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Angular Momentum And Geometric Algebra


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#18 Dubbelosix

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Posted 11 March 2019 - 12:43 PM

Extensive updates.

 

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#19 Dubbelosix

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Posted 11 March 2019 - 08:10 PM

I was searching for dialogue which doesn't rely on displaying math, but is able to describe this situation of torsion in general relativity in the most concise way I could, as it went like the following from the previous article, if you don't understand something, or if you have any questions, I will take time out right now for next few days to reply.

 

Bivector gravity will lead to a better understanding of the treatment of general relativity. Since no priori is given makes the issue of a vanishing torsion mathematically ad hoc, which really leaves us two approaches;

 

1. It can be understood as a corrective term describing a degree of symmetry that must arise (since the rotation space) in the bivector gravity assures it is part of the Poincaire group of spacetime symmetries (as expected when spinning masses are implicated). In this instance, both the symmetric and antisymmetric terms survive.

 

2. That there is a gauge fixing of torsion to zero, so in this approach we end up with standard general relativity without the torsion, whereas fixing the curvature to zero, implies the remaining telaparallel transport.

 

An additional consequence of 1. is from what I found independently, that bivector gravity by definition implies both the symmetric and antisymmetric parts (as a priori) from geometric algebra - which means this could be used to make progress in how we treat general relativity.

 

I use now a quotation from a paper (provided below), ''There is no doubt that quantum mechanics has seized hold of a beautiful element of truth and that it will be a touchstone for a future theoretical basis in that it must be deducible as a limiting case from that basis, just as electrostatics is deducible from the Maxwell equations of the electromagnetic field or as thermodynamics is deducible from statistical mechanics. I do not believe that quantum mechanics will be the starting point in the search for this basis, just as one cannot arrive at the foundations of mechanics from thermodynamics or statistical mechanics.”

 

- Einstein (1936)

 

The definition of a bivector gravity (must imply) a symmetric and antisymmetric geometric product, without any ad hoc statements about what should and should not remain. The torsion is a corrective term, whether for quantum systems or perhaps for systems with significant gravity.

 
arxiv.org

Edited by Dubbelosix, 11 March 2019 - 08:11 PM.


#20 Dubbelosix

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Posted 13 March 2019 - 05:34 AM

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#21 Dubbelosix

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Posted 17 March 2019 - 09:06 PM

New blog post, expanding on creative idea's

 

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#22 Dubbelosix

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Posted 18 March 2019 - 11:10 AM

Compiled about five blog posts into one, this will be the first paper I write on the topic. Any more investigations, will be kept in blog posts until I write a second paper up. This blog post contains all the relevant information I have learned, from other work published, including mostly independent research.

 

https://bivector.quo...ivector-Gravity



#23 Dubbelosix

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Posted 20 March 2019 - 10:22 PM

Let's take a look at a development made concerning a gravitational analogue of the electromagnetic field density in terms of geometric (Clifford) algebra.

 

The Gravitational Field Analogue

 

Let's take another look at a different aspect of electromagnetism which seems to also have a geometric dependence. 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

 

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

 

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]

 

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]

 

(note the d’Alembertian has been absorbed by the charge [math] \hbar c[/math] 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

 

There would be a few minor differences, since permeability or permittivity have to be replaced by their gravitational counterparts and the charge density would no longer be defined as

 

[math]\frac{Jc}{V}[/math]

 

but instead as

 

[math]\frac{Gm^2}{V}[/math]

 

We wouldn’t need to do too much work to find its analogue for the field densities, since this has already been provided to us through the work of Heaviside who constructed the most well-known versions of gravielectromagnetism - the gravitational analogue of the electromagnetic flux density looks like

 

[math]\mathbf{H} = - \frac{c^2}{4 \pi G}(\Gamma \times \Omega)[/math]

 

We need to be careful if we want to update these idea’s into a geometric interpretation since the torsion field [math]\Omega[/math] notation is reserved for only the antisymmetric part of the usual Einstein field equations.

 

[math]-\frac{c^2}{4 \pi G}[/math]

 

is the gravitational permeability [math]\frac{1}{\mu_G}[/math]. The Heaviside vector allows us to write the energy density flux as

 

[math]\mathbf{U} = \frac{1}{4 \pi G}(\Gamma^2 \times c^2\Omega^2)[/math]

 

Reference to my work:

 

https://bivector.quo...ivector-Gravity


Edited by Dubbelosix, 25 March 2019 - 07:32 AM.


#24 VictorMedvil

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Posted 21 March 2019 - 12:13 AM

You are trying to model Gravity like electromagnetism, there are many anologs between Electromagnetism and Gravity but Gravity is a combination of all the forces, I realize you are just making a theory of Electromagnetism and Gravity but maybe you should look for inspiration from all 4 fours combined rather than work on it in parts such as you are doing not to say this is the wrong approach but gravity is Energy-Stress from Mass and the other 3 forces maybe you should treat it as such, try a equation in this form. 

 

(Energy Stress) = Mass + SNF + EM + WNF

 

or

 

Angular Momentum  with the same.

 

It may give you a more exact solution to Tuv , Guv  , and Rij

 

I guess what I am saying is stop treating gravity just like electromagnetism you will find it isn't just like it. Have you even consider the effect the Strong and Weak nuclear force have on gravity and Angular Momentum, dubbel?

 

You work alot of unifying Electromagnetism and Gravity but little to the other two forces, is what I am saying.


Edited by VictorMedvil, 21 March 2019 - 12:25 AM.


#25 Dubbelosix

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Posted 21 March 2019 - 12:28 AM

The analogue of gravity to electromagnetism, is called gravitoelectromagnetic theory - it is a linearized form of gravity which according to what I have read, works pretty well.

 

But this is not the crux of what I am doing - you are right, matter contributes to the stress energy, and I have been exploring geometric dependence which so far has suggested to me, that perhaps everything is reducible to geometry. This is in same respect to Wheeler who envisioned this a long while ago. I am only now just appreciating how beautiful the idea is, because I don't think the similarities between gravity and electromagnetism are in any way, coincidental. There may appear to be a geometric explanation to all the physical forces we know and that to me, surely cannot be coincidental.



#26 VictorMedvil

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Posted 21 March 2019 - 12:30 AM

The analogue of gravity to electromagnetism, is called gravitoelectromagnetic theory - it is a linearized form of gravity which according to what I have read, works pretty well.

 

But this is not the crux of what I am doing - you are right, matter contributes to the stress energy, and I have been exploring geometric dependence which so far has suggested to me, that perhaps everything is reducible to geometry. This is in same respect to Wheeler who envisioned this a long while ago. I am only now just appreciating how beautiful the idea is, because I don't think the similarities between gravity and electromagnetism are in any way, coincidental. There may appear to be a geometric explanation to all the physical forces we know and that to me, surely cannot be coincidental.

 

I guess it is your work and your theory, Continue then your works on Electromagnetism and Gravity are good I was just wondering what you would do with the SNF and WNF when you merged them in. This approach until you add them will never yield a Grand Unified Field theory you know. You will always be at the same level as Einstein and the Theory of General Relativity which is just Electromagnetism and Gravity but I do admit that gravity the electromagnetism are quite the same but not exactly. Gravity is much like Electromagnetisms without a negative charge unless you count negative energy theories but when you try to apply them you will always run into problems when you attempt to write them in the same language and be missing parts of gravity in my opinion, gravity is so much expansive than electromagnetism being that EM is just for charged particles and gravity is for anything with energy, but if you treat energy like charge it doesn't quite give the same results as charged particles. In any case, that is my opinion on gravito-electromagnetism theories.

 

For instance, in your theory how would you explain something like a neutrino? See that would be an annoying gravito-electromagnetism particle to explain.


Edited by VictorMedvil, 21 March 2019 - 12:42 AM.


#27 Dubbelosix

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Posted 21 March 2019 - 11:50 AM

Let's be clear, I am not saying gravity is identical to electromagnetism, what I am saying is that geometry seems to pop up in the structure  of all fields - gravity has some fundamental difference to the rest of the forces, for instance, for starters, it is not a real force. Therefore, I think gravity is a description, or a blueprint in how to unify, but is not a fundamental force itself.



#28 Flummoxed

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Posted 21 March 2019 - 12:12 PM

Let's be clear, I am not saying gravity is identical to electromagnetism, what I am saying is that geometry seems to pop up in the structure  of all fields - gravity has some fundamental difference to the rest of the forces, for instance, for starters, it is not a real force. Therefore, I think gravity is a description, or a blueprint in how to unify, but is not a fundamental force itself.

 

Kaluza Klein used 5 dimensions to try and unify Relativity and Electromagnetism, you appear to be only using 4. Why have you not thought about using the Kaluza Klein approach. https://en.wikipedia...a–Klein_theory 



#29 Dubbelosix

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Posted 21 March 2019 - 12:44 PM

Kaluza Klein used 5 dimensions to try and unify Relativity and Electromagnetism, you appear to be only using 4. Why have you not thought about using the Kaluza Klein approach. https://en.wikipedia...a–Klein_theory 

 

Good question - well there are probably a lot of reasons to not think of these types of models. I understand why Kaluza did it - it was because adding new dimensions into a theory allows you to have freedom for parameters that previously could not be taken into account. And I must admit, it seems appealing that five dimensional space can retrieve all the relevant information, but we would need to come to accept some extra luggage - such as compactification, the crucial component in string theory and hyperdimension theory. Kaluza also made his theory appealing in another way, such as it was completely classical - I too strongly believe that gravity is purely classical, but it hasn't stopped scientists from searching for a quantum interpretation of it, including Kaluza's.

 

This is a matter of ... do we need another dimension for gravity? The answer is no, because we understand gravity as a curvature of spacetime perfectly well, without additional dimensions, which could arguably complicate things.



#30 Dubbelosix

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Posted 21 March 2019 - 12:56 PM

In fact, the intuitive notion of geometry arises from four dimensions concretely. The fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle): In gravitationally-warped spacetime the motion through time manifests as motion through space.



#31 Flummoxed

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Posted 22 March 2019 - 03:15 AM

In fact, the intuitive notion of geometry arises from four dimensions concretely. The fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle): In gravitationally-warped spacetime the motion through time manifests as motion through space.

 

Maybe but it does not include entanglement effects. ie Particles separated by space sharing the same wave function responding ftl cant be explained by just the normal 4 dimensions. Most of the latest alternative theories on gravity do include extra dimensions, Kaluza Klein may have been one of the first, but today with the holographic and, related emergent gravity theories etc, I strongly suspect there is more to space than the reflection in 4 dimensions that we perceive :)  



#32 Dubbelosix

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Posted 22 March 2019 - 11:43 AM

Yes entanglement needs to be understood, but I am not sure anyone really does, including me.



#33 Dubbelosix

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Posted 22 March 2019 - 12:42 PM

Let's take a dimensional look at the equations. The gravitational analogue of the electromagnetic field density looks like

 

[math]\mathbf{H} = - \frac{c^2}{4 \pi G}(\Gamma \times \Omega)[/math]

 

where

 

[math]-\frac{c^2}{4 \pi G}[/math]

 

is the gravitational permeability [math]\frac{1}{\mu_G}[/math]. The Heaviside vector allows us to write the energy density

 

[math]\mathbf{U} = \rho c^2 = \frac{c^2}{4 \pi G} [\Gamma \times c^{-1}\ \Omega][/math]

 

where this time [math]\frac{1}{4 \pi G}[/math] is the permittivity. The equation we would obtain is

 

[math]\gamma_0 \mathbf{U}_{\mu \nu} = \frac{1}{\mu_G}\ [(\mathbf{\nabla}_{\mu} \cdot \Gamma_{\nu} + \nabla_{\nu} \cdot \Gamma_{\mu})\gamma_0 + 2i \vec{\sigma} \cdot (\Gamma_{\mu} \times c^{-1}\ \Omega_{\nu})]^k \gamma_k[/math]

 

where \mathbf{U} is the energy density of the gravitational field. To make sense of the dimenions, the gravitational field has units of acceleration, when weighted by G, it gives a mass over radius squared, the remaining derivative gives a density.


Edited by Dubbelosix, 23 March 2019 - 04:13 AM.