View source for Tensor contraction

{{Short description|Operation in mathematics and physics}}
{{for|the module-theoretic construction of tensor fields and their contractions|tensor product of modules#Example from differential geometry: tensor field}}

In [[multilinear algebra]], a '''tensor contraction''' is an operation on a [[tensor]] that arises from the [[dual system|canonical pairing]] of a [[vector space]] and its [[dual vector space|dual]]. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the [[summation convention]] to a pair of dummy indices that are bound to each other in an expression.  The contraction of a single [[mixed tensor]] occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In [[Einstein notation]] this summation is built into the notation. The result is another [[tensor]] with order reduced by 2.

Tensor contraction can be seen as a [[generalization]] of the [[trace (linear algebra)|trace]].

== Abstract formulation ==
Let ''V'' be a vector space over a [[field (mathematics)|field]] ''k''. The core of the contraction operation, and the simplest case, is the canonical pairing of ''V'' with its [[Dual space|dual vector space]] ''V''<sup>∗</sup>. The pairing is the [[linear map]] from the [[tensor product]] of these two spaces to the field ''k'':

: <math> C : V \otimes V^* \rightarrow k </math>

corresponding to the [[bilinear form]]

: <math> \langle v, f \rangle = f(v) </math>

where ''f'' is in ''V''<sup>∗</sup> and ''v'' is in ''V''. The map ''C'' defines the contraction operation on a tensor of type {{nowrap|(1, 1)}}, which is an element of <math>V \otimes V^* </math>. Note that the result is a [[scalar (mathematics)|scalar]] (an element of ''k''). In [[dimension of a vector space|finite dimension]]s, using the [[natural isomorphism]] between <math>V \otimes V^* </math> and the space of linear maps from ''V'' to ''V'',<ref name="natural iso">Let {{nowrap|L(''V'', ''V'')}} be the space of linear maps from ''V'' to ''V''. Then the natural map

: <math>V^* \otimes V \rightarrow L(V,V) </math>

is defined by

: <math>f \otimes v \mapsto g ,</math>

where {{nowrap|1=''g''(''w'') = ''f''(''w'')''v''}}. Suppose that ''V'' is finite-dimensional. If {''v''<sub>''i''</sub>} is a [[basis (vector space)|basis]] of ''V'' and {''f''<sup>''i''</sup>} is the corresponding dual basis, then <math>f^i \otimes v_j</math> maps to the transformation whose matrix in this basis has only one nonzero entry, a 1 in the ''i'',''j'' position. This shows that the map is an isomorphism.</ref> one obtains a basis-free definition of the [[trace (linear algebra)|trace]].

In general, a [[tensor]] of type {{nowrap|(''m'', ''n'')}} (with {{nowrap|''m'' ≥ 1}} and {{nowrap|''n'' ≥ 1}}) is an element of the vector space

: <math>V \otimes \cdots \otimes V \otimes V^{*} \otimes \cdots \otimes V^{*}</math>

(where there are ''m'' factors ''V'' and ''n'' factors ''V''<sup>∗</sup>).<ref name="fulton_harris">{{cite book |first1=William |last1=Fulton |author-link1=William Fulton (mathematician) |first2=Joe |last2=Harris |author-link2=Joe Harris (mathematician) |title=Representation Theory: A First Course |series=[[Graduate Texts in Mathematics|GTM]] |volume=129 |publisher=Springer |location=New York |year=1991 |isbn=0-387-97495-4 |pages=471–476 }}</ref><ref name="warner">{{cite book |first=Frank |last=Warner |title=Foundations of Differentiable Manifolds and Lie Groups |series=[[Graduate Texts in Mathematics|GTM]] |volume=94 |publisher=Springer |location=New York |year=1993 |isbn=0-387-90894-3 |pages=54–56 }}</ref> Applying the canonical pairing to the ''k''th ''V'' factor and the ''l''th ''V''<sup>∗</sup> factor, and using the identity on all other factors, defines the (''k'', ''l'') contraction operation, which is a linear map that yields a tensor of type {{nowrap|(''m'' − 1, ''n'' − 1)}}.<ref name="fulton_harris"/> By analogy with the {{nowrap|(1, 1)}} case, the general contraction operation is sometimes called the trace.

== Contraction in index notation ==

In [[tensor index notation]], the basic contraction of a vector and a dual vector is denoted by

: <math> \tilde f (\vec v) = f_\gamma v^\gamma, </math>

which is shorthand for the explicit coordinate summation<ref name="physics">In physics (and sometimes in mathematics), indices often start with zero instead of one. In four-dimensional spacetime, indices run from 0 to 3.</ref>

: <math> f_\gamma v^\gamma = f_1 v^1 + f_2 v^2 + \cdots + f_n v^n </math>

(where {{math|''v''<sup>''i''</sup>}} are the components of {{math|''v''}} in a particular basis and {{math|''f''<sub>''i''</sub>}} are the components of {{math|''f''}} in the corresponding dual basis).

Since a general mixed [[dyadic tensor]] is a linear combination of decomposable tensors of the form <math>f \otimes v</math>, the explicit formula for the dyadic case follows: let

: <math> \mathbf{T} = T_{j}^i \mathbf{e}_i \otimes \mathbf{e}^j </math>

be a mixed dyadic tensor. Then its contraction is

: <math> T_{j}^i  \mathbf{e}_i \cdot \mathbf{e}^j = T_{j}^i  \delta_i {}^j
= T_{j}^j= T_{1}^1 + \cdots + T_{n}^n </math>.

A general contraction is denoted by labeling one [[Covariance and contravariance of vectors|covariant]] index and one [[Covariance and contravariance of vectors|contravariant]] index with the same letter, summation over that index being implied by the [[summation convention]]. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor ''T'' of type (2,2) on the second and third indices to create a new tensor ''U'' of type (1,1) is written as

: <math> T^{ab} {}_{bc} = \sum_{b}{T^{ab}{}_{bc}} = T^{a1} {}_{1c} + T^{a2} {}_{2c} + \cdots + T^{an} {}_{nc} = U^a {}_c .</math>

By contrast, let

: <math> \mathbf{T} = \mathbf{e}^i \otimes \mathbf{e}^j </math>

be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted,{{clarification|What is "dotted" supposed to mean here? Since it is not a contraction, as is explicitly stated, then what is its definition? In particular, how is the tensor <math>g^{ij}</math> intended to be different from the tensor <math>\mathbf{T}</math>? Right now the difference only looks formal, i.e. different notation for what must otherwise be the same object.|date=May 2020}} the result is the contravariant [[metric (mathematics)|metric tensor]],

: <math> g^{ij} = \mathbf{e}^i \cdot \mathbf{e}^j </math>,

whose rank is 2.

== Metric contraction ==
{{see also|Raising and lowering indices#An example from Minkowski spacetime}}
As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of an [[inner product]] (also known as a [[Metric tensor|metric]]) ''g'', such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as ''[[metric contraction]]''.<ref name="o'neill">{{cite book |first=Barrett |last=O'Neill |title=Semi-Riemannian Geometry with Applications to Relativity |publisher=Academic Press |year=1983 |page=86 |isbn=0-12-526740-1 }}</ref>

== Application to tensor fields ==

Contraction is often applied to [[tensor fields]] over spaces (e.g. [[Euclidean space]], [[manifold]]s, or [[scheme (mathematics)|schemes]]{{fact|date=April 2015}}). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. if ''T'' is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field) ''U'' at a point ''x'' is given by

: <math>U(x) = \sum_{i} T^{i}_{i}(x)</math>

Since the role of ''x'' is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.

Over a [[Riemannian manifold]], a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, the [[Ricci tensor]] is a non-metric contraction of the [[Riemann curvature tensor]], and the [[scalar curvature]] is the unique metric contraction of the Ricci tensor.

One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold<ref name="o'neill"/> or the context of sheaves of modules over the structure sheaf;<ref name="hartshorne">{{cite book |first=Robin |last=Hartshorne |author-link=Robin Hartshorne |title=Algebraic Geometry |location=New York |publisher=Springer |year=1977 |isbn=0-387-90244-9 }}</ref> see the discussion at the end of this article.

=== Tensor divergence ===

As an application of the contraction of a tensor field, let ''V'' be a [[vector field]] on a [[Riemannian manifold]] (for example, [[Euclidean space]]). Let <math> V^\alpha {}_{\beta}</math> be the [[covariant derivative]] of ''V'' (in some choice of coordinates). In the case of [[Cartesian coordinates]] in Euclidean space, one can write

: <math> V^\alpha {}_{\beta} = {\partial V^\alpha \over \partial x^\beta}. </math>

Then changing index ''β'' to ''α'' causes the pair of indices to become bound to each other, so that the derivative contracts with itself to obtain the following sum:

: <math> V^\alpha {}_{\alpha} = V^0 {}_{0} + \cdots + V^n {}_{n}, </math>

which is the [[divergence]] div ''V''. Then

: <math> \operatorname{div} V = V^\alpha {}_{\alpha} = 0 </math>

is a [[continuity equation]] for ''V''.

In general, one can define various divergence operations on higher-rank [[tensor fields]], as follows. If ''T'' is a tensor field with at least one contravariant index, taking the [[covariant differential]] and contracting the chosen contravariant index with the new covariant index corresponding to the differential results in a new tensor of rank one lower than that of ''T''.<ref name="o'neill"/>

== Contraction of a pair of tensors ==

One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensors ''T'' and ''U''. The [[tensor product]] <math>T \otimes U</math> is a new tensor, which, if it has at least one covariant and one contravariant index, can be contracted. The case where ''T'' is a vector and ''U'' is a dual vector is exactly the core operation introduced first in this article.

In tensor index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.

For example, matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant.  Let <math> \Lambda^\alpha {}_\beta </math> be the components of one  matrix and let <math> \Mu^\beta {}_\gamma </math> be the components of a second matrix.  Then their multiplication is given by the following contraction, an example of the contraction of a pair of tensors:

: <math> \Lambda^\alpha {}_\beta \Mu^\beta {}_\gamma = \Nu^\alpha {}_\gamma </math>.

Also, the [[interior product]] of a vector with a [[differential form]] is a special case of the contraction of two tensors with each other.

== More general algebraic contexts ==

Let ''R'' be a [[commutative ring]] and let ''M'' be a finite free [[module (mathematics)|module]] over ''R''. Then contraction operates on the full (mixed) tensor algebra of ''M'' in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the canonical pairing is still perfect in this case.)

More generally, let ''O''<sub>X</sub> be a [[sheaf (mathematics)|sheaf]] of commutative rings over a [[topological space]] ''X'', e.g. ''O''<sub>X</sub> could be the [[structure sheaf]] of a [[complex manifold]], [[analytic space]], or [[scheme (mathematics)|scheme]]. Let ''M'' be a [[locally free sheaf]] of modules over ''O''<sub>X</sub> of finite rank. Then the dual of ''M'' is still well-behaved<ref name="hartshorne"/> and contraction operations make sense in this context.

== See also ==

* [[Tensor product]]
* [[Partial trace]]
* [[Interior product]]
* [[Raising and lowering indices]]
* [[Musical isomorphism]]
* [[Ricci calculus]]

== Notes ==
{{reflist}}

== References ==

* {{cite book |first1=Richard L. |last1=Bishop |author-link1=Richard L. Bishop |first2=Samuel I. |last2=Goldberg |title=Tensor Analysis on Manifolds |location=New York |publisher=Dover |year=1980 |isbn=0-486-64039-6 |url-access=registration |url=https://archive.org/details/tensoranalysison00bish }}
* {{cite book |first=Donald H. |last=Menzel |author-link=Donald Howard Menzel |title=Mathematical Physics |publisher=Dover |location=New York |year=1961 |isbn=0-486-60056-4 }}

{{tensors}}

[[Category:Tensors]]