Tensor Type
Given a tensor field on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or Minkowski metric), one can raise or lower indices to change a type (a, b) tensor to a (a + 1, b − 1) tensor (raise index) or to a (a − 1, b + 1) tensor (lower index), where the notation (a, b) has been used to denote the tensor order a + b with a upper indices and b lower indices.
One does this by multiplying by the covariant or contravariant metric tensor and then contracting indices, meaning two indices are set equal and then summing over the repeated indices (applying Einstein notation). See examples below.
Read more about this topic: Raising And Lowering Indices
Famous quotes containing the word type:
“There is a limit to the application of democratic methods. You can inquire of all the passengers as to what type of car they like to ride in, but it is impossible to question them as to whether to apply the brakes when the train is at full speed and accident threatens.”
—Leon Trotsky (1879–1940)