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BasicDDI

BasicDDI
is a reserved word in xTras. It is used to label the basic dimensional dependent identities associated to different covariant derivatives.
  • Like curvature tensors, the name of the basic DDI associated to a covariant derivative CD is BasicDDICDdim, where dim is the dimension of the manifold of CD.
  • The basic DDI is automatically defined when ConstructDDIs is called, but not before.
  • The basic DDI is the generalized Kronecker delta with 2 (d+1) indices, where d is the dimension of the manifold of the associated covariant derivative. It is antisymmetric in its first d+1 indices, and antisymmetric in its last, and has a pairwise symmetry upon the interchange of these two sets of indices.
  • The basic DDI encodes the antisymmetrization of d+1 delta functions, which is identically zero in d dimensions. This relation is stored in BasicDDIRelations.
  • The basic DDI is completely traceless.
  • The basic DDI lives in the rectangular Young diagram consisting of two columns and d+1 rows, which is zero in d dimensions.
EXAMPLESCLOSE ALL
Basic Examples   (1)
Say we have a 3-dimensional manifold with metric g and covariant derivative CD, the basic DDI is given by
This is a rank-8 tensor which is identically zero in 3 dimensions. Its expression in terms of metrics is
The basic DDI is antisymmetric in its first 4 indices, and its last:
It is symmetric under the simultaneous interchange of the first and last set of indices:
It is completely traceless:
Say we have a 3-dimensional manifold with metric g and covariant derivative CD, the basic DDI is given by
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This is a rank-8 tensor which is identically zero in 3 dimensions. Its expression in terms of metrics is
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Out[2]=
The basic DDI is antisymmetric in its first 4 indices, and its last:
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Out[3]=
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It is symmetric under the simultaneous interchange of the first and last set of indices:
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It is completely traceless:
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