Ricci calculus

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 and differential geometry in the early twentieth century.
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 degree (or 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.
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:
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-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.
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.
It is in special cases (that the metric tensor is everywhere equal to the identity matrix) 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 formula_3 for the product of matrices can sometimes be understood as examples of this – but in general the notation requires that the distinction between upper and lower indices is observed and 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).
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:
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" () 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 on the tangent bundle of a differentiable manifold is called an affine connection.
A connection is a called 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:
which is equivalent to
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 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 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 over the identity mapping of the base manifold, and so its trace is an invariant.
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 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 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: