Penrose graphical notation

In the language of multilinear algebra, each shape represents a multilinear function. The lines attached to shapes represent the inputs or outputs of a function, and attaching shapes together in some way is essentially the composition of functions.

In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to abstract upper and lower indices of tensors respectively. Connecting lines between two shapes corresponds to contraction of indices. One advantage of this notation is that one does not have to invent new letters for new indices. This notation is also explicitly basis-independent.[3]

Each shape represents a matrix, and tensor multiplication is done horizontally, and matrix multiplication is done vertically.

The metric tensor is represented by a U-shaped loop or an upside-down U-shaped loop, depending on the type of tensor that is used.

metric tensor ${\displaystyle g^{ab}}$

metric tensor ${\displaystyle g_{ab}}$

The Levi-Civita antisymmetric tensor is represented by a thick horizontal bar with sticks pointing downwards or upwards, depending on the type of tensor that is used.

${\displaystyle \varepsilon _{ab\ldots n}}$

${\displaystyle \varepsilon ^{ab\ldots n}}$

${\displaystyle \varepsilon _{ab\ldots n}\,\varepsilon ^{ab\ldots n}}$${\displaystyle =n!}$

structure constant ${\displaystyle {\gamma _{\alpha \beta }}^{\chi }=-{\gamma _{\beta \alpha }}^{\chi }}$

The structure constants (${\displaystyle {\gamma _{ab}}^{c}}$) of a Lie algebra are represented by a small triangle with one line pointing upwards and two lines pointing downwards.

Contraction of indices is represented by joining the index lines together.

Kronecker delta ${\displaystyle \delta _{b}^{a}}$

Dot product ${\displaystyle \beta _{a}\,\xi ^{a}}$

${\displaystyle g_{ab}\,g^{bc}=\delta _{a}^{c}=g^{cb}\,g_{ba}}$

Symmetrization of indices is represented by a thick zig-zag or wavy bar crossing the index lines horizontally.

Symmetrization ${\displaystyle Q^{(ab\ldots n)}}$ (with ${\displaystyle {}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}}$)

Antisymmetrization of indices is represented by a thick straight line crossing the index lines horizontally.

Antisymmetrization ${\displaystyle E_{[ab\ldots n]}}$ (with ${\displaystyle {}_{E_{ab}=E_{[ab]}+E_{(ab)}}}$)

The covariant derivative (${\displaystyle \nabla }$) is represented by a circle around the tensor(s) to be differentiated and a line joined from the circle pointing downwards to represent the lower index of the derivative.

covariant derivative ${\displaystyle 12\nabla _{a}\left(\xi ^{f}\,\lambda _{fb[c}^{(d}\,D_{gh]}^{e)b}\right)}$ ${\displaystyle =12\left(\xi ^{f}(\nabla _{a}\lambda _{fb[c}^{(d})\,D_{gh]}^{e)b}+(\nabla _{a}\xi ^{f})\lambda _{fb[c}^{(d}\,D_{gh]}^{e)b}+\xi ^{f}\lambda _{fb[c}^{(d}\,(\nabla _{a}D_{gh]}^{e)b})\right)}$

The Ricci and Bianchi identities given in terms of the Riemann curvature tensor illustrate the power of the notation

Notation for the Riemann curvature tensor

Ricci tensor ${\displaystyle R_{ab}=R_{acb}^{\ \ \ c}}$

Ricci identity ${\displaystyle (\nabla _{a}\,\nabla _{b}-\nabla _{b}\,\nabla _{a})\,\mathbf {\xi } ^{d}}$${\displaystyle =R_{abc}^{\ \ \ d}\,\mathbf {\xi } ^{c}}$

Bianchi identity ${\displaystyle \nabla _{[a}R_{bc]d}^{\ \ \ e}=0}$