**A bit of General Relativity and Causality theory**

One feature of general relativity is that the space-time is modelled as a Lorentzian manifold. The Lorentzian metric on the manifold has signature (-+...+), and thus distinguish between *time-like*, *space-like*, and *null* directions. A $C^1$ curve in your manifold is said to be time-like if its tangent vector is always time-like (similarly space-like/null).

For physical reasons we expect our manifold to be time-orientable: the set of time-like vectors in the tangent space has two connected components. We assume there is a continuous (non-vanishing) time-like vector field, which denotes the direction of "future".

A point $p$ is said to be in the chronological future of another point $q$ if there is a time-like, futurely directed curve originating from $q$ to $p$.

**Asymptotic regions**

For (possibly outdated by modern cosmological observation) physical reasons, one may assume that the local geometry of space-time far away from (large) gravitating bodies is nearly flat. Let $M$ be a Lorentzian manifold which for now we assume to be diffeomorphic to $\mathbb{R}\times \Sigma$, with the restriction of the metric on the $\Sigma$ leaves Riemannian. An asymptotic end of $\Sigma$ is a connected component of $\Sigma \setminus K$ for some compact set $K$ that is diffeomorphic to $\mathbb{R}^d\setminus B_1$. This induces a nice coordinate system on $M\setminus (\mathbb{R}\times K)$ (which allows us to talk about the radius function $r$). We make the flatness assumption (which can be justified by possibly enlarging $K$) that in each asymptotic region, future (or null) directed null rays $\gamma$ that remain in the asymptotic region for all parameter $s\in [s_0,\infty)$ will have $\lim_{s\to\infty} r(\gamma) = \infty$.

Fix now one asymptotic region, call it $M_\infty$. Its corresponding *domain of outer communications* $D_\infty$ consists of all points of $M$ which can be reached from $M_\infty$ by both a future time-like curve and a past time-like curve.

Intuitively speaking, each domain of outer communications is a connected region which can "communicate" with the asymptotic region through exchange of light-signals. In particular, it lies outside any black holes or white holes.

**Worm holes**

A simple description of a worm-hole space-time is just a connected Lorentzian manifold with multiple domains of outer communication.

For them to be of physical interest, we require there to exist a time-like or null curve originating from one domain of outer communication and ending in another. This represents the ability to send signals or even travelers between two "almost disjoint" regions of the universe.

So far it should be clear that you need some knowledge of **differential/pseudo-Riemannian geometry** at the very least to study worm holes.

The mathematical study of wormholes essentially boils down to writing down

- Sufficient conditions to rule out their existence
- Explicit examples of wormhole solutions

Let me start with the latter to motivate the former. The most classical solution to Einstein's equation in general relativity that admits a wormhole is the Reissner-Nordstrom charged black hole solution. This solution models a charged black hole, and the explicit analytic extension of the solution to the interior of the black hole reveal some possible structure. In particular, observers who fall into the black hole may not necessarily encounter a singularity in finite time (as opposed to the situation in Schwarzschild black holes). And *based on the analytic extension*, it is possible that such an observer will emerge after some time into another domain of outer communications by being spat out of a white whole.

Two key things to notice:

- This worm hole is "one-way only"
- I kept using the phrase
*analytic extension*

As it turns out, under fairly generic conditions on the properties of the universe and the matters residing within it (in particular the dominant energy condition which also played a strong role in the Positive Mass Theorem of Schoen and Yau), worm holes can "only be one way". (Actually, with certain additional assumptions worm holes cannot even exist.) Results of this type are captured (as easy corollaries) under the name "Topological Censorship". The original version of the result states that the domain of outer communication must be (under certain reasonable physical assumptions) homeomorphic to $\mathbb{R} \times (\mathbb{R}^d\setminus B_1)$. A recent improvement by Galloway and collaborators states that (under reasonable physical assumptions including *global hyperbolicity* [more on which later]) any future directed time-like or null curve that starts from one asymptotic region and ends in another (a priori possibly different) asymptotic region must be entirely contained in one domain of outer communication. (Which basically states that wormholes don't exist unless you violate one of the physical assumptions; and if you remove global hyperbolicity, you can possibly obtain one-way worm holes.)

(There is also a much more classical result in the reverse direction: if a space-time admits a Cauchy hypersurface [a special type of a spatial slice] that has non-trivial topology, it must be geodesically incomplete [which suggests that there should be a black hole].)

The second issue at play is this analytic extension business. If you know a bit about the wave equation, please recall that it has the *finite speed of propagation* property. That is, when solving the initial value problem, the value of the solution at any given point in space-time depends can only on a compact sub-set of the initial space-like slice. (Or, in physical description, information does not travel faster than the speed of light.) Applying this to Einstein's equation of general relativity, one obtains the notion of "Maximal Globally Hyperbolic Extension" corresponding to a set of given initial data: the largest set in the space-time which is completely determined by the given initial data.

A priori, it is not clear whether the maximal globally hyperbolic extension can be further extended as a solution to the Einstein's equations. (Of course, such an extension will be non-unique, but it is interesting to ask whether they exist at all.) And in fact, in certain situations, *more* extension can be constructed by (real) analytic continuation of the solution. The case of Reissner-Nordstrom is one particular one, and it is due to this possible extension that we have the wormhole behaviour.

An open problem in mathematical general relativity is whether one can rule-out such extensions "generically". The "strong cosmic censorship conjecture" posits that the answer is positive: that the existence of such extensions requires the space-time to be conspiratorially nice.

Going back to the issue of constructing solutions/examples: as any example, to be physically meaningful, will be required to solve Einstein's equation, topological methods tend to be less useful in this context (unless you mean Gromov's h-principle), as geometry and analysis tend to be more "rigid" structures.

In classical mathematical GR, as you can see, topology has not come into play strongly. (The "problems" introduced by topology tend to be considered as pathological; lots of work were done in the 60s and 70s to discuss sufficient conditions to rule out such pathologies.) Differential/pseudo-Riemannian geometry and mathematical analysis (esp of PDEs) are generally used more heavily.

But that is not to say that topology has no role. It is more that the role of topology is much less clear, beyond the more simple applications of point-set topology, and some slightly less trivial applications based on connections between Riemannian geometry and topology.

In conclusion, let me give you some possible references if your student can be satisfied by *black holes* instead of *wormholes*. Some classical results to look into that in some way involves topology:

- The Singularity Theorem of Hawking and Penrose (see, Wald,
*General Relativity*; O'Neill, *Semi-Riemannian Geometry will applications to relativity*)
- The Black Hole Topology Theorem (see this paper for the geometric aspects; the topological aspects requires knowing about the Yamabe problem)

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