Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.[1][2][3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper[4] written with his pupil Tullio Levi-Civita in 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 twentieth century.[5]

A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows an efficient expression 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 summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as 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

Space and time coordinates

Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:[6]

• 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, indicated by 0, 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.

Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Otherwise, in general mathematical contexts, any symbols can be used for the indices, generally running over all dimensions 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) = (Ax, Ay, Az) shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression Ai, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are not variable indices, more like "names" for the components. In the context of spacetime, the index value 0 conventionally corresponds to the label t.

Reference to basis

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

to denote a possibly different basis for that index. An example is in Lorentz transformations from one frame of reference to another, where one frame could be unprimed and the other primed, as in:

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

Upper and lower indices

Covariant tensor components

A lower index (subscript) indicates covariance of the components with respect to that index:

Contravariant tensor components

An upper index (superscript) indicates contravariance of the components with respect to that index:

Mixed-variance tensor components

A tensor may have both upper and lower indices:

Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with the generalized Kronecker delta).

Tensor type and degree

The number of each upper and lower indices of a tensor gives its type: a tensor with p upper and q lower indices is said to be of type (p, q), or to be a type-(p, q) tensor.

The number of indices of a tensor, regardless of variance, is called the degree of the tensor (alternatively, its valence, order or rank, although rank is ambiguous). Thus, a tensor of type (p, q) has degree p + q.

Summation convention

Two indices (one upper and one lower) with the same symbol within a term are summed over:

The operation implied by such a summation is called tensor contraction:

More than one index may each occur exactly twice within one term, for example:

As for a non-identity,

is not considered well-formed, that is, it is meaningless because it does not transform like a tensor under a change of basis (change of coordinate system).

Multi-index notation

If a tensor has a list of all upper or lower indices, one shorthand is to use a capital letter for the list:[7]

where I = i1 i2 ⋅⋅⋅ in and J = j1 j2 ⋅⋅⋅ jm.

Sequential summation

A pair of vertical bars | | around a set of all-upper indices or all-lower indices, associated with contraction with another set of indices:[8]

means a restricted sum over index values, where each index is constrained to being strictly less than the next. The vertical bars are placed around either the upper set or the lower set of contracted indices, not both sets. Normally when contracting indices, the sum is over all values. In this notation, the summations are restricted as a computational convenience. This is useful when the expression is completely antisymmetric in each of the two sets of indices, as might occur on the tensor product of a p-vector with a q-form. More than one group can be summed in this way, for example:

When using multi-index notation, an underarrow is placed underneath the block of indices:[9]

where

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:

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.

Correlations between index positions and invariance

This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.[10]

Basis transformation Component transformation Invariance
Covector, covariant vector, dual vector, 1-form
Vector, contravariant vector

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

for all α, β, 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 involved in contractions are called free indices. Indices used in contractions are termed dummy indices, or summation indices.

A tensor equation represents many ordinary (real-valued) equations

The components of tensors (like Aα, Bβγ etc.) are just real numbers. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations. If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents mn equations: each index takes on every value of a specific set of values.

For instance, if

is in four dimensions (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (α, β, δ), there are 43 = 64 equations. Three of these are:

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:

whereas an erroneous change is:

In the first replacement, λ replaced α and μ replaced γ everywhere, so the expression still has the same meaning. In the second, λ did not fully replace α, and μ did not fully replace γ (incidentally, the contraction on the γ index became a tensor product), which 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:

as for an erroneous expression:

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 (once as an upper index and once as a lower index), and thus it is a valid 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

Parentheses, ( ), around multiple indices denotes the symmetrized part of the tensor. When symmetrizing p indices using σ to range over permutations of the numbers 1 to p, one takes a sum over the permutations of those indices ασ(i) for i = 1, 2, 3, …, p, and then divides by the number of permutations:

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

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

The symmetrization is distributive over addition;

Indices are not part of the symmetrization when they are:

• not on the same level, for example;
• within the parentheses and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;

Here the α and γ indices are symmetrized, β is not.

Antisymmetric or alternating part of tensor

Square brackets, [ ], around multiple indices denotes the antisymmetrized part of the tensor. For p antisymmetrizing indices – the sum over the permutations of those indices ασ(i) multiplied by the signature of the permutation sgn(σ) is taken, then divided by the number of permutations:

where δβ1⋅⋅⋅βp
α1⋅⋅⋅αp
is the generalized Kronecker delta of degree 2p, with scaling as defined below.

For example, two antisymmetrizing indices imply:

while three antisymmetrizing indices imply:

as for a more specific example, if F represents the electromagnetic tensor, then the equation

represents Gauss's law for magnetism and Faraday's law of induction.

As before, the antisymmetrization is distributive over addition;

As with symmetrization, indices are not antisymmetrized when they are:

• not on the same level, for example;
• within the square brackets and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;

Here the α and γ indices are antisymmetrized, β is not.

Sum of symmetric and antisymmetric parts

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

as can be seen by adding the above expressions for A(αβ)γ⋅⋅⋅ and A[αβ]γ⋅⋅⋅. This does not hold for other than two indices.

Differentiation

For compactness, derivatives may be indicated by adding indices after a comma or semicolon.[11][12]

Partial derivative

While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor coordinated apply only with a coordinate basis: a basis that is defined through differentiation of the coordinates. Coordinates are typically denoted by xμ, but do not in general form the components of a vector. In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δxμ, can be treated as a contravariant vector. With the same constraints on the space and on the choice of coordinate system, the partial derivatives with respect to the coordinates yield a result that is effectively covariant. Aside from use in this special case, the partial derivatives of components of tensors are useful in building expressions that are covariant, albeit still with a coordinate basis if the partial derivatives are explicitly used, as with the covariant and Lie derivatives below.

To indicate partial differentiation of the components of a tensor field with respect to a coordinate variable xγ, a comma is placed before an appended lower index of the coordinate variable.

This may be repeated (without adding further commas):

These components do not transform covariantly, unless the expression being differentiated is a scalar. This derivative is characterized by the product rule and the derivatives of the coordinates

where δ is the Kronecker delta.

Covariant derivative

To indicate covariant differentiation of any tensor field, a semicolon ( ; ) is placed before an appended lower (covariant) index. Less common alternatives to the semicolon include a forward slash ( / )[13] or in three-dimensional curved space a single vertical bar ( | ).[14]

For a contravariant vector, its covariant derivative is:

where Γαβγ is a Christoffel symbol of the second kind.

For a covariant vector, its covariant derivative is:

For an arbitrary tensor:[15]

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 applied to the metric tensor gμν it gives zero:

The covariant formulation of the directional derivative of any tensor field along a vector vγ may be expressed as its contraction with the covariant derivative, e.g.:

One alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol β. For the case of a vector field Aα:[16]

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 tensor. The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field Xρ may be expressed as[17]

This derivative is characterized by the product rule and the fact that the derivative of the given contravariant vector field Xρ is zero.

The Lie derivative of a type (r, s) relative tensor field Λ of weight w along (the flow of) a contravariant vector field Xρ may be expressed as[18]

Notable tensors

Kronecker delta

The Kronecker delta is like the identity matrix

when multiplied and contracted. The components δα
β
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.[19] Its trace is the dimensionality of the space; for example, in four-dimensional spacetime,

The Kronecker delta is one of the family of generalized Kronecker deltas. The generalized Kronecker delta of degree 2p may be defined in terms of the Kronecker delta by (a common definition includes an additional multiplier of p! on the right):

and acts as an antisymmetrizer on p indices:

Metric tensor

The metric tensor gαβ is used for lowering indices and gives the length of any space-like curve

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

where t is any smooth strictly monotone parameterization of the trajectory. See also line element.

The inverse matrix gαβ of the metric tensor is another important tensor, used for raising indices:

Riemann curvature tensor

If this tensor is defined as

then it is the commutator of the covariant derivative with itself:[20][21]

since the connection Γαβμ is torsionless, which means that the torsion tensor

vanishes.

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

which are often referred to as the Ricci identities.[22]

References

1. Synge J.L.; Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. pp. 6–108.
2. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.
3. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
4. Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen (in French), Springer, 54 (1–2): 125–201, doi:10.1007/BF01454201
5. 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). 10. Berlin: Springer Verlag.
6. C. Møller (1952), The Theory of Relativity, p. 234 is an example of a variation: 'Greek indices run from 1 to 3, Latin indices from 1 to 4'
7. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
8. Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, p. 91, ISBN 0-7167-0344-0
9. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
10. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 61, 202–203, 232. ISBN 0-7167-0344-0.
11. G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
12. Covariant derivative – Mathworld, Wolfram
13. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 298, ISBN 978-1107-602601
14. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 510, §21.5. ISBN 0-7167-0344-0.
15. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 299, ISBN 978-1107-602601
16. D. McMahon (2006). Relativity. Demystified. McGraw Hill. p. 67. ISBN 0-07-145545-0.
17. Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 130
18. Lovelock, David; Hanno Rund (1989). Tensors, Differential Forms, and Variational Principles. p. 123.
19. Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 85
20. Synge J.L.; Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. pp. 83, p. 107.
21. P. A. M. Dirac. General Theory of Relativity. pp. 20–21.
22. Lovelock, David; Hanno Rund (1989). Tensors, Differential Forms, and Variational Principles. p. 84.

Sources

• 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; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
• C. Møller (1952), The Theory of Relativity (3rd ed.), Oxford University Press
• 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