Mathematics – Differential Geometry
Scientific paper
2005-10-14
J.Phys.Conf.Ser. 30 (2006) 152-162
Mathematics
Differential Geometry
11 pages
Scientific paper
10.1088/1742-6596/30/1/018
Algebraic curvature tensors possess generators which can be formed from symmetric or alternating tensors S, A or tensors \theta with an irreducible (2,1)-symmetry. In differential geometry examples of curvature formulas are known which contain generators on the basis of S or A realized by differentiable tensor fields in a natural way. We show that certain curvature formulas for stationary or static space-times contain such differentiable realizations of generators based on \theta. The tensor \theta is connected with the timelike Killing vector field of the space-time. \theta lies in a special symmetry class from the infinite family of irreducible (2,1)-symmetry classes. We determine characteristics of this class. In particular, this class allows a maximal reduction of the length of the curvature formulas. We use a projection formalism by Vladimirov, Young symmetrizers and Littlewood-Richardson products. Computer calculations were carried out by means of the packages Ricci and PERMS.
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