# Abstract index notation

Abstract index notation uses an upper Latin index to represent each contravariant vector component of a tensor, and a lower index to represent each covariant vector (1-form) component. We can see from the preceding figure that this notation is quite compact and clearly indicates the type of each tensor while re-using letters to indicate what “slots” are to be used in the mapping.

The tensor product of two tensors $${S^{a}{}_{b}\otimes T^{c}{}_{d}}$$ is simply denoted $${S^{a}{}_{b}T^{c}{}_{d}}$$, and in this form the operation is sometimes called the tensor direct product. We may also consider a contraction $${T^{ab}{}_{bc}=T^{a}{}_{c}}$$, where two of the components of a tensor operate on each other to create a new tensor with a reduced number of indices. For example, if $${T^{ab}{}_{c}=v^{a}\otimes w^{b}\otimes\varphi_{c}}$$, then $${T^{ab}{}_{b}=\varphi(w)\cdot v^{a}}$$. Taking the tensor direct product of two tensors and then contracting all opposite indices is also called the contraction of the two tensors, i.e. the contraction of $${S^{ab}{}_{c}}$$ and $${T_{def}}$$ is $${C_{cf}=S^{ab}{}_{c}T{}_{abf}}$$. It is easily seen that the contraction of any two symmetric indices with any two anti-symmetric indices yields zero.

A (pseudo) inner product on $${V}$$ is a symmetric bilinear mapping, and thus corresponds to a symmetric tensor $${g_{ab}}$$ called the (pseudo) metric tensor. The isomorphism $${v\in V\mapsto v^{\flat}\in V^{*}}$$ induced by this pseudo inner product is then defined by $${v^{a}\mapsto v_{a}\equiv g_{ab}v^{b}}$$, and is called index lowering. The corresponding pseudo inner product on $${V^{*}}$$ is denoted $${g^{ab}}$$, which provides a consistent index raising operation since we immediately obtain $${g^{ab}g_{ac}g_{bd}=g_{cd}}$$. We also have the relation $${v^{a}=g^{ab}v_{b}=g^{ab}g_{bc}v^{c}\Rightarrow g^{ab}g_{bc}=g^{a}{}_{c}=\delta^{a}{}_{c}}$$, the identity mapping. The inner product of two tensors of the same type is then the contraction of their tensor direct product after index lowering/raising, e.g. $${\left\langle T^{ab},S^{cd}\right\rangle =T^{ab}S_{ab}=T^{ab}g_{ac}g_{bd}S^{cd}}$$.

 Δ It is important to remember that if $${v}$$ is a vector, the operation $${v_{a}v^{a}}$$ implies index lowering, which requires an inner product. In contrast, if $${\varphi}$$ is a 1-form, the operation $${\varphi_{a}v^{a}}$$ is always valid regardless of the presence of an inner product.

A symmetric or anti-symmetric tensor can be formed from a general tensor by adding or subtracting versions with permuted indices. For example, the combination $${\left(T_{ab}+T_{ba}\right)/2}$$ is the symmetrized tensor of $${T}$$, i.e. exchanging any two indices leaves it invariant. The anti-symmetrized tensor $${\left(T_{ab}-T_{ba}\right)/2}$$ changes sign upon the exchange of any two indices, and yields the original tensor $${T_{ab}}$$ when added to the symmetrized tensor. The following notation is common for tensors with $${n}$$ indices, with the sums over all permutations of indices:

$$\displaystyle \textrm{Symmetrization:}\; T_{(a_{1}\dots a_{n})}\equiv\frac{1}{n!}\underset{\pi}{\sum}T_{a_{\pi(1)}\dots a_{\pi(n)}}$$

$$\displaystyle \textrm{Anti-symmetrization:}\; T_{[a_{1}\dots a_{n}]}\equiv\frac{1}{n!}\underset{\pi}{\sum}\textrm{sign}(\pi)T_{a_{\pi(1)}\dots a_{\pi(n)}}$$

This operation can be performed on any subset of indices in a tensor, as long as they are all covariant or all contravariant. Skipping indices is denoted with vertical bars, as in $${T_{\left(a|b|c\right)}=\left(T_{abc}+T_{cba}\right)/2}$$; however, note that vertical bars alone are sometimes used to denote a sum of ordered permutations, as in $${T_{\left|abc\right|}=T_{abc}+T_{bca}+T_{cab}}$$.