I have been reading a lot on geometric algebra. I came to ask whether we had a formula for gravity under this algebra? - it turns out that an electromagnetic geometric algebra does exist but I could not find one for gravity on the net.
I should demonstrate a way for this to work, the geometric form of gravity arises like
$$\nabla_{\mu} \mathbf{D}_{\nu} = \partial_{\mu} \cdot \mathbf{D}_{\nu} + i \sigma \cdot (\Gamma_{\mu} \times \mathbf{D}_{\nu})$$
The interesting thing I noticed from this was the fact that the last term refers to a torsion field. In similar fashion, I had an idea to represent the total angular momentum using geometric algebra and this also gave an energy term that was also related to a torsion:
The expanded form of the unit pseudo-vector such that the geometric interpretation behind angular moment appears like
\begin{align} \nabla_{\mu}\mathbf{J}_{\nu} &= \partial_{\mu} \mathbf{S}_{\nu}\gamma_0 - \vec{v} \cdot (\Gamma_{\mu} \times \mathbf{L}_{\nu})\gamma_1\gamma_2\gamma_3 \\ &= \partial_{\mu} \mathbf{S}^k_{\nu}\gamma_k\gamma_0 - \vec{v} \cdot (\Gamma_{\mu} \times \mathbf{L}^k_{\nu})\gamma_k\gamma_1\gamma_2\gamma_3 \end{align}
The idea was taken further to show it could be written like:
\begin{align} \nabla \gamma_0 \mathbf{D} &= (\nabla^k\gamma_k \gamma_0 - \mathbf{D}^j\gamma_j \gamma_1 \gamma_2 \gamma_3)\gamma_0(\nabla^k \gamma_k \gamma_0 -\mathbf{D}^j \gamma_j \gamma_1\gamma_2 \gamma_3)\\ &= \nabla^k \gamma_k \gamma_0\gamma_0 \nabla^j \gamma_j \gamma_0 - \nabla^k \gamma_k \gamma_1 \gamma_0 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3 \\ &\qquad- \mathbf{D}^k\gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \nabla^j\gamma_j \gamma_0 + \mathbf{D}^k \gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3 \end{align}
and there is a lot to take into account here when solving this.
