Robert's Mistakes

Diagram 3: Penrose tensor diagram notation

Jul 10, 2013

The image shows the Ricci identity written in Penrose's graphical notation. Tensor calculus is a generalization of linear algebra (you know, vectors and matrices) and was developed around 1890 by Gregorio Ricci-Curbastro under the title "absolute differential calculus", and originally presented by Ricci in 1892. (see wikipedia on tensors)

It is the natural framework for general relativity. Besides a great application for them, Einstein's contribution here was his summation convention. It basically tells you to build a sum over each index whenever you see the same index in sub- and superscript used in a symbolic expression. Equations become simpler because we don't need to plaster them with summation signs and that is pretty ingenious.

Okay, but what is a tensor?

The data structure for a simple vector is a list of numbers. Correspondingly, a matrix is a two-dimensional table of numbers. You may have heard that tensors are like matrices, but with more dimensions. While this is certainly true it does neither tell what is gained nor how to write them down. Well, you don't write tensors as (n-dimensional) tables. Not because this would be impossible but because it is too tedious and doesn't help to actually simplify anything (except maybe computer programs).

Instead, tensors are written as formulas involving symbols. One simple example is the Kronecker delta symbol δ^i_j. It's 1 when i == j and 0 otherwise. Using that we can plausibly abbreviate any identity matrix as just δ^i_j (leaving the arguments in place so we can relate them to other symbols). That was just one symbol. 12*δ^i_j would be a multiple of the identity matrix (in this case, a scaling transformation). In practice we have a little zoo of useful and rather simple symbols similar to δ^i_j which allow us to construct the sums we need.

I guess, if you'd try to concoct an electric circuit diagram for a tensor equation written in Einstein's style, it might seem natural to have wires going in and out of symbols, say, flowing from top to bottom. That may be how Penrose arrived at his notation... or not. However, the result is an impressive way to get rid of all these index symbols. They only serve as structural indicators anyway (hence the name, summation indices).

Wikipedia's article struggles to be informative but is still a bit terse. It's also where i got the image from:

http://en.wikipedia.org/wiki/Penrose_graphical_notation

Here's two pages from "The Road to Reality" by Roger Penrose uploaded by someone to illustrate a question on stackexchange:

http://math.stackexchange.com/questions/2917/proof-of-detst-detsdett-in-penrose-graphical-notation

I don't know wether there's an introduction to tensor calculus solely based on diagram notation. But for the curious, here's a more conventional treatment, suggested by +Hamilton Carter:

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050175884_2005173651.pdf