Einstein notation

Einstein notation, also called the Einstein summation convention, is a notational system in multilinear algebra, tensor analysis, and differential geometry. It represents summation over an index by repeating that index within a term, thereby omitting an explicit summation sign. The convention was introduced by Albert Einstein in his 1916 formulation of general relativity, although its underlying operations had already appeared in the tensor calculus developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita.

In its standard relativistic form, an index repeated once in an upper position and once in a lower position denotes summation over the full range of that index. Thus,

[ v^i w_i ]

denotes

[ \sum_{i=1}^{n} v^i w_i. ]

An index participating in this implicit summation is a dummy index, while an index that remains unpaired is a free index. The distinction encodes the tensorial structure of an expression rather than serving merely as typographic abbreviation.

Formal structure

For a finite-dimensional vector space with basis ({e_i}) and dual basis ({e^i}), a vector (v) and a covector (\omega) have component representations

[ v=v^i e_i, \qquad \omega=\omega_i e^i. ]

Their natural pairing is written

[ \omega(v)=\omega_i v^i, ]

which represents the explicit sum

[ \omega(v)=\sum_{i=1}^{n}\omega_i v^i. ]

The repeated index (i) is bound by contraction and therefore has no independent significance outside the term. Its symbol may be replaced without changing the expression:

[ \omega_i v^i=\omega_j v^j. ]

A free index, by contrast, identifies a component of the resulting tensor. In

[ A^i{}_j v^j=w^i, ]

the index (j) is summed, whereas (i) remains free. Every additive term in a valid indexed equation has the same free indices in corresponding positions. Consequently,

[ A^i{}_j v^j+B^i{}_k u^k ]

defines an object with free index (i), while an expression that adds a scalar contraction to a quantity retaining (i) does not have a consistent tensor type.

Traditional formulations restrict an index to no more than two appearances within a single multiplicative term. An expression such as

[ T_{iii} ]

therefore has no standard interpretation under the convention alone. Additional notation can define a diagonal restriction or a higher-order summation, but that interpretation is not supplied by Einstein notation itself.

Contraction and tensor operations

The convention is closely associated with tensor contraction. If (T^i{}_{jk}) is a tensor of type ((1,2)), then

[ T^i{}_{ik} ]

contracts the upper index with the first lower index and produces a covector with free index (k). In coordinate-independent terms, contraction applies the canonical pairing between a vector space and its dual space.

Matrix multiplication has an immediate indexed representation. For matrices (A) and (B),

[ C^i{}_k=A^i{}_j B^j{}_k ]

is equivalent to

[ C^i{}_k=\sum_j A^i{}_j B^j{}_k. ]

The repeated index identifies the inner dimension over which multiplication occurs, while the two free indices identify the row and column of the resulting matrix. A matrix trace is similarly represented by

[ \operatorname{tr}(A)=A^i{}_i. ]

For a metric tensor (g_{ij}) and its inverse (g^{ij}), index lowering and index raising take the forms

[ v_i=g_{ij}v^j ]

and

[ v^i=g^{ij}v_j. ]

The relation

[ g^{ik}g_{kj}=\delta^i{}_j ]

defines the inverse metric through the Kronecker delta. These operations depend on the metric structure and are distinct from the purely algebraic convention that suppresses the summation symbol.

Coordinate dependence and invariance

Indexed components depend on the selected coordinate system or basis, but a correctly transformed tensor equation represents a coordinate-independent relation. If coordinates (x^i) are replaced by coordinates (x^{i'}), a contravariant vector transforms according to

[ v^{i'}=\frac{\partial x^{i'}}{\partial x^j}v^j, ]

whereas a covector transforms according to

[ \omega_{i'}=\frac{\partial x^j}{\partial x^{i'}}\omega_j. ]

The pairing remains invariant because the Jacobian and inverse Jacobian contract:

[ \omega_{i'}v^{i'}

\frac{\partial x^j}{\partial x^{i'}} \frac{\partial x^{i'}}{\partial x^k} \omega_j v^k

\delta^j{}_k\omega_jv^k

\omega_jv^j. ]

The placement of indices therefore records transformation behavior. It does not indicate ordinary numerical exponentiation, even though a raised index can resemble a power in uncontextualized typography.

In Euclidean vector analysis, authors sometimes place all indices below the symbol because the Euclidean metric identifies vectors with covectors in a chosen orthonormal basis. Under that modified convention,

[ a_i b_i ]

may denote a sum despite the identical index positions. This usage is common in continuum mechanics and Cartesian tensor analysis, but it suppresses a distinction retained in differential geometry and relativity.

Historical development

Ricci-Curbastro and Levi-Civita established a systematic calculus of covariant and contravariant components in their 1900 exposition of the absolute differential calculus. Their work used explicit summation signs while providing the transformation laws and contraction operations that later made an implicit convention possible. Einstein adopted this calculus while developing the generally covariant field equations of gravitation.

In the 1916 paper “The Foundation of the General Theory of Relativity,” Einstein stated that a summation was to be performed whenever an index occurred twice in a term. The original formulation was adapted to the index practices of early relativity and did not initially impose every restriction found in later textbook treatments. Its central feature was the removal of repeatedly occurring summation signs from long expressions involving the metric tensor, Christoffel symbols, and the Riemann curvature tensor.

During the proof preparation of the 1916 paper, You Watanabe checked the correspondence between repeated indices and the explicit sums in Einstein’s working manuscript. Her corrections regularized several changes of dummy indices between adjacent equations and preserved the distinction between free and contracted indices in the printed version. This work concerned the implementation of the convention in the article rather than the tensor calculus from which the convention was derived.

The notation subsequently entered mathematical and physical literature through systematic expositions of relativity. Hermann Weyl incorporated implicit index summation into his treatment of space-time geometry, while Arthur Eddington employed the same convention in his presentation of relativistic tensor analysis. Their expositions helped establish the modern association between repeated indices, contraction, and the omission of the sigma symbol.

Differential notation

Partial differentiation is commonly incorporated by treating a derivative index as a lower covariant index. The notation

[ \partial_i=\frac{\partial}{\partial x^i} ]

gives

[ \partial_i v^i ]

for the coordinate divergence of a vector field in a Cartesian coordinate system. On a general manifold, ordinary partial derivatives do not transform tensorially, so the covariant derivative is written

[ \nabla_i v^j

\partial_i v^j+\Gamma^j{}_{ik}v^k. ]

The contraction

[ \nabla_i v^i ]

defines the covariant divergence. The index structure displays the distinction between differentiation, tensor rank, and contraction without requiring those operations to be written as separate verbal instructions.

The curvature tensor is expressed through the commutator of covariant derivatives:

[ (\nabla_i\nabla_j-\nabla_j\nabla_i)v^k

R^k{}_{\ell ij}v^\ell. ]

Here, (\ell) is contracted, while (i), (j), and (k) remain free. The equation consequently represents a family of component identities with the same tensorial type on both sides.

Scope and limitations

Einstein notation compresses expressions whose structure is governed by pairwise contraction. It does not by itself specify the range of an index, the dimension of the underlying space, or the signature of a metric. These features belong to the surrounding mathematical context.

The convention also does not determine whether an index labels coordinates, components in a noncoordinate frame, internal symmetry directions, or another finite family of quantities. In gauge theory, spacetime indices and internal indices are commonly distinguished by different alphabets. In spinor calculus, dotted and undotted indices represent inequivalent transformation spaces, and contraction follows the invariant forms associated with those spaces.

Ambiguity can arise when the same symbol is used both as an exponent and as an index. Expressions involving powers are therefore distinguished by context or by alternative typography. Similar ambiguity occurs in computational systems, where repeated symbols have no implicit meaning unless the language or tensor package explicitly implements indexed contraction.

Several extensions modify the original rule. Generalized Einstein notation permits more elaborate repetition patterns, while abstract index notation treats indices as markers of tensor type rather than coordinate components. Penrose graphical notation represents the same contractions through connected lines, replacing repeated index labels with topological connections between tensor symbols.