Einstein notation - Aetherwisp

Einstein summation notation is a convention used in linear algebra, especially special relativity, general relativity and differential geometry, to make vector notation more compact.

For example, using two four-vectors $a$ and $b$ , the quantity $a_{\mu}b^{\mu}$ implicitly calls for a sum, since the index $\mu$ is repeated and not otherwise defined, so

a_{\mu}b^{\mu}\equiv \sum_{\mu=0}^{3}a_{\mu}b^{\mu}

Whenever the covariance and contravariance of a vector is explicitly written using upper and lower indexes (as shown here), the notation is generally considered to apply only on an index that is repeated once as superscript and once as subscript. For instance, $a_{\mu}b^{\mu}$ would imply a sum, but $a_{\mu}b_{\mu}$ would not.