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Concept
Raising and lowering indices
Résumé
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.
Mathematically vectors are elements of a vector space over a field , and for use in physics is usually defined with or . Concretely, if the dimension of is finite, then, after making a choice of basis, we can view such vector spaces as or .
The dual space is the space of linear functionals mapping . Concretely, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by , so that is a linear map .
Then under a choice of basis , we can view vectors as an vector with components (vectors are taken by convention to have indices up). This picks out a choice of basis for , defined by the set of relations .
For applications, raising and lowering is done using a structure known as the (pseudo-)metric tensor (the 'pseudo-' refers to the fact we allow the metric to be indefinite). Formally, this is a non-degenerate, symmetric bilinear form
In this basis, it has components , and can be viewed as a symmetric matrix in with these components. The inverse metric exists due to non-degeneracy and is denoted , and as a matrix is the inverse to .
Raising and lowering is then done in coordinates. Given a vector with components , we can contract with the metric to obtain a covector:
and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:
This process is called raising the index.
Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in the metric and inverse metric tensors being inverse to each other (as is suggested by the terminology):
where is the Kronecker delta or identity matrix.
Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia.
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