- #1

- TL;DR Summary
- How must we link the Christoffel's symbols and the concept of torsion, accordingly to the fact that the Christoffel's symbols are always symmetric in the two lower indices?

I have a technical problem.

1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices.

2. These symbols have been introduced when studying the preservation of differential forms of degree two. The results which have been obtained at this time in that document have proved to be useful “a posteriori”; for example: in general relativity (GR).

3. When one looks precisely at the formalism of these symbols (second kind; see a recent article - 2007, e.g.: [01; §2 p. 217, (2)]) and exchange the positions of lower indices but impose the equality, one then automatically obtains a condition on the metric.

This fact induces questions for which some help is needed:

a. Although it is hard to believe such restriction: “Does that condition “de facto” restrict the domain of definition of GR?

b. Why do a lot of (many) scientific articles desperately introduce a concept of torsion based on a presumed possible anti-symmetry of these lower indices (which is in total contradiction with 1. and 2.)? Very roughly, something like: T(^a, _b, _c) = Gamma(^a, _b, _c) - Gamma (^a, _c, _b) which should always vanish.

c. Did I misunderstood something essential concerning these difficult topics (preservation of the element of length, connections, metrics)?

I am referring to:

[01] Annales de la Fondation Louis de Broglie, Volume 32 n. 2-3, 2007 : On a completely anti-symmetric Cartan torsion tensor.

Thanks for help.

1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices.

2. These symbols have been introduced when studying the preservation of differential forms of degree two. The results which have been obtained at this time in that document have proved to be useful “a posteriori”; for example: in general relativity (GR).

3. When one looks precisely at the formalism of these symbols (second kind; see a recent article - 2007, e.g.: [01; §2 p. 217, (2)]) and exchange the positions of lower indices but impose the equality, one then automatically obtains a condition on the metric.

This fact induces questions for which some help is needed:

a. Although it is hard to believe such restriction: “Does that condition “de facto” restrict the domain of definition of GR?

b. Why do a lot of (many) scientific articles desperately introduce a concept of torsion based on a presumed possible anti-symmetry of these lower indices (which is in total contradiction with 1. and 2.)? Very roughly, something like: T(^a, _b, _c) = Gamma(^a, _b, _c) - Gamma (^a, _c, _b) which should always vanish.

c. Did I misunderstood something essential concerning these difficult topics (preservation of the element of length, connections, metrics)?

I am referring to:

[01] Annales de la Fondation Louis de Broglie, Volume 32 n. 2-3, 2007 : On a completely anti-symmetric Cartan torsion tensor.

Thanks for help.