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.
is finite, then, after making a choice of basis, we can view such vector spaces as
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
The inverse metric exists due to non-degeneracy and is denoted
, 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.
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.
Possible metrics on real space are indexed by signature
is meant to make it clear that the underlying vector space of
With these rules we can immediately see that an expression such as is well formulated while is not.
The covariant 4-position is given by with components: (where x,y,z are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as in components: To raise the index, multiply by the tensor and contract: then for λ = 0: and for λ = j = 1, 2, 3: So the index-raised contravariant 4-position is: This operation is equivalent to the matrix multiplication Given two vectors,
, we can write down their (pseudo-)inner product in two ways: By lowering indices, we can write this expression as In matrix notation, the first expression can be written as while the second is, after lowering the indices of
, It is instructive to consider what raising and lowering means in the abstract linear algebra setting.
is a map which is linear in both arguments, making it a bilinear form.
is often considered a structure on the vector space, for example an inner product or more generally a metric tensor which is allowed to have indefinite signature, or a symplectic form
is either symmetric or anti-symmetric, but in full generality
denotes an argument whose evaluation is deferred.
has well defined (anti-)symmetry, evaluating on either argument is equivalent (up to a minus sign for anti-symmetry).
Non-degeneracy shows that the partial evaluation map is injective, or equivalently that the kernel of the map is trivial.
, so non-degeneracy is enough to conclude the map is a linear isomorphism.
is a structure on the vector space sometimes call this the canonical isomorphism
and this is enough to define an associated bilinear form on the dual: where the repeated use of
Checking these expressions in coordinates makes it evident that this is what raising and lowering indices means abstractly.
We will not develop the abstract formalism for tensors straightaway.
tensor is written We can use the metric tensor to raise and lower tensor indices just as we raised and lowered vector indices and raised covector indices.
For a (0,2) tensor,[1] twice contracting with the inverse metric tensor and contracting in different indices raises each index: Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index: Let's apply this to the theory of electromagnetism.
The contravariant electromagnetic tensor in the (+ − − −) signature is given by[2] In components, To obtain the covariant tensor Fαβ, contract with the inverse metric tensor: and since F00 = 0 and F0i = − Fi0, this reduces to Now for α = 0, β = k = 1, 2, 3: and by antisymmetry, for α = k = 1, 2, 3, β = 0: then finally for α = k = 1, 2, 3, β = l = 1, 2, 3; The (covariant) lower indexed tensor is then: This operation is equivalent to the matrix multiplication For a tensor of order n, indices are raised by (compatible with above):[1] and lowered by: and for a mixed tensor: We need not raise or lower all indices at once: it is perfectly fine to raise or lower a single index.
have suitable values, for example we cannot lower the index of a