Raising and lowering indices
In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
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.
Multiplying by the contravariant metric tensor gij and contracting produces another tensor with an upper index:
The same base symbol is typically used to denote this new tensor, and repositioning the index is typically understood in this context to refer this new tensor, and is called raising the index, which would be written
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