-3
$\begingroup$

Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow related to quantum physics. But when I read the statement of this theorem on Wikipedia, I immediately came to think of a possible application to general relativity. So, considering that space-time is locally isomorphic to $\mathbb{C}^{2}$, can this theorem explain why there should not be any naked singularity?
Sorry if the question is rather vague.

$\endgroup$

1 Answer 1

4
$\begingroup$

Hartog's theorem states that on $\mathbb{C}^n$, $n\ge 2$, if $f$ is a holomorphic function on $G\backslash K$ where $G\subset \mathbb{C}^n$ is open, $K$ is compact and $G\backslash K$ is connected then $f$ can be holomorphically extended to the whole $G$. In other words, singularities cannot stay bounded in compact sets.

So the anwer is No, for two reasons. The first is that the metric components in general relativity are $C^2$, hence in general not holomorphic. The second is that, in any case, naked singularities are expected, mathematically speaking, at infinity: a curve headed at the singularity escapes every compact set. Think for instance of 1+1 Minkowski spacetime with the usual coordinates (t,x) and with the set $x\le 0$ removed. The line $x=0$ is made by naked singularity points.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.