# Is there a Geometric Algebra for gravity?

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.

## 2 Answers

See Gravity, Gauge Theories and Geometric Algebra by Anthony Lasenby, Chris Doran, Stephen Gull.

Also, Geometric Calculus in Gravity Theory by David Hestenes.

If I recall correctly, Lasenby, Doran and Gull version of Geometric Algebra was different then Hestenes's version - but I can't remember how they differed.

• I think I have read all of Hestene's material, but I will certainly give the first link a good look through. thanks! – Gareth Meredith Mar 1 '19 at 11:03

It doesn’t really add anything new that can’t be done with tensors and forms. More people would take up Hestenes and Dorans advocacy of this kind of mathematical technology hadvthey discovered something significant with it; as it is, they’ve mostly rephrased many old results in a new language.

• This could very well be true, but I am optimistic. – Gareth Meredith Mar 2 '19 at 8:55
• I thought you might be interested, but I have been able to find ''new'' things from it, which I cannot find in literature - not only this, but I had approached this idea independently before reading any related literature for gravity especially, so I constructed a theory which was similar to the one provided to me as a reference in this post, but was qualitatively different in other respects. It uncovers torsion and retrieves Einstein's equations nicely - here is the work I have written - consciousness1.quora.com/Bivector-Gravity-Torsion-part-I – Gareth Meredith Mar 7 '19 at 19:44
• @Mozibur This seems like a rude criticism to make without backing it up at all. What about their fundamental theorem, which strictly generalizes Stokes' theorem of differential forms? What about their extension of the theory of analytic functions and the Cauchy integral theorem to arbitrary dimension? Not new enough for you? What about the greatly simplified methods for solving classical central motion problems? Too pragmatic? ...... If you don't think it's useful that's fine, but that's no reason to discourage others from learning, especially based on an opinion that presents no facts. – Joe Schindler Mar 19 '19 at 22:22