Convex Neighbourhoods in Relativistic Spacetimes

[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.

 

[jim mcnamara]

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

 

[center o bass]

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

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

 

[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.

 

[jim mcnamara]

Thanks for fixing the link.

 

[George Jones]

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.

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 $p$.

 

[center o bass]

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?
t.

This is not what the paper states.

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 $p$.

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?