Expanding manifold with constant boundary

In summary: QFT? I'm not sure if I'm understanding this correctly.In summary, the AdS/CFT correspondence is a conjecture that says that what happens on the boundary space completely determines the physics on the conformal brane.
  • #1
Mike2
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OK. Suppose you have a surface with a closed curve as a boundary. Then suppose that surface grows like a soap bubble but the boundary is stationary like the orifice through which air passes to make the bubble grow. It would seem that the 2D surface grows in both dimensions in the middle of the bubble, but the buble is not growing in at least one dimension along the boundary. What would be the equation for the metric both in the middle and on the boundary and as it approaches the boundary? :cool:

I wonder all this because in a different thread I explore the possibility that matter may be the boundary of an expanding universe. If so, I wonder what distinction there is in the metic of space as it approaches the boundary (particles). I'm kind of thinking that matter may be like a stationary boundary where the growing space must somehow bend and stretch to accommodate a fixed boundary. But the photon particles may be where the boundary grows right along with the surrounding space. I suppose you could have a boundary that has portions that expand and protions that are fixed. :rolleyes:
 
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  • #2
I'd have to think that this would give a similar metric as used in GR. For if the surrounding space has expanded while the boundary surface of mass particles remains fixed, then you would have in effect a contraction of space near mass, just as in GR, right?
 
  • #3
Now I wonder if a manifold with a boundary can have a metric just as easily as a manifold without boundary? If it can, then we can also have a metric on a manifold with a boundary that is distributed into smaller closed boundaries, right?

Thanks.
 
  • #4
Mike2 said:
Now I wonder if a manifold with a boundary can have a metric just as easily as a manifold without boundary? If it can, then we can also have a metric on a manifold with a boundary that is distributed into smaller closed boundaries, right?

Thanks.
I suppose there would be a discontinuity at the boundary points themselves, right? But if there is an undefined metric beyond the boundary, how would that be expressed, by an infinity there, by a zero? Or would we simply specify that the metric is a function defined only for the manifold and not to be extended past the boundary?
 
  • #5
Mike2 said:
I suppose there would be a discontinuity at the boundary points themselves, right? But if there is an undefined metric beyond the boundary, how would that be expressed, by an infinity there, by a zero? Or would we simply specify that the metric is a function defined only for the manifold and not to be extended past the boundary?


I don't see why a singularity is necessary. You could have nice boundary conditions at the boundary, with good limiting behavior of the derivatives. Note that the AdS/CFT research area is about a manifold with an important boundary.
 
  • #6
selfAdjoint said:
I don't see why a singularity is necessary. You could have nice boundary conditions at the boundary, with good limiting behavior of the derivatives. Note that the AdS/CFT research area is about a manifold with an important boundary.
Thanks. Would you be kind enough to tell me a little more about this AdS/CFT research?

I'm stuck here at the top trying to justify my way to the bottom, and I'm not sure how to proceed. So I ask question about how GR or QM can be justified by first principles. It is probably irritating to the rest of you. But those kinds of questions are inevitable. Please be patient.
 
  • #7
AdS means anti deSitter space, a Riemannian manifold with constant negative curvature. It is the boundary of a brane which supports Conformal Field Theory (CFT). The conjecture is that what happens on the boundary space completely determines the physics on the conformal brane. This is called the holographic conjecture because it is analogous to the way a 2-dimensional hologram cvan accurately capture 3-d shapes in the round. This is a very active research program.
 
  • #8
selfAdjoint said:
AdS means anti deSitter space, a Riemannian manifold with constant negative curvature. It is the boundary of a brane which supports Conformal Field Theory (CFT). The conjecture is that what happens on the boundary space completely determines the physics on the conformal brane. This is called the holographic conjecture because it is analogous to the way a 2-dimensional hologram cvan accurately capture 3-d shapes in the round. This is a very active research program.
I read a brief intro to AdS-CFt correspondence at:
http://arxiv.org/PS_cache/hep-th/pdf/0003/0003120.pdf

Most of it over my head, of course. Did I read right that a Quantum Field theory in d+1 was not a quantum theory in the AdS of dimension d? Does a path integral in the d+1 of CFT translate to a different kind of path integral in the d dimensions in AdS? Or does the path integral of CFT not not translate to a path integral of AdS? Wouldn't it be great if we could justify QM in d+1 from a classical view in AdS of d dimensions? That's probably too much to hope for.
 
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  • #9
This is a valuable paper that I wasn't aware of. Thanks for finding it. The paper is written in the mathematical physics tradition of C*-algebras acting on Hilbert space, where path integrals and other quantization techniques don't appear. It specifically proves that the conformal quantum field theory on d-dimensional Minkowski space determines the quantum field theory on d+1-dimensional space. So no, we don't get a free pass from classical theory to quantum theory.


(added after scanning the paper)

In fact he proves that the algebras of local observables are the same, so the quantum theories are identical, although the physical interpretation of the observables is radically different in the two spaces. An example, as he stresses, of the algebraic approach.
 
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FAQ: Expanding manifold with constant boundary

1. What is an expanding manifold with constant boundary?

An expanding manifold with constant boundary is a mathematical concept in differential geometry that describes a space with a boundary that is continuously expanding. This means that the boundary remains fixed in size, while the interior of the space grows larger.

2. How is an expanding manifold with constant boundary different from a regular manifold?

An expanding manifold with constant boundary differs from a regular manifold in that it has a defined boundary that remains fixed in size while the interior of the space expands. In a regular manifold, there is no boundary or the boundary is not fixed in size.

3. What is the significance of studying expanding manifolds with constant boundary?

Studying expanding manifolds with constant boundary has important implications in various fields such as cosmology and thermodynamics. These spaces can be used to model the evolution of the universe or to study the behavior of physical systems with fixed boundaries.

4. Can an expanding manifold with constant boundary exist in the real world?

There is currently no evidence to suggest that an expanding manifold with constant boundary exists in the physical world. However, it is a useful mathematical concept that can aid in understanding certain phenomena and can be used to make predictions about the behavior of physical systems.

5. What are some examples of expanding manifolds with constant boundary?

One example of an expanding manifold with constant boundary is the Robertson-Walker metric, which is used in cosmology to describe the expanding universe. Another example is the de Sitter space, which is a solution to Einstein's field equations and is used to study thermodynamic systems with fixed volume.

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