Problems with the "Folded Paper" Analogy to Describe Wormholes

Factual Questions
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The concept of a wormhole in theoretical physics has often been described with a 2D analogy. The recent movie Interstellar referenced the common the approach of folding a piece of paper, then poking a hole thru both halves to demonstrate how a wormhole could link vastly distant locations in space-time.

Now I have never solved (let alone understood) the equations of General Relativity and can’t say how any of the math works out, but I find a few fundamental problems with the analogy if taken to their logical extremes. I was hoping someone more experienced could shed some light on this. As far as I can tell, these problems make it seem HIGHLY unlikely that wormholes would be practical as a means of travel. Please, try to stay with me here 'til the end…

[ol]
[li]Just fold the paper. In this image you see the folded paper approach as in Interstellar. Assuming distance traveled on paper is analogous to distance traveled in space-time, it is true that the wormhole provided a shortcut between two points in space. However, this requires the “fold” and in this case the fold over an infinitely long line. The amount of mass-energy required to bend space an infinite amount would be infinite. So the scenario as it is, as it was given in Interstellar, is impossible.[/li][li]So what if the paper was flat? Here you’ll see the wormhole pulled out over flat space-time. What’s immediately apparent is that the distance traveled in the wormhole is greater than that between its ends. So this is useless… One thought that occurred is perhaps the curved space-time of the wormhole accelerates a traveler to near lightspeed without much fuel. But to me this defeats the purpose. It’s not much of a shortcut and you’d still deal with time dilation effects as well as the limits of light speed.[/li][li]What if we only partially fold the paper? Here I show a wormhole acting as a shortcut between two artificially created “bulges” in space-time. The first problem here is that the mass-energy required to make these “bulges” would dwarf the mass-energy required for the wormhole itself. The second problem is that you’ve now stretched space-time in all the surrounding areas around the wormhole a proportional amount. That’s defeating the purpose![/li][li]What if the paper was already folded? This image shows the only case where I see an effective use for wormholes. For illustration, the image shows space-time curved only in one direction, but there could easily be multi-directional curvature. If this were the case however, we would be observing severe curvature in our own universe and we’re simply not. The universe as far as we can tell is flat, aside from space-time around massive bodies (dark matter included).[/li][/ol]

I hope I’m wrong! Please give me comments and questions! I hope you at least enjoyed the illustrations. I do it part-time but have had a lot of time on my hands as of late!

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  1. The “folded paper” analogy is only an analogy, and not a particularly good one.

But for what it’s worth, folding the Universe does not necessarily require introducing any curvature, and thus need not require any energy. Curvature, in the sense relevant for general relativity, is an intrinsic property of a space, not extrinsic, and folds do not change intrinsic curvature.

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If you’re not introducing curvature then “what” exactly are you doing to it?

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There’s a difference between so-called intrinsic and extrinsic curvature in gravity theory. Roughly speaking, we say that an object is intrinsically curved if it can’t be “flattened out” without distorting its surface. The surface of the Earth, for example, is intrinsically curved, which is why you can’t make a flat map of the entire Earth’s surface and have to deal with various projections instead. On the other hand, the surface of a cylinder (excluding the end-caps) has no intrinsic curvature; we could cut a slit down its length and flatten it out perfectly. The cylinder therefore only has extrinsic curvature; it’s not a fundamental property of the surface itself, but rather how we’ve chosen to have it sit in a higher-dimensional space (three-dimensional space, in this case.)

General relativity only deals with the intrinsic curvature of spacetime, not the extrinsic curvature. As far as we know, our spacetime could be “embedded” in some higher-dimensional space, just as the cylinder is “embedded” in three-dimensional space. But those higher dimensions are inaccessible to us,[sup]1[/sup] and the workings of gravity in our four spacetime dimensions aren’t affected by extrinsic curvature, only intrinsic curvature.

[sup]1[/sup]If you’ve seen the movie,the beings that save Matthew McConaughey at the end live in this higher-dimensional space, and so can pluck him out of the black hole before his demise. These higher dimensions are referred to as the “bulk” in the movie (and by physicists who hypothesize about such things.)

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As MikeS and Chronos have both pointed out you’ve fallen afoul of equating the intrinsic curvature of a manifold with the extrinsic curvature of an embedding of that manifold in a higher-dimensional manifold (nice drawings btw!)

In particular in each of your diagrams you’ve suppressed a dimension of space (for obvious reasons of the difficulties of representing an arbitrary 3D space in a 2D diagram) and then taken a 2D space and embedded it in 3D Euclidean space.

General relativity deals with 4D spacetime manifolds and the physics of it relates mass-energy distribution and the gravitational field to the intrinsic curvature of those manifolds. I.e. embedding and extrinsic curvature are largely superfluous to the theory and certainly not related in any obvious way to the mass-energy distribution.

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A wormhole is a topological feature of space, but in general relativity deals with spacetime and not space, so we first need to ask ourselves what is space?

In general relativity the best analogy of space is the concept of a Cauchy surface. A spacetime has Cauchy surfaces when it has a certain “nice” property of being what is called globally hyperbolic. It must be noted that “most” (for certain definitions of “most”) spacetimes aren’t globally hyperbolic (and hence don’t have Cauchy surfaces), but it is desirable property and many spacetimes representing physical situations are globally hyperbolic.

Spacetimes which are globally hyperbolic can be foliated (or sliced) into Cauchy surfaces, with each Cauchy surface being like a snapshot of space at a moment in time. However some caution is needed as what exactly is “a moment in time” in relativity can be highly subjective so there will be different ways a spacetime can be foliated. For example in special relativity certain foliations can be taken to represent different sets of inertial observers and the fact that observers don’t agree on foliations results in time dilation and length contraction.

If a spacetime is globally hyperbolic, then all its Cauchy surfaces must have the same topology Σ and the topology of spacetime will simply be R x Σ. So for a globally hyperbolic spacetime, the wormhole being a topological feature of Σ means it is also a topological feature of the whole spacetime.

We may also want to deal with the situations when a wormhole forms where there was no wormhole before. This suggests a change in topology of the Cauchy surfaces, which is impossible, so therefore the spacetime cannot be globally hyperbolic. However we might still be able to find regions of the spacetime which are globally hyperbolic and so have Cauchy surfaces. In this case the wormhole would be a topological feature of a region of spacetime.

The reason that being globally hyperbolic is such a nice feature of a spacetime to have is that it gives you a firm, though mutable, concept of the past, present and future for a spacetime and in particular the ability to predict the future from the past. As the formation of a wormhole necessarily means a breakdown in global hyperbolicity it is worth noting it also means a breakdown in the ability to predict the future from the past, so the formation of a wormhole in general relativity is unpredictable (not in a random way, just in the way that knowledge of the past is not enough information to know the future).

So a wormhole at its most basic is either a topological feature of a spacetime or of some region of a spacetime. The curvature of spacetime is defined upon the topology of spacetime so the topology does act as a limit to the ways in which spacetime can be curved, but beyond that it gets a bit too messy for me.

The idea of wormholes as a practical means of travel is pure science fiction though.