Einstein’s Summation Convention

Contravariant “Upstairs” Vector

Let the spacetime four-vector be \(a\). The transformation of the interial reference frame can be represented for each component indexed as \(\mu\). We can represent this as the linear combination of the bases of \(x^\nu \in \vec x\)

\[\begin{split} \set{x^\mu} \rightarrow \set{\bar x^\mu}\\ \bar a^\mu = \sum_\nu \left(\frac{\partial \bar x^\mu}{\partial x^\nu} \right)a^\nu \end{split}\]

In Enstein’s summation convention, the twice repeated indices are summed across so we can simply that to,

\[ \bar a^\mu = \left(\frac{\partial \bar x^\mu}{\partial x^\nu} \right)a^\nu \]

Covariant “Downstairs” Vector

Let a constant be \(\alpha\), the gradient vector is represented by a the four-derivative operator,

\[ \partial_\mu \equiv \frac{\partial}{\partial x^\mu} = \left[\frac{\partial}{t}, \nabla \right] \]

The gradient of \(\alpha\) is then,

\[ \partial_\mu \alpha \equiv \frac{\partial\alpha}{\partial\bar x^\mu} = \left(\frac{\partial x^\nu}{\partial \bar x^\mu}\right) \frac{\partial \phi}{\partial x^\nu} \]