Einstein notation - Citizendia

For other topics related to Einstein, see Einstein (disambiguation).

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. It was introduced by Albert Einstein in 1916. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical ^[1]

According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values. In typical applications, the indices are 1,2,3 (representing the three dimensions of physical Euclidean space), or 0,1,2,3 or 1,2,3,4 (representing the four dimensions of space-time, or Minkowski space), but they can have any range, even (in some applications) an infinite set. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Set theory, an infinite set is a set that is not a Finite set. Abstract index notation is an improvement of Einstein notation. Abstract index notation is a mathematical notation for Tensors and Spinors that uses indices to indicate their types rather than their components in a particular basis

In general relativity, the Greek alphabet and the Roman alphabet are used to distinguish whether summing over 1,2,3 or 0,1,2,3 (usually Roman, i, j, . General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early . . for 1,2,3 and Greek, $\mu\,$ , $\nu\,$ , . . . for 0,1,2,3). As in sign conventions, the convention used in practice varies: Roman and Greek may be reversed.

When there is a fixed basis, one can work with only subscripts, but in general one must distinguish between superscripts and subscripts; see below.

It is important to keep in mind that no new physical laws or ideas result from using Einstein notation; rather, it merely helps in identifying relationships and symmetries often 'hidden' by more conventional notation.

In some fields, Einstein notation is referred to simply as index notation, or indicial notation. The use of the implied summation of repeated indices is also referred to as the Einstein Sum Convention.

1 Introduction
2 Vector representations
- 2.1 Mnemonics
3 Superscripts and subscripts vs. only subscripts
4 Common operations in this notation
5 Coefficients on tensors and related
6 Vector dot product
7 Vector cross product
8 Abstract definitions
9 Examples
10 See also
11 References

Introduction

The basic idea of Einstein notation is very simple. It allows one to replace something bulky, such as:

$y = c_1x_1+c_2x_2+c_3x_3+ ... + c_nx_n \,$

typically written as:

$y = \sum_{i=1}^n c_ix_i$

with something even simpler, in Einstein notation:

$y = c_i x^i \,$

In Einstein notation, indices such as i in the equation above can appear as either subscripts or superscripts. The position of the index has a specific meaning. It is important, of course, not to interpret an index appearing in the superscript position as if it were an exponent, which is the convention in standard algebra. Here, the superscripted i above the symbol x represents an integer-valued index running from 1 to n.

The virtue of Einstein notation is that an index appearing two or more times in a single term implies summation across that index, so that the summation symbol is unnecessary. Since the summation in effect "eliminates" the index over which the sum is taken, the summation index does not appear on the opposite side of the equals sign.

Vector representations

First, we can use Einstein notation in linear algebra to distinguish easily between vectors and covectors: upper indices represent the components of vectors, while lower indices represent the components of covectors. Linear algebra is the branch of Mathematics concerned with However, vectors themselves (not their components) have lower indices, and covectors have upper indices.

This point is frequently confused.

Given a vector space V and its dual space $V *$ , one represents vectors (elements of V) with subscripts, as in $v_i \in V$ , and covectors with superscripts, as in $w^i \in V^*$ . In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional However, the components of vectors and covectors follow the opposite convention: if $e i$ are a basis for V and $e i$ are the dual basis for $V *$ , then vectors are represented as:

$v = a^i e_i = \begin{bmatrix}a^1\\a^2\\\vdots\\a^n\end{bmatrix}$

and covectors are represented as

$w = a_i e^i = \begin{bmatrix}a_1 & a_2 & \cdots & a_n\end{bmatrix}$

This is because a component of a vector (a coefficient in some basis) is the value of a covector: the coefficient of $e i$ is the value of the corresponding covector in the dual basis: $a i = e i (v)$ . In linear algebra a dual basis is a set of vectors that forms a basis for the Dual space of a vector space Note that $e i$ is a covector, but $a i$ is a scalar. More prosaically, you pair components with vectors; since vectors have lower indices, components have upper indices.

In terms of covariance and contravariance of vectors, lower indices represent (components of!) covariant vectors (covectors), while upper indices represent (components of!) contravariant vectors (vectors): they transform covariantly (resp. For other uses of "covariant" or "contravariant" see Covariance and contravariance. For other uses of "covariant" or "contravariant" see Covariance and contravariance. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an , contravariantly) with respect to change of coordinates.

A particularly confusing notation is to use the same letter both for a (co)vector and its components, as in:

$v = v^i e_i = \begin{bmatrix}v^1\\v^2\\\vdots\\v^n\end{bmatrix}$

$w = w_i e^i = \begin{bmatrix}w_1 & w_2 & \cdots & w_n\end{bmatrix}$

Here $v i$ does not mean "the covector v", but rather, "the components of the vector v".

Mnemonics

"Upper indices go up to down; lower indices go left to right"
You can stack vectors (column matrices) side-by-side:

$\begin{bmatrix}v_1 & \cdots & v_k\end{bmatrix}.$

Hence the lower index indicates which column you are in.

You can stack covectors (row matrices) top-to-bottom:

$\begin{bmatrix}w^1 \\ \vdots \\ w^k\end{bmatrix}$

Hence the upper index indicates which row you are in.

Superscripts and subscripts vs. only subscripts

In the presence of a non-degenerate form (an isomorphism $V \to V^*$ ), (for instance a Riemannian metric or Minkowski metric), one can raise and lower indices. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In mathematics and mathematical physics given a Tensor on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or

A basis gives such a form (via the dual basis), hence when working on $\mathbf{R}^n$ with a fixed basis, one can work with just subscripts. In linear algebra a dual basis is a set of vectors that forms a basis for the Dual space of a vector space

However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see covariance and contravariance of vectors. For other uses of "covariant" or "contravariant" see Covariance and contravariance.

Common operations in this notation

In Einstein notation, the usual element reference $\mathbf{A}_{mn}$ for the $m$ th row and $n$ th column of matrix $\mathbf{A}$ becomes $\mathbf{A}_n^m$ . We can then write the following operations in Einstein notation as follows.

Inner product

Given a row vector $v i$ and a column vector $u i$ of the same size, we can take the inner product $v i u i$ , which is a scalar: it's evaluating the covector on the vector. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

Multiplication of a vector by a matrix

Given a matrix $A^i_j$ and a (column) vector $v j$ , the coefficients of the product $\mathbf{A}v$ are given by $A^i_j v^j$ .

Matrix multiplication

We can represent matrix multiplication as:

$C^i_k = A^i_j \cdot B^j_k$

This expression is equivalent to the more conventional (and less compact) notation:

$\mathbf{C}_{ik} = (\mathbf{A} \cdot \mathbf{B})_{ik} =\sum_{j=1}^N A_{ij} B_{jk}$

Trace

Given a matrix $A^i_j$ , summing over a common index $A^i_i$ yields the trace. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal

Outer product

The outer product of the column vector u by the row vector v yields an M × N matrix A:

$\mathbf{A} = \mathbf{u} \cdot \mathbf{v}$

In Einstein notation, we have:

$A^i_j = u^i \cdot v_j = uv^i_j$

Since i and j represent two different indices, and in this case over two different ranges M and N respectively, the indices are not eliminated by the multiplication. In Linear algebra, the outer product typically refers to the tensor product of two vectors. Both indices survive the multiplication to become the two indices of the newly-created matrix A.

Coefficients on tensors and related

Given a tensor field and a basis (of linearly independent vector fields), the coefficients of the tensor field in a basis can be computed by evaluating on a suitable combination of the basis and dual basis, and inherits the correct indexing. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity We list notable examples.

Throughout, let $e i$ be a basis of vector fields (a moving frame). In Mathematics, a moving frame is a flexible generalization of the notion of an Ordered basis of a Vector space often used to study the extrinsic differential

(covariant) metric tensor

g i j = g (e i, e j)

(contravariant) metric tensor

g i j = g (e i, e j)

Torsion tensor (using the below)

$T^c_{ab} = \Gamma^c_{ab} - \Gamma^c_{ba}-\gamma^c_{ab},$

which follows from the formula

$T = \nabla_X Y - \nabla_Y X - [X,Y].$

Riemann curvature tensor

${R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})$

This also applies for some operations that are not tensorial, for instance:

Christoffel symbols

$\nabla_ie_j=\Gamma_{ij}^ke_k$

where $\nabla_i e_j$ is the covariant derivative. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a Moving frame around a curve In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. Equivalently,

$\Gamma_{ij}^k = e^k\nabla_ie_j$

commutator coefficients

$[e_i,e_j] = \gamma_{ij}^k e_k$

where $[e i, e j]$ is the Lie bracket. Lie bracket can refer to Lie algebra Lie bracket of vector fields Equivalently,

$\gamma_{ij}^k = e^k[e_i,e_j].$

Vector dot product

In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal unit vectors i, j and k. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

$\mathbf{u} = u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k}$

If the basis vectors i, j, and k are instead expressed as e₁, e₂, and e₃, a vector can be expressed in terms of a summation:

$\mathbf{u} = u^1 \mathbf{e}_1 + u^2 \mathbf{e}_2 + u^3 \mathbf{e}_3 = \sum_{i = 1}^3 u^i \mathbf{e}_i$

In Einstein notation, the summation symbol is omitted since the index i is repeated once as an upper index and once as a lower index, and we simply write

$\mathbf{u} = u^i \mathbf{e}_i$

Using e₁, e₂, and e₃ instead of i, j, and k, together with Einstein notation, we obtain a concise algebraic presentation of vector and tensor equations. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually For example,

$\mathbf{u} \cdot \mathbf{v} = \left( \sum_{i = 1}^3 u^i \mathbf{e}_i \right) \cdot \left( \sum_{j = 1}^3 v^j \mathbf{e}_j \right) = (u^i \mathbf{e}_i) \cdot (v^j \mathbf{e}_j)= u^i v^j ( \mathbf{e}_i \cdot \mathbf{e}_j ).$

Since

$\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$

where $\ \delta_{ij}$ is the Kronecker delta, which is equal to 1 when i = j, and 0 otherwise, we find

$\mathbf{u} \cdot \mathbf{v} = u^i v^j\delta_{ij}.$

One can use $\ \delta_{ij}$ to lower indices of the vectors; namely, $\ u_i=\delta_{ij}u^j$ and $\ v_i=\delta_{ij}v^j$ . In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two Then

$\mathbf{u} \cdot \mathbf{v} = u^i v^j\delta_{ij}= u^i v_i = u_j v^j$

Note that, despite $u i = u i$ for any fixed $i$ , it is incorrect to write

$\mathbf{u} \cdot \mathbf{v} = u^iv^i,$

since on the right hand side the index $i$ is repeated both times as an upper index and so there is no summation over $i$ according to the Einstein convention. Rather, one should explicitly write the summation:

$\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^3u^iv^i.$

Vector cross product

For the cross product,

$\mathbf{u} \times \mathbf{v}= \left( \sum_{j = 1}^3 u^j \mathbf{e}_j \right) \times \left( \sum_{k = 1}^3 v^k \mathbf{e}_k \right) = (u^j \mathbf{e}_j ) \times (v^k \mathbf{e}_k)$

$= u^j v^k (\mathbf{e}_j \times \mathbf{e}_k ) = u^j v^k\epsilon^i_{jk} \mathbf{e}_i$

where $\mathbf{e}_j \times \mathbf{e}_k = \epsilon^i_{jk} \mathbf{e}_i$ and $\ \epsilon^i_{jk}=\delta^{il}\epsilon_{ljk}$ , with $ε i j k$ the Levi-Civita symbol defined by:

$\epsilon_{ijk} = \left\{ \begin{matrix} 0 & \mbox{unless } i,j,k \mbox{ are distinct}\\ +1 & \mbox{if } (i,j,k) \mbox{ is an even permutation of } (1,2,3)\\ -1 & \mbox{if } (i,j,k) \mbox{ is an odd permutation of } (1,2,3) \end{matrix} \right.$

One then recovers

$\mathbf{u} \times \mathbf{v} = (u^2 v^3 - u^3 v^2) \mathbf{e}_1 + (u^3 v^1 - u^1 v^3) \mathbf{e}_2 + (u^1 v^2 - u^2 v^1) \mathbf{e}_3$

from

$\mathbf{u} \times \mathbf{v}= \epsilon^i_{jk} u^j v^k\mathbf{e}_i = \sum_{i = 1}^3 \sum_{j = 1}^3 \sum_{k = 1}^3 \epsilon^i_{jk} u^j v^k\mathbf{e}_i$ . In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor

In other words, if $\mathbf{w} = \mathbf{u} \times \mathbf{v}$ , then $w^i \mathbf{e}_i= \epsilon^i_{jk} u^j v^k\mathbf{e}_i$ , so that $\ w^i = \epsilon^i_{jk} u^j v^k$ .

Abstract definitions

In the traditional usage, one has in mind a vector space V with finite dimension n, and a specific basis of V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the dimension of a Vector space V is the cardinality (i Basis vector redirects here For basis vector in the context of crystals see Crystal structure. We can write the basis vectors as e₁, e₂, . . . , e_n. Then if v is a vector in V, it has coordinates $v^1,\dots,v^n$ relative to this basis.

The basic rule is:

$\mathbf{v} = v^i\mathbf{e}_i.$

In this expression, it was assumed that the term on the right side was to be summed as i goes from 1 to n, because the index i does not appear on both sides of the expression. (Or, using Einstein's convention, because the index i appeared twice. )

The i is known as a dummy index since the result is not dependent on it; thus we could also write, for example:

$\mathbf{v} = v^j\mathbf{e}_j.$

An index that is not summed over is a free index and should be found in each term of the equation or formula. Compare dummy indices and free indices with free variables and bound variables. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a

The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals For example, $V\otimes V$ , the tensor product of V with itself, has a basis consisting of tensors of the form $\mathbf{e}_{ij} = \mathbf{e}_i \otimes \mathbf{e}_j$ . Any tensor T in $V\otimes V$ can be written as:

$\mathbf{T} = T^{ij}\mathbf{e}_{ij}$ .

V*, the dual of V, has a basis e¹, e², . . . , eⁿ which obeys the rule

$\mathbf{e}^i (\mathbf{e}_j) = \delta^i_j$ .

Here δ is the Kronecker delta, so $\delta^i_j$ is 1 if i =j and 0 otherwise. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

$\mathrm{Hom}(V,W) = V^* \otimes W$

the row-column coordinates on a matrix correspond to the upper-lower indices on the tensor product.

Examples

Einstein summation is clarified with the help of a few simple examples. Consider four-dimensional spacetime, where indices run from 0 to 3:

$\mathbf{} a^\mu b_\mu = a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3$

$\mathbf{} a^{\mu\nu} b_\mu = a^{0\nu} b_0 + a^{1\nu} b_1 + a^{2\nu} b_2 + a^{3\nu} b_3.$

The above example is one of contraction, a common tensor operation. In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional The tensor $\mathbf{} a^{\mu\nu}b_{\mu}$ becomes a new tensor by summing over the first upper index and the lower index. Typically the resulting tensor is renamed with the contracted indices removed:

$\mathbf{} {s}^{\nu} = a^{\mu\nu}b_{\mu}.$

For a familiar example, consider the dot product of two vectors a and b. The dot product is defined simply as summation over the indices of a and b:

$\mathbf{a}\cdot\mathbf{b} = a^{\alpha}b_{\alpha} = a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3,$

which is our familiar formula for the vector dot product. Remember it is sometimes necessary to change the components of a in order to lower its index; however, this is not necessary in Euclidean space, or any space with a metric equal to its inverse metric (e. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space g. , flat spacetime).

References

Rawlings, Steve. Abstract index notation is a mathematical notation for Tensors and Spinors that uses indices to indicate their types rather than their components in a particular basis Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical In Mathematics and Physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten visual depiction of Multilinear functions "Lecture 10 - Einstein Summation Convention and Vector Identities", Oxford University, 2007-02-01. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1327 - Teenaged Edward III is crowned King of England, but the country is ruled by his mother Queen

^ Einstein, Albert (1916). Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Year 1916 ( MCMXVI) was a Leap year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Leap year "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik.

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