Exploring the Interior/Exterior of a 3 Sphere in GR

[robousy]

Hi,

As I understand things we exist on the 'surface' of a 3 sphere of radius R in the context of general relativity. The popular analogy is that our 3D space can be visualized as the surface of an expanding balloon


I would like to ask if anything 'exists' on the interior/exterior of the balloon.

More precisely I would like to ask if the int/ext contains the vacuum of quantum field theories.

I am interested if zero-point energies can exist on the int/ext of the balloon and I am also interested in calculating casimir energies with our universe representing the boundary conditions (the plates in the popular casimir effect).

Before one can do this clearly the question is can the int/ext of the balloon be considered tangible field theoretic manifolds.

Thanks in advance.

 

[JesseM]

As I understand things we exist on the 'surface' of a 3 sphere of radius R in the context of general relativity. The popular analogy is that our 3D space can be visualized as the surface of an expanding balloon

That's only if the average density of mass/energy is above a certain critical value, "Omega", which gives space positive curvature like the surface of a sphere. If the density is equal to Omega then space is flat like a plane, and if it's below Omega then space is negatively curved, which is usually described as being analogous to the shape of a saddle. See the more detailed explanation and diagrams on page 3 of Ned Wright's cosmology tutorial. Current evidence suggest that space is flat, or very close to it.

I would like to ask if anything 'exists' on the interior/exterior of the balloon.

Not according to general relativity. While there's nothing to say it's impossible that curved 3D space couldn't be "embedded" in some larger 4D space, like the 2D surface of a sphere sitting in our 3D space, mathematically there is no need for such a thing--instead of describing the curvature of a surface with reference to a higher-dimensional "embedding space", it is possible to describe curvature using purely intrinsic features that could be observed by a being confined to the surface (like whether the sum of angles of a triangle drawn on the surface is more, less, or equal to 180 degrees), and general relativity uses only such intrinsic features to describe what it means for space to be curved (see this page on differential geometry, the mathematical basis for general relativity, which talks about the difference between intrinsic and extrinsic descriptions of curvature).

More precisely I would like to ask if the int/ext contains the vacuum of quantum field theories.

Not according to any current theory. I'm not even sure what it would mean for it to "contain" this vacuum, since the vacuum is supposed to refer to properties of the ordinary 3D space we see around us.

 

[robousy]

Thank-you for taking the time out to give such a detailed response Jesse.

Its nice to have an idea and be able to bounce it off of other physicists almost instantaneously - and not have to wait a day to speak to my supervisor or a postdoc!

I had a quick look at the links that you provided and they look like exactly what I am looking for so thanks for that too!