Abstract index notation

Abstract index notation (also referred to as slot-naming index notation)^[1] is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis.^[2] 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.^[3]

Let $V$ be a vector space, and $V^{*}$ its dual space. Consider, for example, an order-2 covariant tensor $h\in V^{*}\otimes V^{*}$ . 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:

h=h(-,-).

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):

h=h_{ab}.

A tensor contraction (or trace) between two tensors is represented by the repetition of an index label, where one label is contravariant (an upper index corresponding to the factor $V$ ) and one label is covariant (a lower index corresponding to the factor $V^{*}$ ). Thus, for instance,

t_{ab}{}^{b}

is the trace of a tensor $t=t_{ab}{}^{c}$ over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or natural pairing) between tensor factors of type $V$ and those of type $V^{*}$ .

^ Kip S. Thorne and Roger D. Blandford (2017). Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press. ISBN 978-0-69115902-7.
^ Roger Penrose (2007). The Road to Reality: A Complete Guide to the Laws of the Universe. Vintage. ISBN 978-0-67977631-4.
^ Roger Penrose and Wolfgang Rindler (1984). Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields. Cambridge University Press. ISBN 978-0-52133707-6.

[Thorne2017-1] Kip S. Thorne and Roger D. Blandford (2017). Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press. ISBN 978-0-69115902-7.

[Penrose2007-2] Roger Penrose (2007). The Road to Reality: A Complete Guide to the Laws of the Universe. Vintage. ISBN 978-0-67977631-4.

[Penrose1984-3] Roger Penrose and Wolfgang Rindler (1984). Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields. Cambridge University Press. ISBN 978-0-52133707-6.

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