Locally Orthonormal Frame

At each point we will also store a matrix \(J\) such that \(J^{T}J=g\) that will map to a locally orthonormal basis. The hodge star operator will be implemented in that basis as

\[\begin{split}\left(\begin{array}{cc} 0 & -1\\ 1 & \phantom{+}0 \end{array}\right)\end{split}\]

Coordinate bases for vectors and forms

\[\frac{\partial}{\partial x^{i}}\qquad dx^{i}\]

Non-coordinate orthonormal bases for vectors and forms

\[\hat{e}_{i}\qquad\hat{\theta}^{i}\]

Transformations to orthornormal basis

\[\hat{e}_{i}=J_{i}^{\phantom{j}j}\frac{\partial}{\partial x^{j}}\quad\hat{\theta}^{i}=J_{\phantom{i}j}^{i}\,dx^{j}\]

where \(J_{i}^{\phantom{j}j}\) and \(J_{\phantom{i}j}^{i}\) are transposed inverses of each other

\[J_{i}^{\phantom{j}k}J_{\phantom{j}k}^{j}=\delta_{i}^{j}\]

The above property is required because the pairing of one forms with vector fields remains invariant

\[\hat{e}_{i}\hat{\theta}^{i}=\frac{\partial}{\partial x^{i}}dx^{i}\]

Relationship to the metric

\[\hat{\theta}^{i}\delta_{ij}\hat{\theta}^{j}=dx^{i}\,g_{ij}\,dx^{j}\]
\[g_{ij}=J_{\phantom{m}i}^{m}\,\delta_{mn}\,J_{\phantom{n}j}^{n}\]

Transforming one-forms

\[\alpha=\alpha_{i}\,dx^{i}=\hat{\alpha}_{i}\,\hat{\theta}^{i}\]
\[\alpha_{i}=\hat{\alpha}_{j}J_{\phantom{i}i}^{j}\]

Transforming two-forms

\[\beta=\beta_{ij}\,dx^{i}\wedge dx^{j}=\hat{\beta}_{ij}\,\hat{\theta}^{i}\wedge\hat{\theta}^{j}\]