Abstract index notation

Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention 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, an order-2 covariant tensor ${\displaystyle h\in V^{*}\otimes V^{*}}$. 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 with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):

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