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), tensor calculus or tensor analysis 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. The basis of modern tensor analysis was developed by Bernhard Riemann in a paper from 1861.
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 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.
Applications Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Élie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
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: is an example of a variation: 'Greek indices run from 1 to 3, Latin indices from 1 to 4' The lowercase Latin alphabet 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 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 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.
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).
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
| || (both occurrences of are lower; would be fine) |- | || ( occurs twice as a lower index; or would be fine). |} 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 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 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.
{| class="wikitable" |- ! ! Basis transformation ! Component transformation ! Invariance |- ! Covector, covariant 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 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).
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 (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (), 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, 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