Riemann Tensor: Questions & Geometric Interpretation

[victorneto]

Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the same vertex A and going to B, using the other route. In B, the vector Vi will change to V'i, such that the variation will be given by:

ΔVⁱ = + (δa) (δb) Rⁱ_jklV^jU^kU^l,

where (δa) (δb) is the area of the infinitesimal rectangle, and Uk and Ul are unit vectors tangent to the trajectories. The question I put to discussion and clarification is as follows. 1) Most authors, B.Shultz, Dunsby, Professor H. Fleming (USP, São Paulo, Brazil), among others, make the deduction of the above formula, using the scheme described above and detailing, used in the use of indexes.

2) Other authors, such as Baez, give the above formula as

ΔVⁱ = - (ε) ² Rⁱ_jklV^jU^kU^l, + (Oε³)

(Certainly incorporates in (Oε³) the approximations for the 1st order that does, in several phases of the calculations).

But Landau, in Theory of the Field, formula (91.5), p. 342, finds the following expression:

ΔVⁱ = - (1/2) Rⁱ_jklV^jU^kU^l, Δfkl,

where Δf^kl is the infinitesimal area bounded by the two paths; that is, represents the product (δa) (δb) U^kU^l

(In some formulas the sign (-) appears in other o (+), without further explanation.)

The formula ΔVⁱ = - (1/2) Rⁱ_jklV^jU^kU^l, V^j Δf^kl,

obtained by Landau incorporates the factor (1/2), was obtained using the powerful Stokes! Theorem, which gives definite and definitive results, while the previously commented formulas were obtained in a geometric tour de force, step by step, and where it does not appear in the final result, said factor (1/2).

I checked all the calculations (that is, I have refined all, to convince myself, and, from the point of view of mathematical procedure, I found nothing to justify the absence of the factor (1/2) obtained through Stokes; it is necessary, as I said above, to make numerous approximations throughout the deductions ...).

But, objectively, I have doubts. I have not found out yet what lies behind these results, or what justifies them, against Landau. And I also understand that the signal that precedes the expressions should not be mere conventions, but the result of the difference between vectors against and covariates. If covariant, (+), if contravariant, (-).

Could you clarify?

 

[victorneto]

.I request the special finesse of moving the post Tensor of Riemann - Geometric interpretation to the forum Differential Geometriy.

Muito grato.