Tensors as multi-dimensional arrays

In a given basis, a pure tensor of type \({(m,n)}\) can be written using component notation in the form

\(\displaystyle v^{1}\otimes\dotsb\otimes v^{m}\otimes\varphi_{1}\otimes\dotsb\otimes\varphi_{n}\equiv T^{\mu_{1}\dots\mu_{m}}{}_{\nu_{1}\dots\nu_{n}}e_{\mu_{1}}\otimes\dotsb\otimes e_{\mu_{m}}\otimes\beta^{\nu_{1}}\otimes\dotsb\otimes\beta^{\nu_{n}}, \)

where the Einstein summation convention is used in the second expression. Note that the collection of terms into \({T}\) is only possible due to the defining property of the tensor product being linear over addition. The tensor product between basis elements is often dropped in such expressions.

A general tensor is a sum of such pure tensor terms, so that any tensor \({T}\) can be represented by a \({\left(m+n\right)}\)-dimensional array of scalars. For example, any tensor of order 2 is a matrix, and as a linear mapping operates on vectors and forms via ordinary matrix multiplication if they are all expressed in terms of components in the same basis. Basis-independent quantities from linear algebra such as the trace and determinant are then well-defined on such tensors.

Δ A potentially confusing aspect of component notation is the basis vectors \({e_{\mu}}\), which are not components of a 1-form but rather vectors, with \({\mu}\) a label, not an index. Similarly, the basis 1-forms \({\beta^{\nu}}\) should not be confused with components of a vector.

The Latin letters of abstract index notation (e.g. \({T^{ab}{}_{cd}}\)) can thus be viewed as placeholders for what would be indices in a particular basis, while the Greek letters of component notation represent an actual array of scalars that depend on a specific basis. The reason for the different notations is to clearly distinguish tensor identities, true in any basis, from equations true only in a specific basis.

Δ In general relativity both abstract and index notation are abused to represent objects that are non-tensorial. We will see this in the chapter on Riemannian manifolds.
Δ Note that if abstract index notation is not being used, Latin and Greek indices are often used to make other distinctions, a common one being between indices ranging over three space indices and indices ranging over four spacetime indices.
Δ Note that “rank” and “dimension” are overloaded terms across these constructs: “rank” is sometimes used to refer to the order of the tensor, which is the dimensionality of the corresponding multi-dimensional array; the dimension of a tensor is that of the underlying vector space, and so is the length of a side of the corresponding array (also sometimes called the dimension of the array). However, the rank of a order 2 tensor coincides with the rank of the corresponding matrix.

An Illustrated Handbook