G's Guide to GR on Banach Spaces

[MathematicalPhysicist]

OK, I started reading GR for mathematicians from Wu and Sachs.

And I see that from the start that they look on finite dimensional linear algebra, has there been any treatment for a general setting?

MP

 

[WannabeNewton]

Are you asking if there have been formulations of GR that replace the usual 4 - dimensional pseudo riemannian manifold with infinite dimensional banach manifolds (note that the notion of a strong riemannian structure finds issue for general infinite dimensional banach manifolds)?

 

[MathematicalPhysicist]

Yes, Rimmenain or semi-riemmanian are based on finite dimensional bases, I think. I didn't finish the textbook, I just started it.

Excuse me for spelling mistakes, haven't had a good sleep for more than two weeks...

 

[TrickyDicky]

Besides the infinite dimensionality issue mentioned by WN, the main problem is trying to model GR's non-linearity with a linear space.

 

[MathematicalPhysicist]

Isn't the manifold that being used (3+1 hyperbolic cone) has finite dimension, i.e you need just finite dimensional linear algebra to deal with this?

 

[WannabeNewton]

Yes, Rimmenain or semi-riemmanian are based on finite dimensional bases, I think. I didn't finish the textbook, I just started it.

Excuse me for spelling mistakes, haven't had a good sleep for more than two weeks...

Well the manifolds themselves have nothing to do with linear algebra in general. The tangent space at every point however is finite dimensional and happens to have the same dimension, as a vector space, that the manifold has as a topological manifold. There is, of course, a physical reason we choose 4 - manifolds, since we are after all modeling space - time, and it happens that Einstein's version is one of the simplest of metric theories of gravity that just happen to agree with experiment. I'm not sure how one would model GR based on an infinite dimensional Banach manifold (or even more general a Frechet manifold) and the problem is that some nice properties of the strong riemannian structure, like inducing an isomorphism between tangent and cotangent space which is taken advantage of regularly, need not carry over in general. Try taking a look at this however: http://link.springer.com/article/10.1007/BF02724475?LI=true. Cheers!

 

[atyy]

http://arxiv.org/abs/1302.6382 (large D, but nothing about Banach space).