For such cases there is no decomposition theorem for the individual scalar, vector and tensor components themselves as that would violate gauge invariance, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance (though not of covariance itself) in ...
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4.1 Decomposition Let Sab be any symmetric 2-index tensor in a spacetime endowed with a metric gab and let ua be a timelike vector field. Without loss of generality, we can assume that ua is nor- malized to ucuc = −1 (otherwise one can always normalize it). Define the 3-metric ...
►exprTypeTraits< ::Foam::sphericalTensor > ►exprTypeTraits< ::Foam::symmTensor > ►exprTypeTraits< ::Foam::tensor > ►exprTypeTraits< ::Foam::vector > ►exprTypeTraits< bool > ►exprValuePointPatchField ►extendedCellToCellStencil ►extendedCellToFaceStencil ►extendedCent...
Parallel transport around a small, closed loop allows calculation of the curvature tensor, which then determines if the surface is curved (Rkjmi≠0) or flat (Rkjmi=0). Next, we determine the change ΔA in the contravariant components of vector A after parallel transport around an arbitrary,...
As already discussed in section 1.9.1, a scalar is a zero-order tensor, and a vector is a first-order tensor. Therefore, for any scalar field ϕ, the gradient operator increases the order of the tensor by one to produce a vector field ∇ϕ. Similarly, if V is a vector field, ...
is that the low-energy limit of string theory possesses scalar modes in addition to the tensor and vector modes. Perhaps one of these modes contributes to the gravitational sector in the way that we propose. Thus,we shall assume from the start that the torsion tensor admits the decomposition...
This PR looks a bit more chaotic than it is because I moved some instances around and then lake shake got very excited (over half of the files changed are just changes to imports). Here is an expla...
II. De?nitions and Notation A linear vector space which carries an irrep of SU3 is fully speci?ed by two integers (p, q) henceforth referred to as the irrep labels. The dimension of the space is d = (p + 1)(q + 1)(p + q + 2)/2. (1) A complete set of d orthogonal ...
2. Metric and massive scalar fieldperturbationUsing the Arnowitt-Deser-Misner (ADM) decompositionand introducing the lapse functionN, the shift vectorNiand the spatial metricgij, we have [14]ds2=−N2c2dt2+gij(dxi+Nidt)(dxj+Njdt);(1)and the IR modification HL action which readsI=∫dtd...