Now that I am no longer on the tablet, I can try and make some quick posts, using latex.

**The Gravielectric Field Analogue**

The electric field is defined as

[math]\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}[/math]

from the usual energy density equation for an electromagnetic field, the energy is obtained

[math]E = \epsilon_0 \int\ \mathbf{E} \cdot \mathbf{E}\ dV = \epsilon_0 \int\ (\nabla \phi + \frac{\partial \mathbf{A}}{\partial t})(\nabla \phi + \frac{\partial \mathbf{A}}{\partial t} )\ dV[/math]

Expanding the right hand side we get

[math]\nabla \phi \nabla \phi + \nabla \phi \frac{\partial \mathbf{A}}{\partial t} + \frac{\partial \mathbf{A}}{\partial t} \nabla \phi + \frac{\partial \mathbf{A}}{\partial t} \frac{\partial \mathbf{A}}{\partial t}[/math]

This underlines, a geometric property to the system - to demonstrate this take a look at two other examples, the Berry curvature and the curvature tensor

the so-called, ‘gauge invariant’ Berry curvature is a tensor which describes how phases relates to the curvature of the system:

[math]F_{ij} = \partial_i, A_j - \partial_jA_i + [A_i,A_j][/math]

In fact there is such a thing as a gravitational phase shift which may or may not have something to do behind all this ''geometry'' in the fields of nature;

https://en.wikiversi...nal_phase_shift

I also recall for the reader the following article

https://en.wikiversi...lectromagnetism

It has identical structure with the non-zero torsion formulation of the field equations (including a non-zero curvature)

[math]R_{ij} = -\partial_i, \Gamma_j + \partial_j\Gamma_i\ + [\Gamma_i, \Gamma_j][/math]

There is an extra term in the electric field derivation above, they will follow the usual commutation laws

[math]\nabla \phi \nabla \phi[/math]

An extra term also arose from a geometric interpretation involving the derivatives of space:

[math]\nabla \gamma_0 \mathbf{D} = \nabla^k \gamma_k \gamma_0\gamma_0 \nabla^j \gamma_j \gamma_0[/math]

[math]- \nabla^k \gamma_k \gamma_1 \gamma_0 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3[/math]

[math]- \mathbf{D}^k\gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \nabla^j\gamma_j \gamma_0[/math]

[math]+ \mathbf{D}^k \gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3[/math]

This alternative calculation yields the identity through [math]\gamma_0\gamma_0[/math] and the remaining [math]\gamma_k\gamma_0[/math] can be from Hestenes work simplified to [math]a_k[/math].

The solution to this, is the same as one that is electromagnetic in nature - the electromagnetic solution is an equation we have looked at in this post,

[math]\gamma_0 \rho^2 = ( \epsilon_0 \mathbf{E} \cdot \mathbf{E} + \frac{1}{\mu_0}\mathbf{B} \cdot \mathbf{B})^2\gamma_0 + 2i \vec{\sigma}\ (\epsilon_0 \mathbf{E} \cdot \mathbf{E} \times \frac{1}{\mu_0}\mathbf{B} \cdot \mathbf{B})^k \gamma_k[/math]

which is nice and we will return to this at another date. Keep in mind the structural differences which I argue in the current approach, and not without reason of course. The torsion is related to the frequency as

[math]\Omega = - \frac{\omega}{2}[/math]

and our Heaviside definition of the torsion was a cross product of the gravitational field with a generic derivative of choice:

[math]-\Gamma \times D = \frac{\partial \Omega}{\partial t}[/math]

and obviously order of cross product matters:

[math]D \times \Gamma = -\frac{\partial \Omega}{\partial t}[/math]

(note the d’Alembertian has been absorbed by the charge \hbar c to create energy terms)

So the derivatives of spacetime would replace relative terms in the equation above. The solution was made apparent following the work from

Electromagnetism using Geometric Algebra versus Components

Notice from the archaic looking tex describing the alternative approach (see post 15) we attached this time the timelike gamma matrix to the definition of Heaviside's torsion. What is remarkable is that while we call this a geometric algebra approach, Heaviside's description of torsion is actually a linearized form of gravimagnetism. Heaviside was the first to notice the torsion relationships in geometroelectromagnetism, but it was Einstein who showed how this had a direct relationship with the gravielectric field predicting the motion of Mercury with even greater precision. It seems crual but somewhat strangely fortuitous for the geometric algebra to produce the torsion elegantly within the cross product.

It seemed more logical for the timelike gamma matrix to be attached to the torsion since it has units of frequency. In geometric algebra, all you need to do to settle the dimensions is introduce the nabla operator (ie. [math]\gamma \wedge \nabla = \frac{1}{c}\frac{\partial}{\partial t}[/math] - in terms of the equation above we have shown in this post, then a number of things cam be simplified, as

[math]\nabla \gamma_0 \mathbf{D} = \nabla^k a_k \nabla^j \gamma_j[/math]

[math]- \nabla^k a_k \gamma_1 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3[/math]

[math]- \mathbf{D}^k a_k \gamma_1 \gamma_2 \gamma_3 \nabla^j\gamma_j[/math]

[math]+ \mathbf{D}^k a_k \gamma_1 \gamma_2 \gamma_3 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3[/math]

In the last term, the spatial gamma matrices take on the form of a sign change [math](\gamma_1,\gamma_2,\gamma_3)^2 = -1[/math] on the last term additionally, which would be reserved for the spin space.