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.

According to this convention, when an index variable appears twice in a single term, once in an upper and once in a lower position, it implies that we are summing over all of its possible values. In typical applications, these are 1,2,3 (for calculations in Euclidean space), or 0,1,2,3 or 1,2,3,4 (for calculations in 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.

Introduction

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

[\mathbf = u_x \mathbf + u_y \mathbf + u_z \mathbf]

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_1 \mathbf_1 + u_2 \mathbf_2 + u_3 \mathbf_3 = \sum_^3 u_i \mathbf_i]

In Einstein notation, an index that is repeated twice in an equation implies a summation, and the summation symbol need not be included.

This allows a concise algebraic presentation of vector and tensor equations. For example,

[\mathbf \cdot \mathbf = \sum_^3 u_i \mathbf_i \cdot \sum_^3 v_j \mathbf_j = u_i \mathbf_i \cdot v_j \mathbf_j ]

or equivalently:

[\mathbf \cdot \mathbf = \sum_^3 \sum_^3 u_i v_j ( \mathbf_i \cdot \mathbf_j ) = u_i v_j ( \mathbf_i \cdot \mathbf_j ) ]

where

[ \mathbf_i \cdot \mathbf_j = \delta_ ]

and [\ \delta_] 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 \cdot \mathbf = u_i v_j\delta_= u_i v_i = u_j v_j ]

For the cross product,

[ \mathbf \times \mathbf= \sum_^3 u_j \mathbf_j \times \sum_^3 v_k \mathbf_k = u_j \mathbf_j \times v_k \mathbf_k = u_j v_k (\mathbf_j \times \mathbf_k ) = \epsilon_ \mathbf_i u_j v_k ]

where [ \mathbf_j \times \mathbf_k = \epsilon_ \mathbf_i] and [\ \epsilon_] is the Levi-Civita symbol defined by:

[\epsilon_ =\left\+1 & \mbox (i,j,k) \mbox (1,2,3), (2,3,1) \mbox (3,1,2)\\-1 & \mbox (i,j,k) \mbox (3,2,1), (1,3,2) \mbox (2,1,3)\\0 & \mboxi=j \mbox j=k \mbox k=i\end\right.]

which recovers

[ \mathbf \times \mathbf = (u_2 v_3 - u_3 v_2) \mathbf_1 + (u_3 v_1 - u_1 v_3) \mathbf_2 + (u_1 v_2 - u_2 v_1) \mathbf_3]

from

[ \mathbf \times \mathbf= \epsilon_ \mathbf_i u_j v_k = \sum_^3 \sum_^3 \sum_^3 \epsilon_ \mathbf_i u_j v_k ].

Additionally, if [ \mathbf = \mathbf \times \mathbf], then [ \mathbf = \epsilon_ \mathbf_i u_j v_k ] and [\ w_i = \epsilon_ 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:

[\beginw_1 = \epsilon_ u_j v_k\\w_2 = \epsilon_ u_j v_k\\w_3 = \epsilon_ u_j v_k\end]

Alternatively, this could be expressed as

[ \mathbf \times \mathbf= \mathbf \cdot \epsilon \cdot \mathbf]

but, this isn't the notation Einstein used.

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_ = \mathbf_i \otimes \mathbf_j]. Any tensor T in [V\otimes V] can be written as:

[\mathbf = T^\mathbf_].

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

[\mathbf^i (\mathbf_j) = \delta_^j].

Here δ is the Kronecker delta, so [\delta_^j] is 1 if i =j  and 0 otherwise.

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^ b_\mu = a^ b_0 + a^ b_1 + a^ b_2 + a^ b_3.]

The above example is one of contraction, a common tensor operation. The tensor [ \mathbf a^b_] 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 ^ = a^b_.]

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\cdot\mathbf = a^b_ = 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).

Miscellanea

In some fields, Einstein notation is referred to simply as index notation.

When an index is repeated three or more times, it means that there is a mistake somewhere.

See also

• Bra-ket notation

 

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