DefTensor
• DefTensor[tensor[-a, b, D, ...], deps] defines tensor to be a tensor field with dependencies deps (one or a list of several manifolds and/or parameters) and index structure as given by the abstract indices -a, b, D, ...
• DefTensor[tensor[-a, b, D, ...], deps, syms] in addition defines the tensor field with permutation symmetries syms, given as a generating set (head GenSet) or a strong generating set (head StrongGenSet).
• Currently tensor must be a valid Mathematica symbol, not in use in the present session with any other meaning. In the future a pattern will also be valid to allow for "indexed names".
• The slot structure of the tensor (SlotsOfTensor) is stored as {-M1, M1, M2, ...}, where indices a,b belong to manifold M1, index D belongs to manifold M2, etc.
• A tensor field has dependencies (DependenciesOfTensor), that can be either manifolds and/or parameters. The order in the list deps is irrelevant because it is overwritten using the private function SortDependencies.If the tensor has no dependencies then use {}.A tensor is always a tensor field on the manifolds corresponding to its indices; that is, we consider that a nonscalar tensor cannot be a constant on a given manifold. This is because we need additional structure to show that a tensor field does not depend on that manifold (for example a vector field, in order to take Lie derivatives).
• The symmetries of a tensor (SymmetryGroupOfTensor)are given using group-theoretical notation. For example a symmetric tensor T[a,b] would have a symmetry group described by GenSet[Cycles[{1,2}]] or GenSet[Perm[{a,b}], etc. There are three commands that generate strong generating sets in three frequent cases: Symmetric, Antisymmetric and RiemannSymmetry. See the documentation for the package xPerm`. Symmetries are always stored as strong generating sets, and permutations are always stored in Cycles notation, using numbers.
• Tensors defined with indices on 1-dim vbundles are automatically defined as totally symmetric on those groups of indices of the same vbundle having the same character. The presence of a metric in that vbundle does not introduce symmetries mixing characters.
• Tensorial densities are represented using Ashtekar's tilde notation, with a number of tildes equal to the weight of the density (WeightOfTensor) above (below) the symbol if the weight is positive (negative). The tildes are colored according to the basis used to form the density.
• Possible values for the Dagger option are Real, Imaginary, Complex and Hermitian.
• Options:
Dagger Real behaviour under complex conjugation
ForceSymmetries False whether symmetry for mixed up/down
FrobeniusQ False does it obery the Frobenius condition?
Info {"tensor", ""} information on the defined tensor
Master Null master of new symbol
OrthogonalTo {} orthogonality to a list of vectors
PrintAs Identity string or function of symbol for output
ProjectedWith {} list of projectors
ProtectNewSymbol $ProtectNewSymbols whether to protect new symbol or not
TensorID {} further properties of the tensor
VanishingQ False wheter the tensor vanishes or not
WeightOfTensor 0 weight of tensor as a density
• Special messages:
DefTensor::wrongsym = "Symmetry properties are inconsistent with indices of tensor.", thrown when the given symmetries involve both up-indices and down-indices and the option ForceSymmetries is not set to True.
DefTensor::nodummy = "Tensor cannot be defined with dummy indices", thrown when there are paired indices in the tensor to be defined.
DefTensor::zero = "Symmetry makes tensor zero. Use VanishingQ instead.", thrown when the tensor is detected to be zero as a consequence of its symmetry properties.
• See: Section 4.4.
• See also: UndefTensor
• New in version 0.
• Last update: 4-XI-2007 for version 0.9.3 of xTensor`.
| Created by Mathematica (May 16, 2008) | |