Motivation
In linear algebra, a finite-dimensional vector space is isomorphic to its dual space (the space of linear functionals mapping the vector space to its base field), but not canonically isomorphic to it. This is to say that given a fixed basis for the vector space, there is a natural way to go back and forth between vectors and linear functionals: vectors are represented in the basis by column vectors, and linear functionals are represented in the basis by row vectors, and one can go back and forth by transposing. However, without a fixed basis, there is no way to go back and forth between vectors and linear functionals. This is what is meant by that there is no canonical isomorphism. On the other hand, a finite-dimensional vector space endowed with a non-degenerate bilinear form is canonically isomorphic to its dual. The canonical isomorphism is given by : . The non-degeneracy of means exactly that the above map is an isomorphism. An example is where and is theDiscussion
Let be a (pseudo-)Riemannian manifold. At each point , the map is a non-degenerate bilinear form on the tangent space . If is a vector in , its ''flat'' is the covector : in . Since this is a smooth map that preserves the point , it defines a morphism of smooth vector bundles . By non-degeneracy of the metric, has an inverse at each point, characterized by : for in and in . The vector is called the ''sharp'' of . The sharp map is a smooth bundle map . Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each in , there are mutually inverse vector space isomorphisms between and . The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if is a vector field and is a covector field, : and : .In a moving frame
Suppose is a moving tangent frame (see also smooth frame) for the tangent bundle with, as dual frame (see also dual basis), the moving coframe (a ''moving tangent frame'' for the ''cotangent bundle'' ; see also coframe) . Then the pseudo-Riemannian metric, which is a 2-covariant tensor field, can be written locally in this coframe as using Einstein summation notation. Given a vector field and denoting , its flat is : . This is referred to as lowering an index, because the components of are written with an upper index , whereas the components of are written with a lower index . In the same way, given a covector field and denoting , its sharp is : , where are the components of the inverse metric tensor (given by the entries of the inverse matrix to ). Taking the sharp of a covector field is referred to as ''raising an index''.Extension to tensor products
The musical isomorphisms may also be extended, for each , to an isomorphism between the bundle : of tensors and the bundle of tensors. Here can be positive or negative, so long as and . Lowering an index of an tensor gives a tensor, while raising an index gives a . Which index is to be raised or lowered must be indicated. For instance, consider the tensor . Raising the second index, we get the tensor : In other words, the components of are given by : Similar formulas are available for tensors of other orders. For example, for a tensor , all indices are raised by: : For a tensor , all indices are lowered by: : For a mixed tensor of order , all lower indices are raised and all upper indices are lowered by : Well-formulated expressions are constrained by the rules of Einstein summation notation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as is well formulated while is not.Extension to ''k''-vectors and ''k''-forms
In the context of exterior algebra, an extension of the musical operators may be defined on and its dual , and are again mutual inverses: : : defined by : : In this extension, in which maps ''k''-vectors to ''k''-covectors and maps ''k''-covectors to ''k''-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated: This works not just for ''k''-vectors in the context of linear algebra but also for ''k''-forms in the context of a (pseudo-)Riemannian manifold: : :Vector bundles with bundle metrics
More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.Trace of a tensor
Given a tensor , we define the ''trace of through the metric tensor '' by Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric. The trace of an tensor can be taken in a similar way, so long as one specifies which two distinct indices are to be traced. This process is also called contracting the two indices. For example, if is an tensor with , then the indices and can be contracted to give an tensor with components :Example computations
In Minkowski spacetime
The covariant 4-position is given by : with components: : (where ,, 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 : : and for : : So the index-raised contravariant 4-position is: : This operation is equivalent to the matrix multiplication : Given two vectors, and , 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 , :In electromagnetism
For a (0,2) tensor, 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 byNB: Some texts, such as: , will show this tensor with an overall factor of −1. This is because they used the negative of the metric tensor used here: , see metric signature. In older texts such as Jackson (2nd edition), there are no factors of since they are using Gaussian units. Here SI units are used. : In components, : To obtain the covariant tensor , contract with the inverse metric tensor: : and since and , this reduces to : Now for , : : and by antisymmetry, for , : : then finally for , ; : The (covariant) lower indexed tensor is then: : This operation is equivalent to the matrix multiplication :See also
* Duality (mathematics) * * Einstein notation * Flat (music) and Sharp (music) about the signs and * Hodge star operator * Metric tensor * Vector bundleCitations
References
* * * {{Manifolds Differential geometry Riemannian geometry Riemannian manifolds Symplectic geometry