Tears in spacetime shielded by strings

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Hi all,

Alright so while reading through some stuff, I came across something peciluar, string/M-theory state that rips in spacetime are a natural phenomenon and happen all the time. "Edward Witten showed that strings, which have spatial extant, can travel while encircling the tear as it takes place. This, in effect, cancels out the potentially disastrous consequences of an actual rip in space. This conclusion relies on the fact that the two-dimensional worldsheet a string "sweeps out" as it moves essentially shields the rest of the universe from the potential effects. Moreover, there is always a string available to provide the shield - according to Feynman's sum-over-paths method of quantum mechanics calculations, all objects travel from one place to another by traveling along all possible trajectories. This ensures that a infinite number of strings will be passing by the tear when it occurs, thus protecting the universe from the effects of the singularity."

Now as crazy as that sounds I want to find a way to remove this shield, hypothetically this should cause a cosmic catastrophe but let's not worry about that at the moment :) so how can one go about doing that? Possibly with a lot of energy but strings themselves are made of energy so how does that work out? Also another request, can anyone point me to the paper where Ed. Witten shows this concept? Thanks a lot :)

- Vikram
 

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  • #2
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The essay you were reading is based on chapter 11 of Brian Greene's The Elegant Universe, which describes how complementary papers by http://arxiv.org/abs/hep-th/9301042" [Broken] came to be written.
string/M-theory state that rips in spacetime are a natural phenomenon and happen all the time
To understand the context of these papers, you must first remember that mathematically or as a matter of possibility, string theory can describe strings propagating on infinitely many different geometric backgrounds, which differ by size and shape of the extra dimensions, the fluxes, and so on. These papers describe circumstances in which the topology of the extra dimensions does change. In our universe, or in the visible part of it, that topology should be stable. Even the geometry of the extra dimensions shouldn't be changing. We are in a long-term stable state and that is why galaxies had the chance to form.

Furthermore, we don't actually know the topology of the extra dimensions. String theory has not yet produced a specific model which gives us back the masses and charges of the known particles - physicists are still going through all the possibilities. So we don't even know exactly what effects are stabilizing the topology, and therefore we don't know what we would have to do to destabilize them. But presumably, even a whole galaxy's worth of energy concentrated into a small region is not enough to do it, because the universe is full of supermassive black holes and physics remains stable.

The paper by Witten is quite technical and is only incidentally about topology change (section 5.5). To understand it, you first have to understand a standard switch of perspective in string theory, from "strings in space" to "fields on the string". Unfortunately I can't see an elementary exposition of this idea on the web that I'm happy with. But basically: The string is an object which moves around in nine dimensions of space, so each point on the string has a nine-dimensional coordinate vector (it would be ten, if you count the time coordinate as well). So if you change your perspective to "inside" the string, it's as if, at each point on the string, you have a set of nine numbers. (Actually, it becomes eight numbers, once you allow for rescaling the "interior", "length" coordinate inside the string. The tenth number, corresponding to the time coordinate in the space outside the string, disappears for the same reason, except you are rescaling the internal time inside the string.) Anyway, it turns out that mathematically, a string moving through ten space-time dimensions can be represented as eight "scalar" fields in two space-time dimensions - space and time inside the string. (Then you have some other, fermionic fields inside the string which make it a "super"string.) This is the "worldsheet" perspective.

In the end, superstrings moving through a particular geometric background, like four large space-time dimensions with six extra space dimensions forming a Calabi-Yau manifold, are equivalent to a "worldsheet" theory in two space-time dimensions. It's nothing more than a description from inside the string, a description from the string's perspective. But this is the perspective which originally made string theory tractable.

I think what is happening in Witten's paper, is that he is showing that strings on a particular type of background are described, from the worldsheet perspective, by an already known type of field theory. That connection is the main topic of the paper! But in section 5.5, he looks at a type of phase transition in that field theory, and shows that it corresponds to a topology change in the Calabi-Yau manifold.
 
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mitchell, thanks a lot your helpful reply. I looked around for that paper but couldn't find it because i didn't know what it was called, I do have the moduli space paper written by Greene though. I am just pulling some ideas together to write something but for that I needed to know what that swapping/shielding phenomenon is called, now I see from what you described (if I picked it correctly):

"Anyway, it turns out that mathematically, a string moving through ten space-time dimensions can be represented as eight "scalar" fields in two space-time dimensions - space and time inside the string. (Then you have some other, fermionic fields inside the string which make it a "super"string.) This is the "worldsheet" perspective."

This notion of containing the spacetime inside the string is how the shielding works right? (please pardon my non-technical language) So then as it moves through space this effect must be temporary, but since there are an infinite number of these strings passing around, there must be an infinite number of shields that permanently make the fabric of spacetime stable. That ties back to your idea of our universe being stable but I see the problem in trying to remove this shield, ridiculous amounts of energy would be needed since ever supermassive black holes can't do it. So if energy can't do it then there must be another way, perhaps I should reformulate the problem and then see what happens. Anyhow, thanks a lot for your description and the link to papers :)
 
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No, you don't have it yet, but you should be able to get there.

First, the worldsheet perspective is not so important here. I mentioned it only to explain Witten's paper. It just means that you're doing your calculations from a position onboard the string - feeling the space-time pass through the string, if you want a metaphor, rather than the string passing through the space-time. But to understand these ideas about topology change, you definitely need to go back to the space-time perspective, and think of the strings moving through space.

Next, when Brian Greene says this is a tear in space, usually he's not talking about something as disruptive as tearing a piece of paper in half, but about changes in the connectivity of high-dimensional space which still leave it "smooth" overall. The "tears" happen to the circles and spheres which describe the connectivity.

http://en.wikipedia.org/wiki/Simply_connected_space" [Broken] for a lower-dimensional example. The torus has two distinct types of circle on it - one goes around the donut hole, the other one goes through it. But on the surface of a sphere, every circle is contractible to a point. So if you turned a torus into a sphere, by contracting a "through-the-hole circle" to a point, and then pulling it apart at that point, those circle-types get broken up. That's what happens in these Calabi-Yau topology changes.

The physics produced by string theory on a particular geometry is largely determined by that connectivity structure of the extra dimensions, because the strings can wrap around the circle-types (technically called 1-cycles), and p-dimensional branes wrap around p-cycles (spheres, hyperspheres, etc). So if the connectivity structure changes, some types of strings and branes are no longer possible, or new ones might become possible, and so the set of interacting entities which make up fundamental physics will change.

We need one more conceptual ingredient, which is mirror symmetry. This was the discovery that Calabi-Yau spaces come in pairs which, although topologically different, give rise to the same physics. (Technically, there are "Hodge numbers" which indirectly provide information about the cycles, and in mirror manifolds, the Hodge numbers are swapped around.)

Finally, we can now discuss the flop transitions a little more precisely. The papers by Witten and by Aspinwall et al discuss them from different sides of mirror symmetry.

In the flop transition, the Calabi-Yau manifold develops singularities due to contraction of cycles. In Aspinwall et al, they consider a mirror of this manifold which does not develop singularities. Now remember, the mirror-pair manifolds look the same from the perspective of observable physics. So in Aspinwall et al, they deal with the singularity by switching to another equivalent perspective in which no singularity occurs.

In Witten's paper, he doesn't consider the mirror manifold. Instead, he remains in the original manifold, but looks at higher-order quantum processes. I don't know how much you know about quantum theory, but the standard way to think about quantum processes since Feynman is in terms of a "sum over histories" which derives the probability of a particular outcome by summing over all the ways you can get there from the starting condition. This includes "virtual particles" which get emitted and then absorbed in the course of a history; they only exist temporarily, but they still affect the overall probability. The same thing happens in string theory.

I believe that what Witten shows (in a subsection of 5.5 called "The instanton sum") is that virtual strings wrapping the topological cycles which don't contract to a point dominate the probabilities, in a way which drowns out the effects arising from strings wrapping the cycle which does contract during the topology change. (I say I "believe" this because I haven't actually understood that part of Witten's paper, I'm just relying on an interpreting statement by Greene in http://arxiv.org/abs/hep-th/9702155" [Broken] - page 80, first paragraph, last sentence.) That is the "shielding".
 
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  • #5
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In the flop transition, the Calabi-Yau manifold develops singularities due to contraction of cycles. In Aspinwall et al, they consider a mirror of this manifold which does not develop singularities. Now remember, the mirror-pair manifolds look the same from the perspective of observable physics. So in Aspinwall et al, they deal with the singularity by switching to another equivalent perspective in which no singularity occurs.

In Witten's paper, he doesn't consider the mirror manifold. Instead, he remains in the original manifold, but looks at higher-order quantum processes. I don't know how much you know about quantum theory, but the standard way to think about quantum processes since Feynman is in terms of a "sum over histories" which derives the probability of a particular outcome by summing over all the ways you can get there from the starting condition. This includes "virtual particles" which get emitted and then absorbed in the course of a history; they only exist temporarily, but they still affect the overall probability. The same thing happens in string theory.

I believe that what Witten shows (in a subsection of 5.5 called "The instanton sum") is that virtual strings wrapping the topological cycles which don't contract to a point dominate the probabilities, in a way which drowns out the effects arising from strings wrapping the cycle which does contract during the topology change. (I say I "believe" this because I haven't actually understood that part of Witten's paper, I'm just relying on an interpreting statement by Greene in http://arxiv.org/abs/hep-th/9702155" [Broken] - page 80, first paragraph, last sentence.) That is the "shielding".
Thanks, this is an interesting subject and you are explaining it masterfully! So this seems like an example where an interesting physical phenomenon is explained in two completely different ways. On the one hand, a symmetry is exploited to remove a singularity. On the other hand, it is shown that quantum amplitudes are insensitive to the type of singularities in question.

Torquil Sørensen
 
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