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# Poincare Symmetry And Chirality Coefficient Torsion

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R_μv = [∇_μ,∇_v] = (∂^k_μv γ_kγ_1γ_2γ_3 + iσ ⋅ Γ^k_μv γ_k γ_o)^2

Using

(γ_1γ_2γ_3)^2 = - 1

Distributing (γ_1γ_2γ_3)

We get for one connection

∇ = ∂^k γ_k (γ_1γ_2γ_3)^2 + iσ ⋅ Γ^k γ_k γ_o γ_1γ_2γ_3

iγ_o γ_1γ_2γ_3 = γ^5

Which yields

∇ = - ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5

Edited by Dubbelosix
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unlabeled variables (particularly when they are missing unit space) are the easiest ones to do the wrong operations on. Just saying.

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unlabeled variables (particularly when they are missing unit space) are the easiest ones to do the wrong operations on. Just saying.

You wish me to label variables? It's spacetime algebra written with standard relativity connections.

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Spin without gravity? A connection which is flat but with a non-vanishing torsion? Torsion is all but connected and so it is not hard to understand why some scientists have explored versions which retain both terms

'' In the theory of teleparallelism, one encounters a connection, the Weitzenböck connection, which is flat (vanishing Riemann curvature) but has a non-vanishing torsion. The flatness is exactly what allows parallel frame fields to be constructed.''

Wiki

Edited by Dubbelosix
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An interesting diction from a stack exchange contributor;

'' Notice that torsion is usually only defined for curves at points where curvature does not vanish, so to make sense of this some assumption in that direction is needed.'' – Mariano Suárez-Álvarez

Torsion of a curve measures the planarity of the curve. That is, the curve is planar (i.e. it lies on a plane) if and only if its torsion is identically equal to zero. From wiki

'' A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane.

The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and non-zero is a helix. The torsion is positive for a right-handed helix and is negative for a left-handed one.''

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So for torsion to survive naysayers, it seems a 'helix spacetime' model should at least be on the table, especially when torsion cannot be easily done away under spacetime albebra involving all space translations (ie. Poincare symmetries) which should hold in nature.

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And if this is the way forward, a helix spacetime model also exploits helixity of electrons along the presence of a curved path. In the wiki article it says that a helix is a curvature in 3d space, which is in fact owed to the existence of a fourth dimension - time is after all manifested an observable in three-dimensional Space under GR.

Edited by Dubbelosix
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It seems after some reading, helixity of spacetime is a relatively hot topic.

https://www.researchgate.net/publication/226728082_Quantization_of_helicity_on_a_compact_spacetime

And many more papers can be found. The goal of my work is to show not only is spin important, torsion is a natural consequence. .. Otherwise we live in a strange universe where the twisting of spacetime is unrelated to what we call curvature.

Edited by Dubbelosix
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It seems after some reading, helixity of spacetime is a relatively hot topic.

https://www.researchgate.net/publication/226728082_Quantization_of_helicity_on_a_compact_spacetime

And many more papers can be found. The goal of my work is to show not only is spin important, torsion is a natural consequence. .. Otherwise we live in a strange universe where the twisting of spacetime is unrelated to what we call curvature.

I agree with this Dubbel that Spin causes Torsion, the twisting of space-time is naturally caused by spinning massive objects and the curvature being bent in the direction of spin is what does Torsion.

Edited by VictorMedvil
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If the math tools are right, if this is one connection

∇ = - ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5

Then the curvature tensor is

[∇, ∇] = - (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5) (- ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5)

= (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5) (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5)

This can be expanded but certain algebra rules need to be accordingly done for instance, the square of the Pauli spin vector spits back the identity operator which is attached to the torsion.

Edited by Dubbelosix
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In the formal creation of the toy model of the gravitational connections of the pseudo field showed the Heaviside version of torsion arising from the antisymmetric part of the bivector model.

∂D = ∂ ⋅ D - iσ ⋅ (Γ x D)

= ∂ ⋅ D + iσ ⋅ (∂Ω/∂t)

-Γ x D = ∂Ω/∂t

While the torsion has units of inverse time, it occurred to me a while back that the timelike gamma may be connected to it - it is just a bonus that it is attached to the same definition for torsion also under general relativity.

Introducing correct subscripts and gamma matrices it would look like

∂_μD_v = ∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (Γ_μ x D^k_v)γ_kγ^o

= ∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (∂Ω_μv/∂t) γ_kγ^o

Hestenes under his brilliant work shows that the spacetime split can be simplified

a_k = γ_kγ^o

Though he is probably unaware of my own work. The order is also important;

γ_kγ^o = - γ^oγ_k

Moving on we could expand

∂_μγ^oD_v

= (∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (Γ_μ x D^k_v)γ_kγ^o) γ^o (∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (Γ_μ x D^k_v)γ_kγ^o)

= (∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (∂Ω_μv/∂D) γ_kγ^o) γ^o (∂_μ ⋅ D^k_v γ_kγ^1γ_2γ^3 + iσ ⋅ (∂Ω_μv/∂t) γ_kγ^o)

When expanded we inexorably get the two timelike gamma matrices which produces the 4x4 identity operator which can be understood as a commutator:

[γ^k, γ^o] = γ^kγ^o + γ^kγ^o = 2η_μv I(4)

Based on what I knew from here, I considered the following ~ starting with a curvature tensor

R_μv = [∇_μ,∇_v] = (∂^k_μv γ_kγ_1γ_2γ_3 + iσ ⋅ Γ^k_μv γ_k γ_o)^2

Using

(γ_1γ_2γ_3)^2 = - 1

Distributing (γ_1γ_2γ_3)

We get for one connection

∇ = ∂^k γ_k (γ_1γ_2γ_3)^2 + iσ ⋅ Γ^k γ_k γ_o γ_1γ_2γ_3

iγ_o γ_1γ_2γ_3 = γ^5

Which yields

∇ = - ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5

The new covariant deravative I suggest is

∇ = - ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5

Then the curvature tensor is

[∇, ∇] = - (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5) (- ∂^k γ_k + σ ⋅ Γ^k γ_k γ^5)

= (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5) (∂^k γ_k + σ ⋅ Γ^k γ_k γ^5)

This can be expanded but certain algebra rules need to be accordingly done for instance, the square of the Pauli spin vector spits back the identity operator which is attached to the torsion. Expanding we get:

= ∂^k_μ γ_k∂^k_v γ_k

+ ∂^k_μ γ_k (σ ⋅ Γ^k) γ_k γ^5

+ (σ ⋅ Γ^k_μ) γ_k γ^5 ∂^k_v γ_k

+ i^2 σ^2 ⋅ [Γ^k_μ γ_k , Γ^k_v γ_k [γ^5]^2]

We remind ourselves of the identity matrices for the chirality and Pauli vector. The identity in such a case can be thus removed from the equation but we will retain the gamma five matrices since they submit to commutation ~

[γ^5, γ^k] = γ^5γ^k + γ^kγ^5 = 0

On top of that, we have a space and time commutator as well which may satisfy;

[Γ(x), Γ(t) ]

Which may have a non-zero solution.

Edited by Dubbelosix
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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

$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$

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

$E = \epsilon_0 \int\ \mathbf{E} \cdot \mathbf{E}\ dV = \epsilon_0 \int\ (\nabla \phi + \frac{\partial \mathbf{A}}{\partial t})(\nabla \phi + \frac{\partial \mathbf{A}}{\partial t} )\ dV$

Expanding the right hand side we get

$\nabla \phi \nabla \phi + \nabla \phi \frac{\partial \mathbf{A}}{\partial t} + \frac{\partial \mathbf{A}}{\partial t} \nabla \phi + \frac{\partial \mathbf{A}}{\partial t} \frac{\partial \mathbf{A}}{\partial t}$

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:

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

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.wikiversity.org/wiki/Gravitational_phase_shift

I also recall for the reader the following article

https://en.wikiversity.org/wiki/Physics/Essays/Fedosin/Gravitoelectromagnetism

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

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

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

$\nabla \phi \nabla \phi$

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

$\nabla \gamma_0 \mathbf{D} = \nabla^k \gamma_k \gamma_0\gamma_0 \nabla^j \gamma_j \gamma_0$

$- \nabla^k \gamma_k \gamma_1 \gamma_0 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3$

$- \mathbf{D}^k\gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \nabla^j\gamma_j \gamma_0$

$+ \mathbf{D}^k \gamma_k \gamma_1 \gamma_2 \gamma_3 \gamma_0 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3$

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

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,

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

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

$\Omega = - \frac{\omega}{2}$

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

$-\Gamma \times D = \frac{\partial \Omega}{\partial t}$

and obviously order of cross product matters:

$D \times \Gamma = -\frac{\partial \Omega}{\partial t}$

(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. $\gamma \wedge \nabla = \frac{1}{c}\frac{\partial}{\partial t}$ - in terms of the equation above we have shown in this post, then a number of things cam be simplified, as

$\nabla \gamma_0 \mathbf{D} = \nabla^k a_k \nabla^j \gamma_j$

$- \nabla^k a_k \gamma_1 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3$

$- \mathbf{D}^k a_k \gamma_1 \gamma_2 \gamma_3 \nabla^j\gamma_j$

$+ \mathbf{D}^k a_k \gamma_1 \gamma_2 \gamma_3 \mathbf{D}^j \gamma_j \gamma_1 \gamma_2 \gamma_3$

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

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