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In mathematics, Ricci calculus is the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–96 and subsequently popularized in a paper written with his pupil Tullio Levi-Civita at the turn of 1900.

Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early 20th century.

This article summarizes the rules of index notation and manipulation for tensors and tensor fields.

Introduction

Tensors and tensor fields can be expressed in terms of their components (meaning scalar coefficients of a tensor basis), and operations on tensors and tensor fields can be expressed in terms of operations on their components. Basis-independent tensor fields and operations are the focus of the Ricci calculus, and the notation leverages this to allow compact expressions of such tensor fields and operations.

While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays.

A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. The number of indices equals the order of the tensor.

For compactness and convenience, the notational convention implies certain things, notably that of summation over repeated indices and of universal quantification of free indices.

Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.

Notation for indices

Basis-related distinctions

Space-time split

Where a distinction is to be made between the space-like basis elements and a time-like element, this is conventionally done through indices as follows:

• The lowercase Latin alphabet a, b, c... is used to indicate restriction to 3-dimensional Euclidean space, which take values 1, 2, 3 for the spatial components, and the time-like element is shown separately.
• The lowercase Greek alphabet α, β, γ... is used for 4-dimensional spacetime, which typically take values 0 for time components and 1, 2, 3 for the spatial components.

else in general mathematical contexts, any symbol can be used, independent of the dimension of the vector space.

Coordinate and index notation

The author(s) will usually make it clear whether a subscript is intended as an index or as a label.

For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector A = (A₁, A₂, A₃) = (A_x, A_y, A_z) shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression A_i, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are not indices, more like "names" for the components.

Reference to coordinate systems

Indices themselves may be labelled using diacritic-like symbols, such as a hat (^), bar (¯), tilde (), or prime (′)

${\displaystyle X_{\hat {\phi }}\,,Y_{\bar {\lambda }}\,,Z_{\tilde {\eta }}\,,T_{\mu '}\cdots }$

to denote a possibly different basis (and hence coordinate system) for that index, for example in Lorentz transformations from one frame of reference to another, one frame could be unprimed and the other primed.

This is not to be confused with van der Waerden notation for spinors, which uses hats and overdots on indices to reflect the chiralty of a spinor.

Raised and lowered indices

Covariant tensor components

A lower index (subscript) indicates covariance of the components with respect to that index: ${\displaystyle A_{\alpha \beta \gamma \cdots }}$

Contravariant tensor components

An upper index (superscript) indicates contravariance of the components with respect to that index: ${\displaystyle A^{\alpha \beta \gamma \cdots }}$

Mixed-variance tensor components

A tensor may have both upper and lower indices: ${\displaystyle A_{\alpha }{}^{\beta }{}_{\gamma }{}^{\delta \cdots }}$

Summation

Two indices (one raised and one lowered) with the same symbol are summed over: ${\displaystyle A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }}$ or ${\displaystyle A^{\alpha }B_{\alpha }\equiv \sum _{\alpha }A^{\alpha }B_{\alpha }\,.}$

The operation of setting two indices equal, and hence an immediate summation, is called tensor contraction:

${\displaystyle A_{\alpha }B^{\beta }\rightarrow A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\,.}$

More than one index may be repeated twice, but only twice and within one term, for example:

${\displaystyle A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,,}$

as for a non-identity

${\displaystyle A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\not \equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,}$

is not considered well-formed, that is, it is meaningless.

Multi-index notation

If a tensor has a list of indices all raised or lowered, one shorthand is to use a capital letter for the list:

${\displaystyle A_{i_{1}\cdots i_{n}}B^{i_{1}\cdots i_{n}j_{1}\cdots j_{m}}C_{j_{1}\cdots j_{m}}\equiv A_{I}B^{IJ}C_{J}}$

where I = i₁ i₂ ... i_n and J = j₁ j₂ ... j_m.

Sequential summation

Two vertical bars | | around a set of indices (with a contraction):

${\displaystyle A_{|\alpha \beta \gamma |\cdots }B^{\alpha \beta \gamma \cdots }=\sum _{\alpha }\sum _{\beta }\sum _{\gamma }A_{\alpha \beta \gamma \cdots }B^{\alpha \beta \gamma \cdots }}$

denotes the summation in which each preceding index is counted up to (and not including) the value of the next index:

${\displaystyle \alpha <\beta <\gamma \,.}$

Only one group of the repeated set of indices has the vertical bars around them (the other contracted indices do not). More than one group can summed in this way:

${\displaystyle A_{|\alpha \beta \gamma |}{}^{|\delta \epsilon \cdots \lambda |}B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda |\mu \nu \cdots \zeta |}C^{\mu \nu \cdots \zeta }=\sum _{\alpha }\sum _{\beta }\sum _{\gamma }\sum _{\delta }\sum _{\epsilon }\cdots \sum _{\lambda }\sum _{\mu }\sum _{\nu }\cdots \sum _{\zeta }A_{\alpha \beta \gamma }{}^{\delta \epsilon \cdots \lambda }B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda \mu \nu \cdots \zeta }C^{\mu \nu \cdots \zeta }}$

where

${\displaystyle \alpha <\beta <\gamma \,,\quad \delta <\epsilon <\cdots <\lambda \,,\quad \mu <\nu \cdots <\zeta \,.}$

This is useful to prevent over-counting in some summations, when tensors are symmetric or antisymmetric.

Alternatively, using the capital letter convention for multi-indices, an underarrow is placed underneath the block of indices:

${\displaystyle A_{\underset {\rightharpoondown }{P}}{}^{\underset {\rightharpoondown }{Q}}B^{P}{}_{Q{\underset {\rightharpoondown }{R}}}C^{R}=\sum _{\underset {\rightharpoondown }{P}}\sum _{\underset {\rightharpoondown }{Q}}\sum _{\underset {\rightharpoondown }{R}}A_{P}{}^{Q}B^{P}{}_{QR}C^{R}}$

where

${\displaystyle {\underset {\rightharpoondown }{P}}=|\alpha \beta \gamma |\,,\quad {\underset {\rightharpoondown }{Q}}=|\delta \epsilon \cdots \lambda |\,,\quad {\underset {\rightharpoondown }{R}}=|\mu \nu \cdots \zeta |}$

Raising and lowering indices

By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:

${\displaystyle B^{\gamma }{}_{\beta \cdots }=g^{\gamma \alpha }A_{\alpha \beta \cdots }}$ and ${\displaystyle A_{\alpha \beta \cdots }=g_{\alpha \gamma }B^{\gamma }{}_{\beta \cdots }}$

The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.

General outlines for index notation and operations

Tensors are equal if and only if every corresponding component is equal, e.g. tensor A equals tensor B if and only if

${\displaystyle A^{\alpha }{}_{\beta \gamma }=B^{\alpha }{}_{\beta \gamma }}$

for all α, β and γ. Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to dimensional analysis).

Free and dummy indices

Indices not in contractions are called free indices.

Indices in contractions are termed dummy indices, or summation indices.

If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents m equations: each has a specific set of index values.

For instance, if

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }=T^{\alpha }{}_{\beta }{}_{\delta }}$

is in 4-dimensions (that is, each index runs from 0 to 3 or 1 to 4), then because there are three free indices (α, β, δ), there are 4 = 64 equations:

${\displaystyle A^{0}B_{1}{}^{0}C_{00}+A^{0}B_{1}{}^{1}C_{10}+A^{0}B_{1}{}^{2}C_{20}+A^{0}B_{1}{}^{3}C_{30}+D^{0}{}_{1}{}E_{0}=T^{0}{}_{1}{}_{0}}$

${\displaystyle A^{1}B_{0}{}^{0}C_{00}+A^{1}B_{0}{}^{1}C_{10}+A^{1}B_{0}{}^{2}C_{20}+A^{1}B_{0}{}^{3}C_{30}+D^{1}{}_{0}{}E_{0}=T^{1}{}_{0}{}_{0}}$

${\displaystyle A^{1}B_{2}{}^{0}C_{02}+A^{1}B_{2}{}^{1}C_{12}+A^{1}B_{2}{}^{2}C_{22}+A^{1}B_{2}{}^{3}C_{32}+D^{1}{}_{2}{}E_{2}=T^{1}{}_{2}{}_{2}}$

(followed by 61 more equations, each with various other choices of α, β, δ)...

This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation.

Indices are replaceable labels

Replacing any index symbol throughout by another leaves the tensor equation unchanged (provided there is no conflict with other symbols already used). This can be useful when manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-civita symbol (see also below). An example of a correct change is:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\rightarrow A^{\lambda }B_{\beta }{}^{\mu }C_{\mu \delta }+D^{\lambda }{}_{\beta }{}E_{\delta }}$

as for a possible invalid change:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\nrightarrow A^{\lambda }B_{\beta }{}^{\gamma }C_{\mu \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\,.}$

In the first replacement, λ replaced α and γ replaced μ everywhere, so the expression still has the same meaning. In the second, λ did not fully replace α and similarly for γ and μ, and is entirely inconsistent for reasons shown next.

Indices are the same in every term

The same indices on each side of a tensor equation always appear in the same (upper or lower) position throughout every term, except for indices repeated in a term (which implies a summation over that index), for example:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }=T^{\alpha }{}_{\beta }{}_{\delta }}$

as for a possible invalid expression:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D_{\alpha }{}_{\beta }{}^{\gamma }E^{\delta }}$

In other words, non-repeated indices must be of the same type in every term of the equation. In the above identity α, β, δ line up throughout and γ occurs twice in one term due to a contraction (correctly once as an upper index and once as a lower index), so it's a valid as an expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is inconsistent.

Brackets and punctuation used once where implied

When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.

If the brackets enclose covariant indices - the rule applies only to all covariant indices enclosed in the brackets, not to any contravariant indices which happen to be placed intermediately between the brackets.

Similarly if brackets enclose contravariant indices - the rule applies only to all enclosed contravariant indices, not to intermediately placed covariant indices.

Symmetric and antisymmetric parts

Symmetric part of tensor

Round brackets ( ) around some or all indices denotes the symmetrized part of the tensor: the sum of the tensor components with those indices permuted, then divided by the number of permutations.

For two symmetrizing indices, there are two indices to sum over and permute:

${\displaystyle A_{(\alpha \beta )\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }+A_{\beta \alpha \gamma \cdots }\right)}$

as an example,

${\displaystyle A_{(\alpha }B^{\beta }{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }+A_{\gamma }B^{\beta }{}_{\alpha }\right).}$

For three symmetrizing indices, there are three indices to sum over and permute:

${\displaystyle A_{(\alpha \beta \gamma )\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }+A_{\alpha \gamma \beta \delta \cdots }+A_{\gamma \beta \alpha \delta \cdots }+A_{\beta \alpha \gamma \delta \cdots }\right).}$

For p symmetrizing indices – sum over all components with those indices permuted:

${\displaystyle A_{(\alpha _{1}\alpha _{2}\cdots \alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {1}{p!}}\sum _{\mathrm {permute} \,(\alpha _{1},\alpha _{2},\cdots ,\alpha _{p})}A_{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha _{p}\cdots \alpha _{q}}.}$

Antisymmetric part of tensor

Square brackets around some or all indices denotes the antisymmetrized part of the tensor. To antisymmetrize, a total is formed with one term per permutation, components arising from an even permutation of the indices are added, while components arising from an odd permutation of the indices are subtracted, then the total is divided by the number of permutations.

For two antisymmetrizing indices:

${\displaystyle A_{\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }-A_{\beta \alpha \gamma \cdots }\right)}$

For three antisymmetrizing indices:

${\displaystyle A_{\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }-A_{\alpha \gamma \beta \delta \cdots }-A_{\gamma \beta \alpha \delta \cdots }-A_{\beta \alpha \gamma \delta \cdots }\right)}$

as an example,

${\displaystyle F_{}={\dfrac {1}{3!}}\left(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }-\partial _{\gamma }F_{\beta \alpha }-\partial _{\beta }F_{\alpha \gamma }-\partial _{\alpha }F_{\gamma \beta }\right)}$

For p antisymmetrizing indices – sum over the permutations ${\displaystyle \sigma }$ of those indices multiplied by the signature of the permutation, divided by the number of permutations:

${\displaystyle {\begin{aligned}A_{\alpha _{p+1}\cdots \alpha _{q}}&={\dfrac {1}{p!}}\sum _{\sigma }\operatorname {sgn}(\sigma )A_{\sigma (\alpha _{1})\cdots \sigma (\alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}\\&={\dfrac {1}{(n-p)!}}\varepsilon _{\alpha _{1}\dots \alpha _{p}\,\beta _{1}\dots \beta _{n-p}}{\dfrac {1}{p!}}\varepsilon ^{\gamma _{1}\dots \gamma _{p}\,\beta _{1}\dots \beta _{n-p}}A_{\gamma _{1}\dots \gamma _{p}\alpha _{p+1}\cdots \alpha _{q}}\\\end{aligned}}}$

where n is the dimensionality of the underlying vector space.

Symmetry and antisymmetry sum

Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices:

${\displaystyle A_{\alpha \beta \gamma \cdots }=A_{(\alpha \beta )\gamma \cdots }+A_{\gamma \cdots }}$

Differentiation

For compactness, derivatives may be indicated by adding indices after a comma or semicolon.

Partial derivative

To indicate partial differentiation of a tensor field with respect to a coordinate variable ${\displaystyle x^{\gamma }}$, a comma is placed before an added lower index of the coordinate variable. This may repeated (without adding further commas):

${\displaystyle A_{\alpha \beta \cdots ,\gamma }=\partial _{\gamma }A_{\alpha \beta \cdots }={\dfrac {\partial }{\partial x^{\gamma }}}A_{\alpha \beta \cdots }}$

${\displaystyle A_{\alpha \beta \cdots ,\gamma \delta }=\partial _{\delta }\partial _{\gamma }A_{\alpha \beta \cdots }={\dfrac {\partial ^{2}}{\partial x^{\delta }\partial x^{\gamma }}}A_{\alpha \beta \cdots }\,.}$

These components do not transform covariantly. This derivative is characterized by the product rule and the derivatives of the coordinates

${\displaystyle x^{\alpha }{}_{,\gamma }=\delta _{\gamma }^{\alpha }\,}$

where δ is the Kronecker delta.

Covariant derivative

To indicate covariant differentiation of any tensor field, a semicolon ( ; ) or a forward slash ( / , less common) is placed before an added lower (covariant) index.

For a contravariant vector: ${\displaystyle A^{\alpha }{}_{;\beta }=A^{\alpha }{}_{,\beta }+\Gamma ^{\alpha }{}_{\gamma \beta }A^{\gamma }}$ where ${\displaystyle \Gamma ^{\alpha }{}_{\beta \gamma }\,}$ is a Christoffel symbol of the second kind.

For a covariant vector: ${\displaystyle A_{\alpha ;\beta }=A_{\alpha ,\beta }-\Gamma ^{\gamma }{}_{\alpha \beta }A_{\gamma }\,.}$

For an arbitrary tensor:

${\displaystyle {\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma }=T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }\,&+\Gamma ^{\alpha _{1}}{}_{\delta \gamma }T^{\delta \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}+\cdots +\Gamma ^{\alpha _{r}}{}_{\delta \gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\delta }{}_{\beta _{1}\cdots \beta _{s}}\\&-\,\Gamma ^{\delta }{}_{\beta _{1}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\delta \beta _{2}\cdots \beta _{s}}-\cdots -\Gamma ^{\delta }{}_{\beta _{s}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\delta }\,.\end{aligned}}}$

Defining the multi-indices (using the capital-letter convention above):

${\displaystyle {\begin{aligned}&A=\alpha _{1}\alpha _{2}\cdots \alpha _{r}\\&A_{1}=\delta \alpha _{2}\cdots \alpha _{r}\\&A_{2}=\alpha _{1}\delta \cdots \alpha _{r}\\&\quad \vdots \\&A_{r}=\alpha _{1}\alpha _{1}\cdots \alpha _{r-1}\delta \\\end{aligned}}\quad {\begin{aligned}&B=\beta _{1}\beta _{2}\cdots \beta _{s}\\&B_{1}=\delta \beta _{2}\cdots \beta _{s}\\&B_{2}=\beta _{1}\delta \cdots \beta _{s}\\&\quad \vdots \\&B_{s}=\beta _{1}\beta _{1}\cdots \beta _{s-1}\delta \\\end{aligned}}}$

the covariant derivative can be written

${\displaystyle T^{A}{}_{B;\gamma }=T^{A}{}_{B,\gamma }+\sum _{i}\Gamma ^{\alpha _{i}}{}_{\delta \gamma }T^{A_{i}}{}_{B}-\sum _{j}\Gamma ^{\delta }{}_{\beta _{j}\gamma }T^{A}{}_{B_{j}}}$

and this is not the summation convention, so for this case the summation signs are explicitly stated. The components of this derivative of a tensor field transform covariantly, and hence form another tensor field. This derivative is characterized by the product rule and the fact that the derivative of the metric ${\displaystyle g_{\mu \nu }\,}$ is zero:

${\displaystyle g_{\mu \nu ;\gamma }=0\,.}$

The covariant formulation of the directional derivative of any tensor field along a vector ${\displaystyle v^{\gamma }}$ may be expressed as its contraction with the covariant derivative, e.g.:

${\displaystyle v^{\gamma }A_{\alpha ;\gamma }\,.}$

Lie derivative

The Lie derivative is another derivative that is covariant, but which should not be confused with the covariant derivative. It is defined even in the absence of a metric. The Lie derivative of a type-(r,s) tensor field ${\displaystyle T}$ along (the flow of) a contravariant vector field ${\displaystyle X^{\rho }}$ may be expressed as

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}=X^{\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }\,&-X^{\alpha _{1}}{}_{,\gamma }T^{\gamma \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -X^{\alpha _{r}}{}_{,\gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\gamma }{}_{\beta _{1}\cdots \beta _{s}}\\&+X^{\gamma }{}_{,\beta _{1}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\gamma \beta _{2}\cdots \beta _{s}}+\cdots +X^{\gamma }{}_{,\beta _{s}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\gamma }\,.\end{aligned}}}$

or defining the multi-indices

${\displaystyle {\begin{aligned}&A=\alpha _{1}\alpha _{2}\cdots \alpha _{r}\\&A_{1}=\gamma \alpha _{2}\cdots \alpha _{r}\\&A_{2}=\alpha _{1}\gamma \cdots \alpha _{r}\\&\quad \vdots \\&A_{r}=\alpha _{1}\alpha _{1}\cdots \alpha _{r-1}\gamma \\\end{aligned}}\quad {\begin{aligned}&B=\beta _{1}\beta _{2}\cdots \beta _{s}\\&B_{1}=\gamma \beta _{2}\cdots \beta _{s}\\&B_{2}=\beta _{1}\gamma \cdots \beta _{s}\\&\quad \vdots \\&B_{s}=\beta _{1}\beta _{1}\cdots \beta _{s-1}\gamma \end{aligned}}}$

the Lie derivative can be written

${\displaystyle ({\mathcal {L}}_{X}T)^{A}{}_{B}=X^{\gamma }T^{A}{}_{B,\gamma }\,-\sum _{i}X^{\alpha _{i}}{}_{,\gamma }T^{A_{i}}{}_{B}+\sum _{j}X^{\gamma }{}_{,\beta _{j}}T^{A}{}_{B_{j}}}$

analagous to the covariant derivative. This derivative is characterized by the product rule and the fact that the derivative of the given contravariant vector field ${\displaystyle X^{\rho }}$ is zero.

${\displaystyle ({\mathcal {L}}_{X}X)^{\rho }=^{\rho }=0\,.}$

The Lie derivative of a type-(r,s) relative tensor field ${\displaystyle \Lambda }$ of weight ${\displaystyle w\,}$ along (the flow of) a contravariant vector field ${\displaystyle X^{\rho }}$ may be expressed as

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}\Lambda )^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}=X^{\gamma }\Lambda ^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }\,&-X^{\alpha _{1}}{}_{,\gamma }\Lambda ^{\gamma \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -X^{\alpha _{r}}{}_{,\gamma }\Lambda ^{\alpha _{1}\cdots \alpha _{r-1}\gamma }{}_{\beta _{1}\cdots \beta _{s}}\\&+X^{\gamma }{}_{,\beta _{1}}\Lambda ^{\alpha _{1}\cdots \alpha _{r}}{}_{\gamma \beta _{2}\cdots \beta _{s}}+\cdots +X^{\gamma }{}_{,\beta _{s}}\Lambda ^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\gamma }\\&+wX^{\gamma }{}_{,\gamma }\Lambda ^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}\,.\end{aligned}}}$

Components of tensors

Given a tensor field and a basis (of linearly independent vector fields), the coefficients of the tensor field in a basis can be determined by evaluating a suitable combination of the basis and dual basis, and inherits the correct indexing. Some examples are listed below.

Basis and dual basis vectors

Throughout, let e_i be vector fields that constitute a vector basis at each point (a moving frame) for the tangent space, and ε be the dual basis (also called the cobasis) of the cotangent space, so that

${\displaystyle {\boldsymbol {\epsilon }}^{i}{\mathbf {e}}_{j}=\delta ^{i}{}_{j}}$

where δ_j is the Kronecker delta (see below).

Tensor product

Given two vectors

${\displaystyle \mathbf {a} =a^{i}{\mathbf {e}}_{i},\quad \mathbf {b} =b^{j}{\mathbf {e}}_{j},}$

their tensor product has the components:

${\displaystyle \mathbf {a} \otimes \mathbf {b} =a^{i}b^{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}$

and whichever vector comes first has the first index of the tensor, i.e. the above is not the same as

${\displaystyle \mathbf {b} \otimes \mathbf {a} =b^{i}a^{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}$

In the case of covectors, or synonymously 1-forms,

${\displaystyle {\boldsymbol {\alpha }}=\alpha _{i}{\boldsymbol {\epsilon }}^{i},\quad {\boldsymbol {\beta }}=\beta _{j}{\boldsymbol {\epsilon }}^{j},}$

we have

${\displaystyle {\boldsymbol {\alpha }}\otimes {\boldsymbol {\beta }}=\alpha _{i}\beta _{j}{\boldsymbol {\epsilon }}^{i}\otimes {\boldsymbol {\epsilon }}^{j}}$

similarly

${\displaystyle {\boldsymbol {\beta }}\otimes {\boldsymbol {\alpha }}=\beta _{i}\alpha _{j}{\boldsymbol {\epsilon }}^{i}\otimes {\boldsymbol {\epsilon }}^{j}}$

Exterior product

The wedge product is an antisymmetrized tensor product of vectors:

${\displaystyle \mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} \cdots =a^{}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\cdots }$

or 1-forms:

${\displaystyle {\boldsymbol {\alpha }}\wedge {\boldsymbol {\beta }}\wedge {\boldsymbol {\gamma }}\cdots =\alpha _{}\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}\otimes \mathbf {e} ^{k}\cdots }$

Tensor components

A tensor T as a multilinear function of 1-forms α, β, γ... and vectors a, b, c... corresponds to a contraction between the components of the tensor and the input vectors and 1-forms, resulting in another tensor; the notational correspondence is:

• Covariant tensor: ${\displaystyle T(\mathbf {a} ,\mathbf {b} ,\mathbf {c} ,\cdots )=T_{ijk\cdots }a^{i}b^{j}c^{k}\cdots }$
• Contravariant tensor: ${\displaystyle T({\boldsymbol {\alpha }},{\boldsymbol {\beta }},{\boldsymbol {\gamma }},\cdots )=T^{ijk\cdots }\alpha _{i}\beta _{j}\gamma _{k}\cdots }$
• Mixed tensor: ${\displaystyle T({\boldsymbol {\alpha }},{\mathbf {b}},{\boldsymbol {\gamma }},\cdots )=T^{i}{}_{j}{}^{k\cdots }\alpha _{i}b^{j}\gamma _{k}\cdots }$

hence the specific components of a tensor can be found by inserting the basis and dual basis vectors:

• Covariant tensor: ${\displaystyle T(\mathbf {e} _{i},\mathbf {e} _{j},\mathbf {e} _{k},\cdots )=T_{ijk\cdots }}$
• Contravariant tensor: ${\displaystyle T({\boldsymbol {\epsilon }}^{i},{\boldsymbol {\epsilon }}^{j},{\boldsymbol {\epsilon }}^{k},\cdots )=T^{ijk\cdots }}$
• Mixed tensor: ${\displaystyle T({\boldsymbol {\epsilon }}^{i},\mathbf {e} _{j},{\boldsymbol {\epsilon }}^{k},\cdots )=T^{i}{}_{j}{}^{k\cdots }}$

The tensor product of basis vectors gives the basis tensor elements, contracting with the components gives the tensor:

• Covariant tensor: ${\displaystyle T_{ijk\cdots }{\boldsymbol {\epsilon }}^{i}\otimes {\boldsymbol {\epsilon }}^{j}\otimes {\boldsymbol {\epsilon }}^{k}\cdots }$
• Contravariant tensor: ${\displaystyle T^{ijk\cdots }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\cdots }$
• Mixed tensor: ${\displaystyle T^{i}{}_{j}{}^{k\cdots }\mathbf {e} _{i}\otimes {\boldsymbol {\epsilon }}^{j}\otimes \mathbf {e} _{k}\cdots }$

analogous to vectors and 1-forms presented above.

This kind of calculation, i.e. "algebraic", also applies for some geometric operations in which appear coefficients that are not tensorial, for instance:

Christoffel symbols of the second kind

${\displaystyle \nabla _{i}\mathbf {e} _{j}=\Gamma ^{k}{}_{ij}\mathbf {e} _{k}}$

where ${\displaystyle \nabla _{i}\mathbf {e} _{j}\equiv \nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}\equiv \mathbf {e} _{j;i}}$ is the covariant derivative above, defined by the del operator ${\displaystyle \nabla }$ (also called Koszul connection on the tangent vector bundle).

Equivalently,

${\displaystyle \Gamma ^{k}{}_{ij}=\mathbf {e} ^{k}(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j})\,}$

Commutator coefficients

${\displaystyle =\gamma ^{k}{}_{ij}\mathbf {e} _{k}}$

where ${\displaystyle }$ is the Lie bracket as below, equivalently

${\displaystyle \gamma ^{k}{}_{ij}=\mathbf {e} ^{k}()\,.}$

Notable tensors

Kronecker delta

The Kronecker delta is like the identity matrix

${\displaystyle \delta _{\beta }^{\alpha }\,A^{\beta }=A^{\alpha }\,}$

${\displaystyle \delta _{\nu }^{\mu }\,B_{\mu }=B_{\nu }\,}$

when multiplied and contracted. The components ${\displaystyle \delta _{\beta }^{\alpha }\,}$ are the same in any basis and form an invariant tensor of type (1,1), i.e. the identity of the tangent bundle over the identity mapping of the base manifold, and so its trace is an invariant. The dimensionality of spacetime is its trace:

${\displaystyle \delta _{\rho }^{\rho }=\delta _{0}^{0}+\delta _{1}^{1}+\delta _{2}^{2}+\delta _{3}^{3}=4\,}$

in four-dimensional spacetime.

Metric tensor

The metric tensor gives the length of any space-like curve

${\displaystyle {\text{ Length }}=\int _{y_{1}}^{y_{2}}{\sqrt {g_{\alpha \beta }{\frac {dx^{\alpha }}{dy}}{\frac {dx^{\beta }}{dy}}}}\,dy\,}$

where y is any smooth strictly monotone parameterization of the path. It also gives the duration of any time-like curve

${\displaystyle {\text{ Duration }}=\int _{t_{1}}^{t_{2}}{\sqrt {{\frac {-1}{c^{2}}}g_{\alpha \beta }{\frac {dx^{\alpha }}{dt}}{\frac {dx^{\beta }}{dt}}}}\,dt\,}$

where t is any smooth strictly monotone parameterization of the trajectory.

The inverse matrix (also indicated with a g) of the metric tensor is another important tensor

${\displaystyle g^{\alpha \beta }g_{\beta \gamma }=\delta _{\gamma }^{\alpha }\,.}$

Riemann curvature tensor

If this tensor is defined as

${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\Gamma ^{\rho }{}_{\nu \sigma ,\mu }-\Gamma _{\mu \sigma ,\nu }^{\rho }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }\,,}$

then it is the commutator of the covariant derivative with itself:

${\displaystyle A_{\nu ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }\,,}$

since the connection ${\displaystyle \Gamma ^{\alpha }{}_{\beta \mu }\,}$ is torsionless, which means that the torsion tensor ${\displaystyle \Gamma ^{\lambda }{}_{\mu \nu }-\Gamma ^{\lambda }{}_{\nu \mu }\,}$ vanishes.

This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows

${\displaystyle {\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma \delta }-T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\delta \gamma }=\,&-R^{\alpha _{1}}{}_{\rho \gamma \delta }T^{\rho \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -R^{\alpha _{r}}{}_{\rho \gamma \delta }T^{\alpha _{1}\cdots \alpha _{r-1}\rho }{}_{\beta _{1}\cdots \beta _{s}}\\&+\,R^{\sigma }{}_{\beta _{1}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\sigma \beta _{2}\cdots \beta _{s}}+\cdots +R^{\sigma }{}_{\beta _{s}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\sigma }\,\end{aligned}}}$

which are often referred to as the Ricci identities.

See also

• Differential form
• Exterior calculus
• Penrose graphical notation
• Ricci decomposition

References

1. Ricci, Gregorio; Levi-Civita, Tullio (1900), "Méthodes de calcul différentiel absolu et leurs applications" (PDF), Mathematische Annalen, 54 (1–2), Springer: 125–201, doi:10.1007/BF01454201 {{citation}}: Unknown parameter |month= ignored (help)
2. Schouten, Jan A. (1924). R. Courant (ed.). Der Ricci-Kalkül - Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimmensional differential geometry). Grundlehren der mathematischen Wissenschaften (in German). Vol. 10. Berlin: Springer Verlag.
3. Synge J.L., Schild A. (1949). first Dover Publications 1978 edition. pp. 6–108. {{cite book}}: Missing or empty |title= (help)
4. J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
5. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
6. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
7. Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
8. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
9. G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
10. Covariant derivative – Mathworld, Wolfram
11. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 298, ISBN 978-1107-602601
12. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 299, ISBN 978-1107-602601
13. Bishop, R.L.; Goldberg, S.I. (1968), p. 130 {{citation}}: Missing or empty |title= (help)
14. Lovelock, David (1989). p. 123. {{cite book}}: Missing or empty |title= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
15. J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 76. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
16. Bishop, R.L.; Goldberg, S.I. (1968), p. 85 {{citation}}: Missing or empty |title= (help)
17. Synge J.L., Schild A. (1949). first Dover Publications 1978 edition. pp. 83, p. 107. {{cite book}}: Missing or empty |title= (help)
18. P. A. M. Dirac. General Theory of Relativity. pp. 20–21.
19. Lovelock, David (1989). p. 84. {{cite book}}: Missing or empty |title= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

Books

• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Lovelock, David (1989) . Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
• J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
• D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
• T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601

• Differential geometry
• Tensors