In
mathematics, Ricci calculus constitutes the rules of index notation and manipulation for
tensors and
tensor fields on a
differentiable manifold, with or without a
metric tensor or
connection. 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 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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
and
differential geometry in the early twentieth century.
A component of a tensor is a
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the underlying
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. The number of indices equals the degree (or order) of the tensor.
For compactness and convenience, the Ricci calculus incorporates
Einstein notation, which 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
Basis-related distinctions
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:
*The lowercase
Latin alphabet
The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and the ...
is used to indicate restriction to 3-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, 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
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
is used for 4-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
, 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 shows a direct correspondence between the subscripts 1, 2, 3 and the labels , , . In the expression , is interpreted as an index ranging over the values 1, 2, 3, while the , , subscripts are only labels, not variables. In the context of spacetime, the index value 0 conventionally corresponds to the label .
Reference to basis
Indices themselves may be ''labelled'' using
diacritic
A diacritic (also diacritical mark, diacritical point, diacritical sign, or accent) is a glyph added to a letter or to a basic glyph. The term derives from the Ancient Greek (, "distinguishing"), from (, "to distinguish"). The word ''diacriti ...
-like symbols, such as a
hat (ˆ),
bar
Bar or BAR may refer to:
Food and drink
* Bar (establishment), selling alcoholic beverages
* Candy bar
* Chocolate bar
Science and technology
* Bar (river morphology), a deposit of sediment
* Bar (tropical cyclone), a layer of cloud
* Bar (un ...
(¯),
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
In theoretical physics, Van der Waerden notation refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard in twistor theory and supersymmetry. It is named after Bartel Leendert van der Waerden.
Do ...
for
spinors, which uses hats and overdots on indices to reflect the chirality of a spinor.
Upper and lower indices
Ricci calculus, and
index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are ''not'' exponents, even though they may look as such to the reader only familiar with other parts of mathematics.
In the special case that the metric tensor is everywhere equal to the identity matrix, it is possible to drop the distinction between upper and lower indices, and then all indices could be written in the lower position. Coordinate formulae in linear algebra such as
for the product of matrices may be examples of this. But in general, the distinction between upper and lower indices should be maintained.
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
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\t ...
).
Tensor type and degree
The number of each upper and lower indices of a tensor gives its ''type'': a tensor with upper and lower indices is said to be of type , or to be a type- 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 has degree .
Summation convention
The same symbol occurring twice (one upper and one lower) within a term indicates a pair of indices that are summed over:
:
The operation implied by such a summation is called
tensor contraction:
:
This summation may occur more than once within a term with a distinct symbol per pair of indices, for example:
:
Other combinations of repeated indices within a term are considered to be ill-formed, such as
:
The reason for excluding such formulae is that although these quantities could be computed as arrays of numbers, they would not in general transform as tensors under a change of basis.
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:
:
where and .
Sequential summation
A pair of vertical bars around a set of all-upper indices or all-lower indices (but not both), associated with contraction with another set of indices when the expression is
completely antisymmetric in each of the two sets of indices:
:
means a restricted sum over index values, where each index is constrained to being strictly less than the next. 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:
:
where
:
Raising and lowering indices
By contracting an index with a non-singular
metric tensor, the
type
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* Ty ...
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 where 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
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of on ...
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.
The
Kronecker delta is used,
see also below.
:
General outlines for index notation and operations
Tensors are equal
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
every corresponding component is equal; e.g., tensor equals tensor if and only if
:
for all . Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to
dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as ...
).
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 , 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 free indices, and if the dimensionality of the underlying vector space is , the equality represents equations: each index takes on every value of a specific set of values.
For instance, if
:
is in
four dimensions Four Dimensions may refer to:
* ''Four Dimensions'' (Don Patterson album), 1968
* ''Four Dimensions'' (Lollipop F album), 2010
See also
*''Four Dimensions of Greta
''Four Dimensions of Greta'' is a 1972 British sex comedy film directed and pro ...
(that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (), there are 4
3 = 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, replaced and replaced ''everywhere'', so the expression still has the same meaning. In the second, did not fully replace , and did not fully replace (incidentally, the contraction on the index became a tensor product), which is entirely inconsistent for reasons shown next.
Indices are the same in every term
The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation the free indices are the same on each side. Dummy indices (which implies a summation over that index) need not be the same, 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, line up throughout and 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 lines up, and do not, and 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 indices using to range over permutations of the numbers 1 to , one takes a sum over the
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
s of those indices for , 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 and indices are symmetrized, is not.
Antisymmetric or alternating part of tensor
Square brackets, /nowiki>">nbsp;/nowiki>, around multiple indices denotes the ''anti''symmetrized part of the tensor. For antisymmetrizing indices – the sum over the permutations of those indices multiplied by the
signature of the permutation is taken, then divided by the number of permutations:
:
where is the
generalized Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\t ...
of degree , with scaling as defined below.
For example, two antisymmetrizing indices imply:
:
while three antisymmetrizing indices imply:
:
as for a more specific example, if 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 and indices are antisymmetrized, 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 and . This does not hold for other than two indices.
Differentiation
For compactness, derivatives may be indicated by adding indices after a comma or semicolon.
Partial derivative
While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor components with respect to coordinates apply only with a
coordinate basis: a basis that is defined through differentiation with respect to the coordinates. Coordinates are typically denoted by , but do not in general form the components of a vector. In flat spacetime with linear coordinatization, a tuple of ''differences'' in coordinates, , 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 do not in general transform covariantly, but 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, exterior and Lie derivatives below.
To indicate partial differentiation of the components of a tensor field with respect to a coordinate variable , 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 is the
Kronecker delta.
Covariant derivative
The covariant derivative is only defined if a
connection is defined. For any tensor field, a ''
semicolon
The semicolon or semi-colon is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought. When a ...
'' () placed before an appended lower (covariant) index indicates covariant differentiation. Less common alternatives to the semicolon include a ''
forward slash'' () or in three-dimensional curved space a single vertical bar ().
The covariant derivative of a scalar function, a contravariant vector and a covariant vector are:
:
:
:
where are the connection coefficients.
For an arbitrary tensor:
:
An alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol . For the case of a vector field :
:
The covariant formulation of the
directional derivative of any tensor field along a vector may be expressed as its contraction with the covariant derivative, e.g.:
:
The components of this derivative of a tensor field transform covariantly, and hence form another tensor field, despite subexpressions (the partial derivative and the connection coefficients) separately not transforming covariantly.
This derivative is characterized by the product rule:
:
Connection types
A
Koszul connection
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The mo ...
on the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
of a
differentiable manifold is called an
affine connection.
A connection is a
metric connection when the covariant derivative of the metric tensor vanishes:
:
An
affine connection that is also a metric connection is called a
Riemannian connection. A Riemannian connection that is torsion-free (i.e., for which the
torsion tensor vanishes: ) is a
Levi-Civita connection.
The for a Levi-Civita connection in a coordinate basis are called
Christoffel symbols of the second kind.
Exterior derivative
The exterior derivative of a totally antisymmetric type tensor field with components (also called a
differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold. In a coordinate basis, it may be expressed as the antisymmetrization of the partial derivatives of the tensor components:
:
This derivative is not defined on any tensor field with contravariant indices or that is not totally antisymmetric. It is characterized by a graded product rule.
Lie derivative
The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor or a connection. The Lie derivative of a type tensor field along (the flow of) a contravariant vector field
may be expressed using a coordinate basis as
:
This derivative is characterized by the product rule and the fact that the Lie derivative of a contravariant vector field along itself is zero:
:
Notable tensors
Kronecker delta
The Kronecker delta is like the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
when multiplied and contracted:
:
The components are the same in any basis and form an invariant tensor of type , i.e. the identity of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
over the
identity mapping of the
base manifold
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
, and so its trace is an invariant.
Its
trace is the dimensionality of the space; for example, in four-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
,
:
The Kronecker delta is one of the family of generalized Kronecker deltas. The generalized Kronecker delta of degree may be defined in terms of the Kronecker delta by (a common definition includes an additional multiplier of on the right):
:
and acts as an antisymmetrizer on indices:
:
Torsion tensor
An affine connection has a torsion tensor :
:
where are given by the components of the Lie bracket of the local basis, which vanish when it is a coordinate basis.
For a Levi-Civita connection this tensor is defined to be zero, which for a coordinate basis gives the equations
:
Riemann curvature tensor
If this tensor is defined as
:
then it is the
commutator of the covariant derivative with itself:
:
since the connection 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''.
Metric tensor
The metric tensor is used for lowering indices and gives the length of any
space-like curve
:
where is any
smooth strictly monotone
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
parameterization of the path. It also gives the duration of any
time-like curve
:
where is any smooth strictly monotone parameterization of the trajectory. See also ''
Line element''.
The
inverse matrix of the metric tensor is another important tensor, used for raising indices:
:
See also
*
Abstract index notation
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeh ...
*
Connection
*
Exterior algebra
*
Differential form
*
Hodge star operator
*
Holonomic basis
*
Metric tensor
*
Penrose graphical notation
*
Regge calculus
*
Ricci decomposition
*
Tensor (intrinsic definition)
*
Tensor calculus
*
Tensor field
Notes
References
Sources
*
*
*
*
*
*
*
*
*
{{tensors
Calculus
Differential geometry
Tensors