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!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...
The purpose of "G's Guide to GR on Banach Spaces" is to provide a comprehensive and accessible guide to General Relativity (GR) on Banach Spaces, a mathematical framework often used in the field of physics.
The guide is primarily aimed at physicists and mathematicians who have a basic understanding of GR and are interested in learning about its application on Banach Spaces. However, it can also be useful for anyone with a strong background in mathematics who is interested in learning about GR.
Some key concepts covered in the guide include the basic principles of GR, Banach spaces and their properties, the mathematical formalism of GR on Banach Spaces, and applications of this framework to various problems in physics.
A strong understanding of undergraduate-level mathematics, including calculus, linear algebra, and differential equations, is recommended for understanding the guide. Familiarity with basic concepts of general relativity, such as the equivalence principle and the Einstein field equations, is also helpful.
No, the guide is intended to serve as an introductory resource and may not cover all aspects of GR on Banach Spaces. Further reading and research may be necessary for a complete understanding of this topic.