Abstract index notation is a mathematical notation for tensors
that uses indices to indicate their types, rather than their
components in a particular basis. The indices are mere
placeholders, not related to any fixed basis, and in particular
are non-numerical. The notation was introduced by Roger Penrose
as a way to use the formal aspects of the Einstein summation
convention in order to compensate for the difficulty in
describing contractions and covariant differentiation in modern
abstract tensor notation, while preserving the explicit
covariance of the expressions involved. Let V be a vector space,
and V* its dual. Consider, for example, a rank 2 covariant
tensor h ∈ V* ⊗ V*. Then h can be identified with a bilinear
form on V. In other words, it is a function of two arguments in
V which can be represented as a pair of slots: Abstract index
notation is merely a labelling of the slots by Latin letters,
which have no significance apart from their designation as
labels of the slots (i.e., they are non-numerica