Spacetime Algebra - Reciprocal Frame

Reciprocal Frame

Associated with the orthogonal basis is the reciprocal basis for all =0,...,3, satisfying the relation

$\gamma_\mu \cdot \gamma^\nu = {\delta_\mu}^\nu$ .

These reciprocal frame vectors differ only by a sign, with, and for k =1,...,3.

A vector may be represented in either upper or lower index coordinates with summation over =0,...,3, according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.

$\begin{align}a \cdot \gamma^\nu &= a^\nu \\ a \cdot \gamma_\nu &= a_\nu\end{align}$

Read more about this topic: Spacetime Algebra

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