How Can We Construct a Wormhole?

[qinglong.1397]

How to embed a wormhole?

http://www.physics.louisville.edu/wkomp/teaching/spring2006/589/final/wormholes.pdf"

Please download the paper and then look at Fig. 1(b).

I want to know, how we can construct such a wormhole. Would you please help find the metric describing the universe with the wormhole?

Thank you guys!

 

[zhermes]

It's an embedded diagram; not an embedded, physical, wormhole. This is simply referring to a means of representing the concept of a wormhole, it is not a new/different entity.
The metric is included in the paper.

 

[qinglong.1397]

It's an embedded diagram; not an embedded, physical, wormhole. This is simply referring to a means of representing the concept of a wormhole, it is not a new/different entity.
The metric is included in the paper.

Thanks for your reply! But the metric is actually describing Fig. 1(a).

 

[zhermes]

This is simply referring to a means of representing the concept of a wormhole, it is not a new/different entity.

Same thing.

 

[qinglong.1397]

Same thing.

Why are they the same? They have different topologies. Fig. 1(a) is an open universe, while (b) is closed if the spacetime still has axisymmetry. For the manifold, different metrics mean different topologies. So I think they are different.

 

[zhermes]

Neither 'a' nor 'b' is locally not necessarily open or closed. The metric is for an overall open spacetime of course. Which manifold? And yes, often different metrics mean different topologies; but more importantly different topologies always mean different metrics---still I don't see what your point is.

'l' is describing the effective radial direction from the mouth of the wormhole. For a worm-hole between two different universes positive and negative 'l' would then connect to metrics for the overall geometry of each, whether they are the same or different. For the same universe, 'l' would instead refer to different regions of the same universe. 'l' is not necessarily the same as the physical space dimensions (e.g. x,y,z in minkowski space, or r in schwarzschild).

 

[qinglong.1397]

Neither 'a' nor 'b' is locally not necessarily open or closed. The metric is for an overall open spacetime of course. Which manifold? And yes, often different metrics mean different topologies; but more importantly different topologies always mean different metrics---still I don't see what your point is.

'l' is describing the effective radial direction from the mouth of the wormhole. For a worm-hole between two different universes positive and negative 'l' would then connect to metrics for the overall geometry of each, whether they are the same or different. For the same universe, 'l' would instead refer to different regions of the same universe. 'l' is not necessarily the same as the physical space dimensions (e.g. x,y,z in minkowski space, or r in schwarzschild).

It should be that topology determines metric. I was wrong. But since the two figures are different--I mean they have different overall topologies--they should have different metrics. I just want to know how to write down the metric for the second figure.

I understand your second paragraph, but I do not think you can embed that metric just like Fig. 1(b).

 

[n1person]

It is my understanding that only in two dimensions will there generally be topological restriction to a given metric, caused by the integral of the ricci scalar being proportional to the Euler characteristic.

$$\int dx^2 \mathcal{R}=\chi$$

However I think that in higher dimensions permissible metrics are not related to permissible topologies.

EDIT:

Another way of looking at the question is that solving the EFEs are fairly local, in that you can look at the metric for that wormhole separately from the rest of space-time assuming you don't care about about the source terms elsewhere.

In general it will be impossible to write a nice metric for any sort of system with more than one source.

 

[qinglong.1397]

It is my understanding that only in two dimensions will there generally be topological restriction to a given metric, caused by the integral of the ricci scalar being proportional to the Euler characteristic.

$$\int dx^2 \mathcal{R}=\chi$$

However I think that in higher dimensions permissible metrics are not related to permissible topologies.

EDIT:

Another way of looking at the question is that solving the EFEs are fairly local, in that you can look at the metric for that wormhole separately from the rest of space-time assuming you don't care about about the source terms elsewhere.

In general it will be impossible to write a nice metric for any sort of system with more than one source.

So we should express the metric piecewise, just like a piecewise function?

 

[HarryRool]

So we should express the metric piecewise, just like a piecewise function?

Sort of. From what I've read it's the so called manifold that is defined piecewise and the metric is defined on the manifold.

The manifold is the breaking up of spacetime into regions together with the definition within each region of a reasonable coordinate system (in which each point gets a unique set of coordinates).

You can see that you need to do that, when you look at the earth. The latitude-longitude coordinate system breaks down at the poles, e.g. the North Pole has ambiguous longitude.
To fix that, you could define a manifold that breaks the surface of the Earth into 3 regions: a small North Pole region, a small South Pole region, and everywhere else.