Einstein's sum convention

The Einstein notation is a convention of notation of mathematical expressions within the Ricci calculus and represents an index notation. This calculus is in the tensor analysis , the differential geometry , and particularly in theoretical physics used. The sum convention was introduced by Albert Einstein in 1916 . With it, the summation symbols are simply omitted to improve the overview and instead a total is made using double indices.

motivation

In matrix and tensor calculus, sums are often formed using indices. For example, the matrix product of two matrices and in components is: ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle (A\cdot B)_{ij}=\sum _{k=1}^{n}A_{ik}\cdot B_{kj}}$

Here the index is totaled from 1 to . If several matrix multiplications, scalar products or other sums occur in one calculation, this can quickly become confusing. With Einstein's summation convention, the calculation from above is then: ${\displaystyle k}$ ${\displaystyle n}$

${\displaystyle (A\cdot B)_{ij}=A_{ik}\cdot B_{kj}}$

Formal description

In the simplest case of the summation convention, the following applies: Duplicate indices within a product are used to sum up. In the theory of relativity, there is an additional rule: It is only added if the index occurs as both an upper (contravariant) and a lower ( covariant ) index.

Above all, the summation convention reduces the writing effort. Sometimes it helps to highlight existing relationships and symmetries that are not so easily recognizable in conventional sums.

Examples

Without considering the index position

The following examples are for matrices with entries and for matching vectors. ${\displaystyle A,B}$ ${\displaystyle n\times n}$ ${\displaystyle A_{ij},B_{ij}}$ ${\displaystyle {\vec {x}}=\left(x_{1},x_{2},\dots ,x_{n}\right),{\vec {y}}=\left(y_{1},y_{2},\dots ,y_{n}\right)}$

• Standard scalar product . ${\displaystyle {\vec {x}}\cdot {\vec {y}}=x_{i}y_{i}}$
• Applying a matrix to a vector: . ${\displaystyle \left(A{\vec {x}}\right)_{i}=A_{ij}x_{j}}$
• Product of several (in this case 4) matrices: . ${\displaystyle (ABCD)_{ij}=A_{il}B_{lm}C_{mn}D_{nj}}$
• Trace of a matrix A: . ${\displaystyle {\text{Spur}}\,A=A_{ii}}$

Taking into account the index position

• Standard scalar product . ${\displaystyle {\vec {x}}\cdot {\vec {y}}=x_{i}y^{i}}$
• The product of two tensors with tensor components and is . ${\displaystyle C_{j}^{i}}$ ${\displaystyle A_{j}^{i}}$ ${\displaystyle B_{j}^{i}}$ ${\displaystyle C_{j}^{i}=A_{k}^{i}B_{j}^{k}}$
• Application of a tensor with components on the sum of the vectors to vector to obtain: . ${\displaystyle A_{j}^{i}}$ ${\displaystyle x^{i},y^{i}}$ ${\displaystyle z^{i}}$ ${\displaystyle z^{i}=A_{j}^{i}\left(x^{j}+y^{j}\right)}$
• A tensor field t in a neighborhood has the representation ${\displaystyle U}$

${\displaystyle t|_{U}=t_{j_{1},\ldots ,j_{s}}^{i_{1},\ldots ,i_{r}}{\frac {\partial }{\partial x^{i_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial x^{i_{r}}}}\otimes \mathrm {d} x^{j_{1}}\otimes \cdots \otimes \mathrm {d} x^{j_{s}}.}$

Here, the index of the object is understood as the lower index. ${\displaystyle {\tfrac {\partial }{\partial x^{i_{1}}}}}$

Individual evidence

1. ↑ Albert Einstein : The basis of the general theory of relativity. In: Annals of Physics. 4th episode, vol. 49 = 354th vol. Of the whole series, number 7 (1916), pp. 770-822, doi : 10.1002 / andp.19163540702 .