Raising and Lowering Indices - Tensor Type

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:

    The perfect detective story cannot be written. The type of mind which can evolve the perfect problem is not the type of mind that can produce the artistic job of writing.
    Raymond Chandler (1888–1959)