The link to the wiki notes describing the frame and metric is:
http://deferentialgeometry.org/#frame metric
but I'll cut and paste the main bits here.
Physically, at every manifold point a frame encodes a map from tangent vectors to vectors in a rest frame. It is very useful to employ the Clifford basis vectors as the fundamental geometric basis vector elements of this rest frame. The ''frame'', then, is a map from the tangent bundle to the Clifford bundle -- a map from tangent vectors to Clifford vectors -- and written as
<br />
\underrightarrow{e} = \underrightarrow{e^\alpha} \gamma_\alpha = \underrightarrow{dx^i} \left( e_i \right)^\alpha \gamma_\alpha<br />
It is a Clifford vector valued 1-form. Using the frame, any tangent vector, $\vec{v}$, on the manifold may be mapped to its corresponding Clifford vector,
<br />
\vec{v} \underrightarrow{e} = v^i \vec{\partial_i} \underrightarrow{dx^j} \left( e_j \right)^\alpha \gamma_\alpha = v^i \left( e_i \right)^\alpha \gamma_\alpha = v^\alpha \gamma_\alpha = v<br />
This frame includes the geometric information usually attributed to a metric. Here, we can compute the scalar product of two tangent vectors at a manifold point using the frame and the Clifford dot product:
<br />
\left( \vec{u} \underrightarrow{e} \right) \cdot \left( \vec{v} \underrightarrow{e} \right) <br />
= u^\alpha \gamma_\alpha \cdot v^\beta \gamma_\beta<br />
= u^\alpha v^\beta \eta_{\alpha \beta} <br />
= u^i \left( e_i \right)^\alpha v^j \left( e_j \right)^\beta \eta_{\alpha \beta}<br />
= u^i v^j g_{ij} <br />
with the use of frame coefficients and the Minkowski metric replacing the use of a metric if desired. Using component indices, the ''metric matrix'' is
<br />
g_{ij} = \left( e_i \right)^\alpha \left( e_j \right)^\beta \eta_{\alpha \beta}<br />
Using Clifford valued forms is VERY powerful -- we can use them to describe every field and geometry in physics.
I had never realized that!