G's Guide to GR on Banach Spaces

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Discussion Overview

The discussion revolves around the application of General Relativity (GR) in the context of Banach spaces, particularly exploring whether GR can be formulated using infinite-dimensional Banach manifolds instead of the traditional finite-dimensional pseudo-Riemannian manifolds. Participants examine the implications of this shift, including issues related to non-linearity and the foundational aspects of manifold theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the existence of formulations of GR that utilize infinite-dimensional Banach manifolds, noting potential issues with strong Riemannian structures in this context.
  • Another participant points out that Riemannian or semi-Riemannian structures are typically based on finite-dimensional bases, suggesting a limitation when considering infinite-dimensional spaces.
  • Concerns are raised regarding the modeling of GR's non-linearity within a linear space, which may complicate the application of infinite-dimensional frameworks.
  • A participant emphasizes that while the manifold used in GR is finite-dimensional, the tangent space at each point is also finite-dimensional, aligning with the manifold's dimensionality.
  • There is a suggestion that the choice of 4-manifolds in GR is motivated by physical considerations, particularly in modeling spacetime, and questions arise about the feasibility of infinite-dimensional models.
  • A reference to an external article is provided for further exploration of the topic, although it does not specifically address Banach spaces.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of infinite-dimensional Banach manifolds to GR, with some highlighting limitations and others questioning the foundational aspects of the current formulations. The discussion remains unresolved regarding the potential for a coherent model of GR in this context.

Contextual Notes

Participants note limitations related to the non-linearity of GR and the properties of strong Riemannian structures that may not extend to infinite-dimensional settings. There are also references to the physical motivations behind the choice of finite-dimensional manifolds in GR.

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

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