# Traversable wormholes: a simple system

From my layman understanding of wormholes, you can only have traversable ones if you take into account the existence of some exotic matter with negative energy relative to the vacuum energy.

If you look for black hole solutions to the Einstein equation, you only need a very massive object. What about the traversable wormholes? What kind of celestial objects would you need for such thing to exist? In other words, what would be the simplest system that would have a traversable wormhole.

This is a related question on SE. They say the exotic matter should curve the space inversely compared to regular matter. I'm not sure what that means? (Antigravity, maybe?)

You are quite correct that exotic matter is required to build a wormhole. But if we ignore this minor problem then wormholes are surprisingly easy to construct.

Some examples of simple wormholes are described in the paper Traversable wormholes: Some simple examples by Matt Visser. My favourite is just to make a cube out of wire i.e. use the wire for the twelve edges of the cube. The only problem is the wire has to be made from exotic matter and it has to be extraordinarily dense exotic matter. The linear density needs to be around $−1.52 \times 10^{43}$ joules/metre. For comparison the mass of the whole Earth is only $+5.4 \times 10^{41}$ joules.

Clearly this is hopelessly unrealistic for two reasons: firstly that exotic matter doesn't exist and secondly that even if it did we'd need the equivalent of several hundred Earth's worth of the stuff. But if we ignore this minor detail then the cube of exotic matter would be a perfectly good traversable wormhole. Anyone passing through the faces and staying clear of the exotic matter at the edges, could pass through the wormhole without experiencing destructive tidal forces.

But we need to be cautious. We know a structure of this type would be a wormhole, but we have no idea what would actually happen when we built it or whether we could use it to connect chosen regions of space. I asked about this in Building a wormhole but it appears no-one knows the answer.

• Thanks for the paper and the explanation. I'm reading and rereading the first paragraph in section 2 (Models) and I still don't understand how he gets the throat from the two copies of space-time with the surface $\Omega\times \rm R$ missing. Aug 22, 2017 at 17:05
• @Magicsowon: this is something that would be best discussed in the chat room. I'm there now for a few minutes, or I'll be in the chat room tomorrow from about 05:00 UTC. Aug 22, 2017 at 17:46
• @Magicsowon: in the mean time I wrote a related answer on the SciFi Stack Exchange that gives a more popular science level description. Aug 22, 2017 at 17:48
• Looking at your 2D answer, I can see what he meant. Aug 22, 2017 at 18:27
• "traversable wormhole" typically refers to a bidirectional Lorentzian wormhole. Is it possible to relax the energy conditions on the throats if the wormhole is unidirectional? (it has a black event horizon in the entrance and a white event horizon on the exit) Sep 2, 2019 at 15:12

The wormhole is a modification of spacetime, and not necessarily that connected to a black hole, where by some "magic" there is more space in a region then ordinarily would be. A case of this odd situation with space is to have two disks of paper where you cut a wedge out of one of them and then glue that wedge into a radial region of the other disk. The first disk is a cone and the second has an odd "floppy" appearance. The first of these if we can smooth out the point on the cone is a dome or part of a sphere while the disk with excess area is a saddle shap once we iron out all the excess area evently through the thing. We are robbing Peter to pay Paul so to speak by taking space from one disk and putting it in the other. The spherical cap or dome has positive curvature and the saddle shaped 2-dimensional space has negative curvature.

The metric for flat Minkowski spacetime is $$ds^2~=~dt^2~-~dr^2~-~r^2(d\theta^2~+~sin^2\theta d\phi^2),$$ and we can modify this to make the Morris-Thorne wormhole with line element $$ds^2~=~dt^2~-~dr^2~-~(r^2~+~\rho^2)(d\theta^2~+~sin^2\theta d\phi^2).$$ Here $\rho$ is the radius of the throat or minimum radius of the bell of the hyperboloid. This add additional surface area to the $2$ sphere defined by $r^2(d\theta^2~+~sin^2\theta d\phi^2)$ with the throad radius $\rho$. So a foliation of two-surfaces making three dimensional space has an appearance in a 2 dimensional projection as the catenoid seen below.

The connection terms are rather odd in a way. For instance two of them are $$\Gamma^\theta_{r\theta r}~=~\Gamma^\phi_{r\phi r}~=~\frac{r}{r^2~+~\rho^2},$$ Where one can derive geodesic equation with an acceleration in the $\theta$ and $\phi$ directions. The curvatures are $$R_{r\theta r\theta}~=~-\frac{\rho^2}{r^2~+~\rho^2},~R_{r\phi r\phi}~=~-\frac{\rho^2sin^2\theta}{r^2~+~\rho^2},~R_{\theta\phi\theta\phi}~=~\rho sin^2\theta.$$ These strange $\theta$ and $\phi$ dependent curvatures are saying there is this negative curvature by “cramming more space” into this region. This negative curvature is what is induced by so called exotic matter.

How does this connect to black holes? We can now modify this model so that $$ds^2~=~\left(1~-~\frac{2m}{r}\right)dt^2~-~\left(1~-~\frac{2m}{r}\right)^{-1}dr^2~-~(r^2~+~\rho^2)(d\theta^2~+~sin^2\theta d\phi^2).$$ If we identify the event horizon with the throat radius of the wormhole then $\rho~=~2m$, and sometimes we have $\rho~=~2m~+~\delta r$ which permits an entering particle to avoid the event horizon. A physically more realistic situation might be where $\rho~<~2m$ so the wormhole throat is inside the black hole. In that way an observer can only wormhole around inside the black hole without causing unfortunate “issues” of chronology and causality in the outside world. We think of this black hole as having been laced with this exotic matter. If this stuff falls towards the black hole is reaches a certain density near the horizon as seen by a distant observer and its negative energy and negative curvature adjusts the structure of the black hole. • This doesn't seem to have much to do with the question.
– user4552
Aug 22, 2017 at 15:59