists) but chronal future and past of this set will include one each of these points and will be disjoint.

Does something rule out this possibility?

• Let $p$ be a point on the boundary of some future set. Why is showing that chronal future and chronal past of $p$ are disjoint enough to prove that this future set is edgless?

• How generic is the situation that Cauchy Horizon of a set separates that part of the causal future of a set which is not a part of its domain of dependence?

Is this something on which intuition can be built or is this a very special scenario?

• Consider the interval $(0,1)$ on the x-axis on the $2-$dimensional Minkowski space where time is the y-axis. For this interval is the Cauchy Horizon the union of the set of points on the lines, $y=-x$ and $y=x-1$ ?

If yes, then I guess this is an example where the Cauchy Horizon is not a part of the domain of dependence of the interval which is the set of points above the given interval and contained in between the above two straight lines.

Is this right?

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For the first question, I fear that you may have not understood the definition completely. Let $p,q\in U$ be the two points. Let $\gamma$ be the future time-like curve joining $p = \gamma(0)$ and $q= \gamma(1)$. I claim that for any $s\in (0,1)$, $\gamma(s)$ is in both the future and past of $U$. Indeed, $\gamma(s)$ is in the future of $p\in U$, hence is in the chronal future of $U$. And $\gamma(s)$ is in the past of $q\in U$, hence is in the chronal past of $U$. Hence the chronological future and the chronological past of $U$ must intersect non-emptily, i.e. they are not disjoint.

Indeed, this shows that if a set $U$ is not achronal, then $I^+(U)$ and $I^-(U)$ must intersect. Hence if the chronological future and past are disjoint, the set $U$ must be achronal.

For question two, I don't think your statement is right. Can you give me a reference (page number in O'Neill will do)?

For question three, isn't that just the definition? The Cauchy Horizon is generally regarded as the boundary of the domain of dependence, no? If that's not the case, can you provide the definition you are using for the Cauchy horizon?

For question four: not quite. The future Cauchy horizon of the set is the union $\{ y = x: x \in [0,1/2] \} \cup \{ y + x = 1: x \in [1/2,1]\}$. The past Cauchy horizon is $\{ y = -x: x \in[0,1/2] \} \cup \{ x-y = 1: x\in[1/2,1]\}$.

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1. That was very stupid of me to not consider the other points on the timelike curve which are not the intersection points! – Anirbit Dec 14 '10 at 11:30
The second argument is what seems to be getting used in Corollary 27 of Chapter 14 in that book to show that boundary of a future set is a hypersrface. I think I was misreading the argument. It only shows edgelessness of of the boundary and doesn't say anything about the future set as I was implying in the question! Sorry. Anyway I would like to know about the intuition behind the idea of edge of an achronal set. Why is this important? – Anirbit Dec 14 '10 at 11:35
3. Thanks for correcting the Cauchy horizons. I was not correctly identifying the future domain of dependence I was mistakenly only looking at points from where at least one past directed causal curve intersects the interval when I should have been looking at points from which every past directed causal curve intersects the interval. In general is there are method to find the domain of dependence and the Cauchy horizons given a set? – Anirbit Dec 14 '10 at 11:43
The point of this open interval example is to illustrate that here the Cauchy horizon is not a part of the causal future of the open interval. This is not very clear to me. – Anirbit Dec 14 '10 at 11:45
This open interval kind of counter-example" (though not clear to me!) make me wary of the kind of intuitive picture of the Cauchy horizon that I tried stating in the third point and hence my question. – Anirbit Dec 14 '10 at 11:47