About the instability of wormholes

[snoopies622]

"The Einstein-Rosen bridge was discovered by Albert Einstein and his colleague Nathan Rosen, who first published the result in 1935. However, in 1962 John A. Wheeler and Robert W. Fuller published a paper showing that this type of wormhole is unstable, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to the other exterior region."

- from Wikipedia's entry on wormholes.

I have not read this paper. Can anyone explain this to me in a little more detail?

 

[snoopies622]

Ok, I got the paper at my local physics library. First question: it looks like the metric for an Einstein-Rosen bridge is the same as the Schwarzschild metric? That's kind of weird. A black hole isn't a wormhole.

 

[PeterDonis]

Ok, I got the paper at my local physics library. First question: it looks like the metric for an Einstein-Rosen bridge is the same as the Schwarzschild metric? That's kind of weird. A black hole isn't a wormhole.

Actually, it sort of is, if you look at the "maximally extended" Schwarzschild spacetime. The easiest way for me to picture it is to look at the Kruskal chart of Schwarzschild spacetime. There are two horizons on this chart, which appear as 45-degree lines, one going "southeast to northwest" (the "antihorizon") and one going "southwest to northeast" (the "horizon"). They meet in the center, and at that point the two exterior regions, which are the "wedges" to the left and right, touch at that single center point for an instant. This can be viewed as a sort of instantaneous "wormhole" between the two exterior regions. AFAIK that instantaneous wormhole is the Einstein-Rosen bridge.

 

[PAllen]

A few other comments on the maximally extended Schwarzschild geometry might be of interest to the OP.

The white hole singularity is in the past of any timelike worldline that extends at all into the white hole interior region. Similarly, the black hole singularity is in the future of any time like world line extending at all into the black hole interior.

Kruskal coordinates also show the white hole horizon collapsing at speed c to a point, from whence the black hole horizon grows at c. This last aspect (coordinate speed of c) will look different in other coordinate schemes, but the geometry won't change. This shows how the interior regions are not static at all, while the exterior regions are static.

 

[snoopies622]

Hey thanks. You've both given me much to think about.

 

[HarryRool]

Ok, I got the paper at my local physics library. First question: it looks like the metric for an Einstein-Rosen bridge is the same as the Schwarzschild metric? That's kind of weird. A black hole isn't a wormhole.

This wormhole FAQ might be of interest to you. Here's an excerpt:

How are wormholes related to black holes?
Unlike a wormhole, a naturally occurring black hole -- one created through stellar collapse -- is not a bridge between two universes (or distant regions within the same universe). There nevertheless exist certain solutions to the Einstein equations of general relativity in which a bridge between universes – a wormhole -- appears to have a black hole at either end. This is the sense in which certain theoretically possible black holes can be said to be wormholes.

Can a wormhole with a back hole at either end be traversed?
No. The bridge between universes remains open for too short a time for any traveler to cross it.

If wormholes that are black holes are not traversable, in what sense do they form an inter-universe or intra-universe connection?
They do in the sense that people from different universes (or from distant parts of the same universe) can both enter the black hole and meet each other. The meeting will likely be interrupted, however, by the violent deaths of both parties in the black hole’s singularity.

 

[snoopies622]

The bridge between universes remains open for too short a time for any traveler to cross it.

Is this the case with all wormholes, or just the Schwarzschild kind? What about something like

$$ds^2 = c^2 dt ^2 - dl^2 - (k^2 + l^2)(d \theta ^2 + sin ^2 \theta d \phi ^2)$$

which is also mentioned in the Wikipedia entry?

 

[PAllen]

It is claimed by many experts (and disputed by some) that a wormhole can be kept open with negative energy to allow passage through it. I have never worked through one of the relevant papers in detail, so I reserve judgment. The relevance, even if true, to our universe is extremely dubious, because exceeding large amounts of negative energy are needed, and the only hints for such in our universe are things like the Casimir effect, which fall within limitations on negative energy derived from quantum mechanics. There is a current research program, making good progress, to re-derive the classical No-Go GR theorems (based on positive energy constraints) based on allowing deviations from such constraints within the inequalities allowed by QM. Assuming this program is successful, we could make the statement the radically new fundamental physics (not just difficult engineering) would be needed to achieve things like traversable wormholes (and alcubiere drive).

 

[snoopies622]

What I'm wondering is why a wormhole like the one I just described in entry #7 would close in the first place. I read somewhere (unfortunately I can't find the essay now) that it's because they're all vacuum solutions, but our solar system isn't a vacuum and yet one can still describe it using the Schwarzschild metric. The presence of planets, asteroids, dust, photons, etc. doesn't cause it to instantly collapse.

 

[PAllen]

What I'm wondering is why a wormhole like the one I just described in entry #7 would close in the first place. I read somewhere (unfortunately I can't find the essay now) that it's because they're all vacuum solutions, but our solar system isn't a vacuum and yet one can still describe it using the Schwarzschild metric. The presence of planets, asteroids, dust, photons, etc. doesn't cause it to instantly collapse.

The metric in #7 is traversible. However, it requires enormous negative energy to create this metric (i.e. if you compute the Einstein tensor from it, which equals the stress energy tensor, you would find large negative energy terms).

 

[snoopies622]

Ah. Well that's what I get for not taking the trouble to acquire the computing power to get to the bottom of these things myself. I thought it had an Einstein tensor of zero.

 

[snoopies622]

So then, is it mathematically proven somewhere that a wormhole with any metric requires negative energy either to create, to hold open, or both?