# Convex (null) neighbourhoods

1. Mar 10, 2015

### center o bass

In general relativistic spacetimes, convex neighbourhoods are guaranteed to exist. As a reminder: a convex neighbourhood $U$ is a neighbourhood $U$ such that for any two points $p$ and $q$ in U there exists a unique geodesic connecting $p$ and $q$ staying within $U$.

With that established, does it somehow follow that within a small enough neighbourhood there exist a unique null-geodesic connecting them?

In this paper the author seem to deduce this in his Proposition 1 -- something that he uses to establish the existence of neighbourhoods for which one can assign coordinates to events by sending and receiving light signals.

Last edited: Mar 10, 2015
2. Mar 10, 2015

### Staff: Mentor

Your link (for me) has problems - Springer.com is often behind a paywall....

3. Mar 10, 2015

### center o bass

Does it work for you now? The title is "On the Radar Method in General Relativistic Spacetimes".

4. Mar 10, 2015

### Matterwave

This doesn't sound right to me, think physically about what you are saying. In SR, a subset of GR, two points which are time-like or space-like connected can't possibly be null connected right?

And indeed in GR, barring the existence of conjugate points, the boundary of the chronological future of a point is exactly the causal future of that point minus the chronological future of that point. In other words, even in GR, for small neighbourhoods, the null connected points to a point lie on the boundary of the chronological future of that point.

5. Mar 10, 2015

### Staff: Mentor

6. Mar 10, 2015

### George Jones

Staff Emeritus
This is not what the paper states.

As Matterwave suggests, consider special relativity as an example.

Let $\gamma$ be a worldine, let $p$ be an event on the worldline, and let $U$ be a neighbourhood that contains $p$. Consider any event $q$ in the neighbourhood $U$ that is not on the the worldline $\gamma$. Then, there exists a unique future-directed null geodseic that starts at $q$ and intersects $\gamma$, and there exists another unique future-directed null geodesic that starts at an event on $\gamma$ and runs to $q$.

If $p$ and $q$ are not lightlike related, these null geodesics do not intersect $\gamma$ at $p$, i.e., these null geodesics do join $p$ and $q$.

The null geodesics might leave the neighbourhood $U$ before they intersect the worldline $\gamma$, but everything can be contained in some larger neighbourhood $V$.

See Fig, 1 from the paper. In Fig. 1, $p$ and $q$ are not lightlike related, and the null geodesics intersect $\gamma$ at $\gamma \left( t_2 \right)$ and $\gamma \left( t_1 \right)$, not at $p$.

7. Mar 11, 2015

### center o bass

I agree with you that what the paper states is that one can find two neighbourhoods $U,V$ where $p\in U \subset V$ such that for any $q \in U\ \text{Im}(\gamma)$ there exists a unique future pointing geodesic, as well as a unique past pointing geodesic -- that stays within $V$ -- connecting $q$ to $\gamma$.

But how do we prove this from convexity?

I would presume that we first take $V$ to be a convex neighbourhood and let $\gamma$ be a worldline of some observer going through $V$. By convexity of $V$, we can then connect any point $q$ in $V - \text{Image}(\gamma)$ to a point $r \in \text{Image}(\gamma)$ by a unique geodesic that stays within $V$. This geodesic might be spacelike, null, or timelike. Now, I imagine sliding the point $r$ along $\gamma$ until this unique geodesic becomes null: I guess the subset $U \subset V$ has the purpose of being those point within $V$ than can be connected by a null geodesic, and not just any geodesic, from $\gamma$. However, I do not see any good arguments on why $V$ has to be such that the geodesic can be made into a null geodesic by sliding $r$ along $\gamma$.

Is there such a reason? And how would I prove the statement more formally?