Abstract index notation is a mathematical notation for tensors and spinors, which uses indices to indicate their type.

Thus the index isn't related to any coordinate system and doesn't take numerical values. It was designed by Roger Penrose to compensate for the difficulty to describe contractions in modern abstract tensor notation, without losing the intrinsically geometric viewpoint.

See also Penrose's graphical notation, Einstein notation.

References

• Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, 2004, has a chapter explaining it.
• Roger Penrose and Wolfgang Rindler, Spinors and space-time, volume I, two-spinor calculus and relativistic fields.

Tensors | Mathematical notation