I'm trying to calculate the connection coefficients of a set of coordinates with Geometric Algebra.

However, following Curvature Calculations with Spacetime Algebra and Spacetime Geometry with Geometric Calculus, by Hestenes, I am stuck when it comes to applying the equation for it

\begin{equation} \omega (g_\mu)= \frac{1}{2}(\gamma_\alpha \wedge D \wedge \gamma^\alpha ) \cdot g_\mu - h^\nu_\mu D \wedge \gamma_\nu \end{equation}

The coordinates I am trying to solve are \begin{align} g_t &= g_1 \gamma_0 + g_2 \gamma_1\\ g_r &= f_2 \gamma_0 + f_1 \gamma_2\\ g_\theta &= r \gamma_\theta\\ g_\phi &= r \sin\theta \gamma_\phi \end{align}

with $\gamma_\mu \cdot \gamma_\nu = \eta_{\mu\nu}$ and $f_1g_1 - f_2g_2=1$.

Can someone explain explicitly how to do these calculations? Thanks!

  • 1
    $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of links, so it is possible to reconstruct links in case of link rot. $\endgroup$
    – Qmechanic
    Jun 1 at 11:48

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