Could GR generalized to non-integer dimension?

[zetafunction]

could GR generalized to non-integer dimension??

let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer dimensions of space time

 

[Al68]

let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer dimensions of space time

What is a "fraction of a dimension"?

 

[JesseM]

What is a "fraction of a dimension"?

Maybe zetafunction is talking about the concept of fractals having fractional dimension--there's a basic conceptual explanation http://math.rice.edu/~lanius/fractals/dim.html, and someone wrote up a good post here, the basic idea being:

The topological definition of dimension is not really very intuitive. It has to do with coverings. The dimension of a space X is m if we cover a space with open sets so as to minimize the amount of overlap, we still have points contained in m+1 sets.

I'd doubt that the topological definition of dimension can be applied to differential geometry to give metric spaces with fractional dimension, which would mean you can't generalize GR to fractional dimensions, but who knows...

 

[DrGreg]

Apparently there are some speculative theories of quantum gravity in which spacetime has a fractal structure. On a macroscopic scale, everything is 4-dimensional as in classical relativity. The suggestion is that as you zoom into quantum scales, the fractal dimension continuously reduces to a lower value, e.g. 2 in the limit. On Planck scales, spacetime becomes a self-similar fractal, it is suggested, so as you zoom in even further there is nothing more to see, just a copy of what you've already seen. Apparently a 2D version of quantum gravity is much easier than higher dimensions. Not that I understand quantum theory, or fractal theory, very well.

Reference: Jurkiewicz, Loll and Ambjorn, "Using Causality to Solve the Puzzle of Quantum Spacetime", Scientific American, July 2008. On page 1, follow the link to "Zooming in on Spacetime".

 

[isabelle]

Fractals are not even manifolds, much less differentiable manifolds. The answer to your question is no, although may be one day someone will find a sweeping generalization of the manifold concept that includes fractals; such a generalization has not been found in mainstream math or physics to date.

 

[Aether]

let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer dimensions of space time

You can complexify dimensions in GR so that the real parts of each dimension are fractional.

 

[Naty1]

I was also thinking along the lines of post #5...but as always approximations might provide an alternative depending on the scales of investigation...

In addition another physical variation is the possibility dimensionality limitations suggested by string theory T duality..insensitivity to size R or 1/R...

And even Planck scale minimums from quantum theory might thwart GR in any formulation to date...
So there seem to be several different views of space/time that are not very smooth nor even classical in concept...more to come, I'm sure...

 

[zetafunction]

But, when applying Dimensional regularization

http://en.wikipedia.org/wiki/Dimensional_regularization

in the sense of T'Hooft and Veltmann , so the dimension -d- is not an integer how can we apply it ??

 

[isabelle]

Dimensional regularization is just a formal trick for taming integrals, it is not equivalent to a geometric consideration of non-integer dimensions: 4 - epsilon etc has symbolic (mathematical) meaning only, not physical meaning, except that the trick leads to correct answers when tested against experiment.