Einstein notation

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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 formulae. It was introduced by Albert Einstein in 1916 ^[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. Abstract index notation is an improvement of Einstein notation.

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, ... for 1,2,3 and Greek, μ, ν, ... for 0,1,2,3). As in sign conventions, the convention used in practice varies: Roman and Greek may be reversed.

Sometimes (as in general relativity), the index is required to appear once as a superscript and once as a subscript; in other applications, all indices are subscripts. See Dual vector space and Tensor product.

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. Additionally, the use of the implied summation of repeated indices is referred to as the Einstein Sum Convention.

1 Introduction
2 Vector representations
3 Matrix representation
4 Matrix multiplication
5 Vector dot product
6 Vector cross product
7 Abstract definitions
8 Examples
9 See also
10 References

[edit] Introduction

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

y = c 1 x 1 + c 2 x 2 + c 3 x 3 + ... + c n x 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.

[edit] Vector representations

First, we can use Einstein notation in linear algebra to distinguish easily between row vectors and column vectors. We could, for example, use superscripted indices to represent the elements of column vectors, and subscripted indices to represent the elements of row vectors. Following this convention, then,

$\mathbf{u} = u^i \ \ \mathrm{for} \ \ i = 1, 2, 3, ... , M$

represents an M × 1 column vector, and

$\mathbf{v} = v_j \ \ \mathrm{for} \ \ j = 1, 2, 3, ... , N$

represents a 1 × N row vector.

In mathematics and theoretical physics, and in particular general relativity, column vectors represent contravariant vectors whereas row vectors represent covariant vectors.

[edit] Matrix representation

Using standard notation, we can generate M × N matrix A by multiplying column vector u by row vector v:

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

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. Both indices survive the multiplication to become the two indices of the newly-created matrix A.

[edit] 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:

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

[edit] Vector dot product

In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal unit vectors i, j and k.

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

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 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. For example,

$\mathbf{u} \cdot \mathbf{v} = \sum_{i = 1}^3 u_i \mathbf{e}_i \cdot \sum_{j = 1}^3 v_j \mathbf{e}_j = u_i \mathbf{e}_i \cdot v_j \mathbf{e}_j$

or equivalently:

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

where

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

and $\ \delta_{ij}$ is the Kronecker delta, which is equal to 1 when i = j, and 0 otherwise. It logically follows that this allows one j in the equation to be converted to an i, or one i to be converted to a j. Then,

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

[edit] Vector cross product

For the cross product,

$\mathbf{u} \times \mathbf{v}= \sum_{j = 1}^3 u_j \mathbf{e}_j \times \sum_{k = 1}^3 v_k \mathbf{e}_k = u_j \mathbf{e}_j \times v_k \mathbf{e}_k = u_j v_k (\mathbf{e}_j \times \mathbf{e}_k ) = \epsilon_{ijk} \mathbf{e}_i u_j v_k$

where $\mathbf{e}_j \times \mathbf{e}_k = \epsilon_{ijk} \mathbf{e}_i$ and $\ \epsilon_{ijk}$ is the Levi-Civita symbol defined by:

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

which 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

$\mathbf{u} \times \mathbf{v}= \epsilon_{ijk} \mathbf{e}_i u_j v_k = \sum_{i = 1}^3 \sum_{j = 1}^3 \sum_{k = 1}^3 \epsilon_{ijk} \mathbf{e}_i u_j v_k$ .

Additionally, if $\mathbf{w} = \mathbf{u} \times \mathbf{v}$ , then $\mathbf{w} = \epsilon_{ijk} \mathbf{e}_i u_j v_k$ and $\ w_i = \epsilon_{ijk} u_j v_k$ . This also highlights that when an index appears once on both sides of the equation, this implies a system of equations instead of a summation:

$\begin{matrix} w_1 = \epsilon_{1jk} u_j v_k\\ w_2 = \epsilon_{2jk} u_j v_k\\ w_3 = \epsilon_{3jk} u_j v_k \end{matrix}$

Alternatively, this could be expressed as

$\mathbf{u} \times \mathbf{v}= \mathbf{u} \cdot \epsilon \cdot \mathbf{v}$

but, this isn't the notation Einstein used.

[edit] Abstract definitions

In the traditional usage, one has in mind a vector space V with finite dimension n, and a specific basis of V. We can write the basis vectors as e₁, e₂, ..., e_n. Then if v is a vector in V, it has coordinates v₁, ..., v_n relative to this basis.

The basic rule is:

v = v_i 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:

v = v_j 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.

In contexts where the index must appear once as a subscript and once as a superscript, the basis vectors e_i retain subscripts but the coordinates become vⁱ with superscripts. Then the basic rule is:

v = vⁱ e_i.

The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. 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.

[edit] 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. The tensor $\mathbf{} a^{\mu\nu}b_{\alpha}$ 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.g., flat spacetime).

[edit] See also

Wikibooks has a book on the topic of

General relativity:Einstein Summation Notation

[edit] References

^ Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. Retrieved on 2006-09-03.

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Einstein notation

From Wikipedia, the free encyclopedia

Contents

[edit] Introduction

[edit] Vector representations

[edit] Matrix representation

[edit] Matrix multiplication

[edit] Vector dot product

[edit] Vector cross product

[edit] Abstract definitions

[edit] Examples

[edit] See also

[edit] References

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