Subject: Fwd: Numerical code and symmetry of spin connection
Date: Sun, 1 Apr 2007 03:42:32 EDT
I have programmed the transformation formula (48) of paper 56 which computes the spin connectin from the tetrade. I used the mixed covariant/contravariant Levi-Citta symbol of eq. (52). It seems that there is a typo in the next to last line (eps 301 must have negative sign).
As a result I get the non vanishing omega elements according to eqs. (55-57) (computed from a tetrad with random numbers, see below). Some questions: 1. Must omega be antisymmetric in a and b? It isn’t in my result (maybe there is still an error in my program). 2. According to my considerations (antisymmetric matrix in a-b with 4 index values mu each) it should be 24 independent components, or 48 non vanishing components including antisymmetry. 3. Is this the most general case, as far as eq. (48) is valid? If yes, memory allocation can be optimized in the code.
a,b,omega: 1 0 -1.45918111318275 a,b,omega: 2 0 -0.675285081099360 a,b,omega: 3 0 0.783896032083388 a,b,omega: 0 1 -1.45918111318275 a,b,omega: 2 1 -0.675285473186179 a,b,omega: 3 1 0.783895639996569 a,b,omega: 0 2 -0.675285081099360 a,b,omega: 1 2 0.675285473186179 a,b,omega: 3 2 -3.920868194323862E-007 a,b,omega: 0 3 0.783896032083388 a,b,omega: 1 3 -0.783895639996569 a,b,omega: 2 3 3.920868194323862E-007
I agree with almost all of these points, there are indeed 24 elements in eqn. (51) of paper 56. The raising and lowering of indices is done with the metric of eq. (50). In eq. (48) the spin connection is antisymmetric in a and b because of the antisymmetry of the eps sup a sub b c symbol. Thus omega sup a sub b is the antisymmetric matrix correpsonding to the axial vector q sup c. This is rotational motion. More geenrally, the spin conection is asymmetric if there is a combination of transaltion and rotation. The spin connection omega sup a sub mu b does nor transform as a tensor, as is well known, but it is always used in Cartan geometry as part of a wedge product, and the complete wedge product transforms as a tensor. I am not sure where the random numbers come in.