Physics[ToCovariant] - rewrite in covariant form the indices of the tensors of a given expression
Physics[ToContravariant] - rewrite in covariant form the indices of the tensors of a given expression
ToCovariant(tensorial_expression, optional ...)
ToContravariant(tensorial_expression, optional ...)
any tensorial expression, or a set, list, equation or matrix of them, onto which the operation is performed
changecharacteroffreeindices = ... : synonym changecharacteroffreeindices, the right-hand side can be true or false (default), to change or not (flip covariant <-> contravariant) the character of the free indices
evaluateexpression = ... : can be true or false (default); to evaluate or not the expression after having manipulated its tensor indices
evaluatetensor = ... : can be true or false (default); to evaluate or not the tensors after manipulating their indices
onlytheseindices = ... : can be any symbol representing a tensor index, or a set or list of them possibly found in tensorial_expressions, to restrict the operation to only those indices
changerepeatedindices = ... : can be true (default) or false, in which case the repeated indices are returned unchanged
quiet = ... : the right-hand side can be true or false (default), to display or not information related to matching keywords
When working with tensors in spaces where the covariant and contravariant tensors' components have a different value (the underlying metric is not Euclidean) one frequently wants to express formulations with some or all of the tensors's indices expressed either in covariant or contravariant form. In previous Maple releases, also in Maple 2021, you can raise or lower free indices multiplying by the metric and performing the contraction. That, however, involves a whole simplification process not always desired, and does not result in flipping the character of repeated indices. The SubstituteTensorIndices is also useful for that purpose but requires changing the indices one by one. Instead, to handle the whole manipulation operation, you can use ToCovariant and ToContravariant.
Several options are available to adjust the operation in different ways, as explained in the Options section above. Perhaps two more relevant ones are changecharacteroffreeindices (default value is false), that can be used to receive an expression where you get all free indices flipping their character, and onlytheseindices = ... to restrict the operation to only some of the indices.
Note that closely related to ToCovariant and ToContravariant, the Physics package includes a SubstituteTensorIndices command.
Set coordinates and a tensor for experimentation; avoid redundant display of functionality using CompactDisplay
⁢Systems of spacetime coordinates are:⁢X=x,y,z,t
A⁡x,y,z,t⁢will now be displayed as⁢A
Consider the following tensorial expression, define it as the components of a new tensor Fμ,ν
Defined objects with tensor properties
The covariant components of Fμ,ν are
Related to the character of indices, note first that, since Maple 2021, when you request the contravariant components of a tensor definition as F⁢μ,ν, the output is already expressed using the contravariant components of the tensors involved, in this case Aμ
The definition of Fμ,ν involves only free and covariant indices; make all the tensors be expressed using covariant indices without changing the mathematical value of the expression
Restrict that operation to only μ
* Partial match of 'only' against keyword 'onlytheseindices'
In the above we see the tensors originally having μ covariant, now with the index in that position contravariant. To achieve the other possible meaning of to contravariant use the option changecharacteroffreeindices
* Partial match of 'changecharacter' against keyword 'changecharacteroffreeindices'
An expression that has free and repeated indices:
There are several tensors with covariant indices, and the free and repeated indices can be determined using Check
The repeated indices per term are: ...,...,..., the free indices are: ...
We see the free indices are μ and β, both covariant. To have all the tensors of this expression (but for the metric) with all their indices contravariant, use
This result is mathematically equal to the starting expression - all that happened is that the covariant versions of the indices were replaced by the contravariant ones at the cost of adding metric factors, so
In addition to the onlytheseindices option, to perform these operations only on the free indices, you can also use changerepeatedindices = false
`.`, CompactDisplay, Coordinates, g_, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, SubstituteTensor, SubstituteTensorIndices, TensorArray
Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
The Physics[ToCovariant] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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