البوابة
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A covariant tensor, denoted with a lowered index (e.g.,  ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor. To examine the transformation properties of a covariant tensor, first consider the gradient
for which
where  . Now let
then any set of quantities  which transform according to
or, defining
according to
is a covariant tensor. Contravariant tensors are a type of tensor with differing transformation properties, denoted  . To turn a contravariant tensor  into a covariant tensor  (index lowering), use the metric tensor  to write
Covariant and contravariant indices can be used simultaneously in a mixed tensخr. In Euclidean spaces, and more generally in flat Riemannian manifolds, a coordinate system can be found where the metric tensor is constant, equal to Kronecker delta
Therefore, raising and lowering indices is trivial, hence covariant and contravariant tensors have the same coordinates, and can be identified. Such tensors are known as Cartesian tensors. A similar result holds for flat pseudo-Riemannian manifolds, such as Minkowski space, for which covariant and contravariant tensors can be identified. However, raising and lowering indices changes the sign of the temporal components of tensors, because of the negative eigenvalue in the Minkowski metric. |
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covariant tensor
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