Inmathematics, especially the usage oflinear algebra inmathematical physics anddifferential geometry,Einstein notation (also known as theEinstein summation convention orEinstein summation notation) is a notational convention that impliessummation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset ofRicci calculus; however, it is often used in physics applications that do not distinguish betweentangent andcotangent spaces. It was introduced to physics byAlbert Einstein in 1916.[1]
According to this convention, when an index variable appears twice in a singleterm and is not otherwise defined (seeFree and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over theset{1, 2, 3},is simplified by the convention to:
The upper indices are notexponents but are indices of coordinates,coefficients orbasis vectors. That is, in this contextx2 should be understood as the second component ofx rather than the square ofx (this can occasionally lead to ambiguity). The upper index position inxi is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see§ Application below). Typically,(x1x2x3) would be equivalent to the traditional(xyz).
Ingeneral relativity, a common convention is that
In general, indices can range over anyindexing set, including aninfinite set. This should not be confused with a typographically similar convention used to distinguish betweentensor index notation and the closely related but distinct basis-independentabstract index notation.
An index that is summed over is asummation index, in this case "i ". It is also called adummy index since any symbol can replace "i " without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
An index that is not summed over is afree index and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "i " in the equation, which is equivalent to the equation.
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.[2] When dealing withcovariant and contravariant vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see§ Superscripts and subscripts versus only subscripts below.
In terms ofcovariance and contravariance of vectors,
They transform contravariantly or covariantly, respectively, with respect tochange of basis.
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and itscomponents, as in:
where is the vector and are its components (not theth covector), is the covector and are its components. The basis vector elements are each column vectors, and the covector basis elements are each row covectors. (See also§ Abstract description;duality, below and theexamples)
In the presence of anon-degenerate form (anisomorphismV →V∗, for instance aRiemannian metric orMinkowski metric), one canraise and lower indices.
A basis gives such a form (via thedual basis), hence when working onRn with aEuclidean metric and a fixedorthonormal basis, one has the option to work with only subscripts.
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; seeCovariance and contravariance of vectors.
In the above example, vectors are represented asn × 1matrices (column vectors), while covectors are represented as1 ×n matrices (row covectors).
When using the column vector convention:
The virtue of Einstein notation is that it represents theinvariant quantities with a simple notation.
In physics, ascalar is invariant under transformations of basis. In particular, aLorentz scalar is invariant under aLorentz transformation. The individual terms in the sum are not. When the basis is changed, thecomponents of a vector change by alinear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.
As for covectors, they change by theinverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
The value of the Einstein convention is that it applies to othervector spaces built fromV using thetensor product andduality. For example,V ⊗ V, the tensor product ofV with itself, has a basis consisting of tensors of the formeij =ei ⊗ej. Any tensorT inV ⊗ V can be written as:
V *, the dual ofV, has a basise1,e2, ...,en which obeys the rulewhereδ is theKronecker delta. Asthe row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
In Einstein notation, the usual element reference for the-th row and-th column of matrix becomes. We can then write the following operations in Einstein notation as follows.
Theinner product of two vectors is the sum of the products of their corresponding components, with the indices of one vector lowered (see#Raising and lowering indices):In the case of anorthonormal basis, we have, and the expression simplifies to:
In three dimensions, thecross product of two vectors with respect to apositively oriented orthonormal basis, meaning that, can be expressed as:
Here, is theLevi-Civita symbol. Since the basis is orthonormal, raising the index does not alter the value of, when treated as a tensor.
The product of a matrixAij with a column vectorvj is:equivalent to
This is a special case of matrix multiplication.
Thematrix product of two matricesAij andBjk is:
equivalent to
For asquare matrixAij, thetrace is the sum of the diagonal elements, hence the sum over a common indexAii.
Theouter product of the column vectorui by the row vectorvj yields anm × n matrixA:
Sincei andj represent twodifferent indices, there is no summation and the indices are not eliminated by the multiplication.
Given atensor, one canraise an index or lower an index by contracting the tensor with themetric tensor,gμν. For example, taking the tensorTαβ, one can lower an index:
Or one can raise an index:
