# In order to build a wormhole . . .

1. Nov 16, 2014

### snoopies622

According to Wikipedia's "Wormhole" essay, this is a metric for a traversable wormhole :
$$ds^2 = -c^2 dt^2 + dl^2 + (k^2 + l^2)(d \theta ^2 + sin ^2 \theta d \phi ^2)$$
If we assume the Einstein relation
$$R_{uv} - \frac {1}{2} R g_{uv} = \frac {8 \pi G}{c^4} T_{uv}$$
what kind of distribution of mass / energy is consistent with this spacetime geometry? (I'm guessing k is a constant and l is something like radius? The essay never bothers to say.)

Last edited: Nov 16, 2014
2. Nov 16, 2014

### space-time

This is the same wormhole metric that I am studying. According to my calculations, in an orthonormal basis the stress energy momentum tensor is as follows:

T00 and T11 = - k2c4 /(8πG) (k2 + l2)2

T22 and T33= k2c4 /(8πG) (k2 + l2)2

Note that the two angular elements are the same as the first two except with an opposite sign.

Every other element is 0.

In a coordinate basis, the temporal and radial elements are the same as above.
T22 = k2c4 /(8πG) (k2 + l2) (coordinate basis)
T33 = k2c4sin2(θ) /(8πG) (k2 + l2) (coordinate)
Every other element is 0.

I hope this helps.

3. Nov 16, 2014

### Staff: Mentor

See this thread, particularly my posts #4 and #5:

https://www.physicsforums.com/threa...-of-the-stress-energy-momentum-tensor.781694/

Note that, for physical interpretation, the best components of the SET to use are the mixed components (with one upper and one lower index); the 0-0 component of the mixed tensor is positive (as opposed to the tensor space-time wrote down, with two lower indexes, and a negative 0-0 component).

4. Nov 16, 2014

### snoopies622

Thank you both, very informative! I guess it's those negative signs which make this kind of wormhole physically . . . improbable?

5. Nov 16, 2014

### pervect

Staff Emeritus
You'd need a time machine (more formally, closed timelike curves) to "construct" a wormhole classically - see reference 5 of "Wormholes, time machines, and the weak energy condition", http://authors.library.caltech.edu/9262/1/MORprl88.pdf.

The same paper suggests that 
using a theory of quantum gravity that goes beyond General Relativity, wormholes might exist as a consequence of an idea known as "quantum foam", and that it might be possible to stabilize one of these quantum foam wormholes, possibly turning them into time machines after you've stabilized them.

You'd also need to violate the "weak energy condition" - which means a negative energy density in some frames - to stabilize the wormholes, assuming that they exist.

Last edited: Nov 17, 2014
6. Nov 16, 2014

### snoopies622

Thanks, pervect. Just to be clear, is the metric studied in that paper
$$ds^2 = - e ^ {2 \phi } dt ^2 + dl ^ 2 + r ^ 2 (d \theta ^2 + sin ^ 2 \theta d \phi ^2 )$$
the same one I sited in my original post, but in a slightly different coordinate system?

Edit: Nah, that makes no sense — never mind.

Last edited: Nov 16, 2014
7. Nov 17, 2014

### pervect

Staff Emeritus
The paper that I'm aware of that discusses that particular metric is " M. S. Morris, K. S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56, 395-412, 1988.". It's got many of the same authors as the paper I mentioned, but it's focus is different.

8. Nov 17, 2014

### snoopies622

Excellent, I will print a copy of it (and the other one) when I next get to the UNH physics library.

9. Nov 17, 2014

### serinorah

I haven't had the time these last several years to go back over the equations for wormholes so could someone please remind me why you would need negative energy to hold one open. The last time I asked a professor about it they explained it badly with something about how you energy expanding as it comes out requires the need for it. This made little sense to me, I hadn't been over the equations in over a year at the time, cause that very same energy was previously compressed on the otherside so shouldn't this balance out?

10. Nov 18, 2014

### snoopies622

Well, for the metric in my original post, the negative term in the stress energy tensor apparently just follows from the Einstein gravity equation. And since the Schwarzschild metric is a vacuum solution, I suppose that a Schwarzschild wormhole should properly contain neither positive nor negative energy. (Now I'm wondering, how many wormhole metrics are there?)

11. Nov 18, 2014

### Staff: Mentor

Correct, the Schwarzschild solution is a vacuum solution; but the "wormhole" in it is only open for an instant, and nothing can pass through it from one "universe" to the other--it closes so fast that even a beam of light gets caught in the black hole before it can get through the wormhole.

12. Nov 19, 2014

### stevebd1

13. Nov 19, 2014

### serinorah

Now I know it's been a while but all of the metrics seem to take into account things coming out of it without also considering the fact that anything leaving would have had to enter first. Am I right in this observation or has the time since having looked over those metrics extensively caused me to miss something?

14. Nov 20, 2014

### snoopies622

Aren't they all time independent? I would think that means that they take into account neither matter going into them nor matter coming out of them.

15. Nov 25, 2014

### pervect

Staff Emeritus
Yes, I have seen discussions of what happens when masses travel through a wormhole (for instance Cramer's science columns, and Visser's book), but not a metric.

A very brief description of what happens: Each end of the wormhole has a mass which can be deduced from the gravitational field at long distances. (I've skimmed over some of the technicalities). When an object passes through the entrance to a wormhole, the continuity conditions require that that end gain the mass that entered, when an object exits through the exit, the exit side looses that mass.

See for instance http://arxiv.org/pdf/astro-ph/9409051.pdf

16. Nov 27, 2014