A gravitomagnetic Spinor Solution to the Dirac Equation

The derivation to get to the following equation is very difficult, but we start with the Spin \(\omega\) And the torsion \(/omega\) as the starting point

\(\omega = - \frac{\Omega}{2} = \frac{\pi^6}{6} \frac{G L B^3e^4m^2}{c^2}\)

Where B is the constant from Bohrs model, I believe it's equivalent to the Boltzmann constant.

L is the angular momentum

e is the charge and m mass with c the speed of light. The result was obtained from a dimensional argument and some simple algebra from Bohrs model where we have

\( \frac{mv^2}{R} = B \frac{e^2}{R^2}\)

I took an equation that I derived two years back when trying to unify electromagnetism with gravity, the resulting equation was

\( \frac{F_N}{F_G} = \frac{1}{G \epsilon \mu} \cdot \frac{nh}{m^2c} \)

I plugged in the Bohr mass before squaring it

\( \frac{1}{m} = B \frac{e^2}{R} \cdot (\frac{\lambda }{h})^2 \)

Which gave

\( \frac{F_N}{F_G} = \frac{1}{G \epsilon \mu} \cdot \frac{nh}{c} \cdot (\frac{4 \pi^2 B e^2 R}{h})^2\)

By noticing that the orbital radius cubed from the Bohr model Is

\( \frac{1}{R^3} = \frac{12 \pi^6 B^3 e^6 m^3}{h^6}\)

And by using the following dimensional analysis by cancelling unwanted terms,

\(hc = Gm^2 =e^2\)

You can work out the same interesting stuff as my line of enquiry led me, so I simplified the search. I used the Bohr radius formula and plugged it into

\(\omega = - \frac{\Omega}{2} = \frac{G L }{2c^2R^3}\)

And it gave

\(\omega = - \frac{\Omega}{2} = \frac{\pi^6}{6} \frac{G L B^3e^4m^2}{c^2}\)

Now the unification of all important terms from spin to gravity and torsion, to electromagmetism can be plugged into the traceless Covariant Derivative spinor formula for the Dirac equation

\(D = \partial - \frac{i}{4} \omega \sigma\)

We'll put in the necessary subscripts at a later date, I am just writing thus up as a preliminary result after stumbling across it last night. Plugging the innovative solutions we get through substitution of the verbein spin connection

The derivation to get to the following equation is very difficult, but the result is

\(\omega = - \frac{\Omega}{2} = \frac{\pi^6}{6} \frac{G L B^3e^4m^2}{c^2}\)

Where B is the constant from Bohrs model, I believe it's equivalent to the Boltzmann constant.

L is the angular momentum

e is the charge and m mass with c the speed of light. The result was obtained from a dimensional argument and some simple algebra from Bohrs model where we have

\( \frac{mv^2}{R} = B \frac{e^2}{R^2}\)

I took an equation that I derived two years back when trying to unify electromagnetism with gravity, the resulting equation was

\( \frac{F_N}{F_G} = \frac{1}{G \epsilon \mu} \cdot \frac{nh}{m^2c} \)

I plugged in the Bohr mass before squaring it

\( \frac{1}{m} = B \frac{e^2}{R} \cdot (\frac{\lambda }{h})^2 \)

Which gave

\( \frac{F_N}{F_G} = \frac{1}{G \epsilon \mu} \cdot \frac{nh}{c} \cdot (\frac{4 \pi^2 B e^2 R}{h})^2\)

By noticing that the orbital radius cubed from the Bohr model Is

\( \frac{1}{R^3} = \frac{12 \pi^6 B^3 e^6 m^3}{h^6}\)

And by using the following dimensional analysis by cancelling unwanted terms,

\(hc = Gm^2 =e^2\)

You can work out the same interesting stuff as my line of enquiry led me, so I simplified the search. I used the Bohr radius formula and plugged it into

\(\omega = - \frac{\Omega}{2} = \frac{G L }{2c^2R^3}\)

And it gave

\(\omega = - \frac{\Omega}{2} = \frac{\pi^6}{6} \frac{G L B^3e^4m^2}{c^2}\)

Now the unification of all important terms from spin to gravity and torsion, to electromagmetism can be plugged into the traceless Covariant Derivative spinor formula for the Dirac equation

\(D = \partial - \frac{i}{4} \omega \sigma\)

We'll put in the necessary subscripts at a later date, I am just writing thus up as a preliminary result after stumbling across it last night. Plugging the innovative solutions we get through substitution of the verbein spin connection

\(D = \partial - i \frac{\pi^6}{24} \frac{G L B^3e^4m^2}{c^2} \sigma\)

This is a natural thing to do as it unifies at last the spin with gravity and magnetism, coupling it with torsion making it part of the full Poincare group.

 

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There may be some corrections need to do later. It's a bit of a rush, never new there's a thirty minutes expired limit to edit.

 

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\( \frac{1}{m} = B \frac{e^2}{R} \cdot (\frac{\lambda }{h})^2 \)

This needs to be squared for clarity, so we have

\( \frac{1}{m^2} = B^2\frac{e^4}{R^2} \cdot (\frac{\lambda }{h})^4 \)

We will later plug this into

\( \frac{F_N}{F_G} = \frac{1}{G \epsilon \mu} \cdot \frac{nh}{m^2c} \)

And do this right.



So if it has affected my result I'll fix it given some time. Just some advice, a post should be given longer to be fixed because work like is often off the top of the head. I don't have my papers at hand so ill do it later and be more careful.

 

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Reiku. Again.

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Yes. I thought the name said it all. By the way, you're such a gentleman at the other site, you know I apply physics and correct scientists all the time there, I have nothing to hide.

 

Besides hasn't my banning expired yet?

 

So far as I know it is permanent.

But we'll soon find out.

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Ok. I know your smug about it... There's no reason unless your jealous about it. You're a moderator at the other science site and will accept my scientific authority to the point you don't even try to correct me. So why you're so hell bent about your power, I have no idea. You'd think by now there would babe aim reasonable respect between us, but clearly not.

 

By hope of springing eternal and not fishing for sympathy but I have a prognosis of a tumour located at the top of my vertebrae. I just wanted one last tumble to log in because it might be my last year. I just wanted to be part of the community while I still have a chance.

 

Anyway, I've fixed it all, you'll see why it has been rewritten as it is, it's much more clearer, except for the fact it's and paste - this will be one if a few threads that discuss the spinor, gravity and torsion.

F_N/F_0 = 1/(Gεµ) nh/(m^2c)

By understanding that we have

F = Nh/(εµ M_2c) (M/R)^2

by empirical dimensional analysis where relativity as the unifying idea is the same as saying it applies in a general sense the upper limit of the Gravitational force.

(m/R)^2 ≡ c^4/G

We remind ourselves that the rotational velocity inverse units of time is also a fundamental relation when seen to preserve torsion, it highlights the importance of Newton's G when prepping a theory that obeys the full Poincare group

ω = - Ω/2 =GL/2c^2R^3

One thing you can naturally do is plug I'm this torsion equation straight into the covariant derivative where the spin-coupling occurs,

Dψ(X,t) = (∂ - i/(4π)ω σ(a,b)) ψ(X,t)

The motivation for this is by inviting a correction term that is suitable for a more realistic model for calculations that can predict small corrections that could answer several problems in quantum theory. The modified derivative is

D ψ(X,t) = (∂ + i/(4π)Ω/2 σ(a,b)) ψ(X,t)

= (∂ - i/(4π)GL/2c^2R^3σ(a,b)) ψ(X,t)

In our next session I will take us through the force equations that featured from the beginning

F_N/F_0 = 1/(Gεµ) nh/(m^2c)

F_N = nh/(εµ M_2c) (M/R)^2

If we inspect the dimensions of these equation, where the first says the gravitational force is caused by all the interesting variables on the left. The second is required to derive the first. By inspecting the dimensions we find an set of interesting solutions under Bohrs incredible theory of the atom

However, what I didn't do before was log in an extra factor if c in the denominator sinceω requires one such factor to have dimensions of inverse length, so it has to be modified properly

Dψ(X,t) = (∂ - i/(4πc)ω σ(a,b)) ψ(X,t)

D ψ(X,t) = (∂ + i/(4π)Ω/2c σ(a,b)) ψ(X,t)

= (∂ - i/(4π)GL/2c^3R^3σ(a,b)) ψ(X,t)

Now the dimensions are spot on for further work.

Bohr obtained two major objects of importance, the Bohr radius and the Bohr inverse mass. He derived the inverse mass from the known classical laws

1/m = mv^2/m^2v^2 ≡(4π ^2Be^2)/h

and his radius formula which when cubed is

1/R^3 = (12π^6B^3e^6 m^3)/h^6

these are standard equations from his model which is still considered accurate for a nuclear charge equal to 1, but we will be inviting wave functions soon. First we identify the mass in my following formula

F_N/F_0 = 1/(Gεµ) nh/(m^2c)

In which we have highlighted because of not only being a dimensionless (and therefore real) observable just so happens to have the mass squared term in the denominator.pluggimg in Bohrs inverse mass term after squaring it yields

after we simplify by staying

hc = e^2

So we cancel these terms out

F_N/F_0 = nh/(Gεµ) (16π ^4B^2e^2)

and rearrange

F_N/F_0 = (16π^4 B^2e^2)/(Gεµ)

And we'll continue tomorrow as it's really late here now.

 

Oh what fresh hell is this?

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You have given us little reason to believe you.

 

Got it

Went through all the equations and all spot on

Only one tiny aspect to complete

What name are you planning to call this new flavour potato chip?

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Yeah, well spotted. It's been awhile since he has slunk back here. Same old Reiku, though...

 

Sorry to hear of your medical issues and I hope everything works out for you. Regardless, nobody wants to read through your meaningless, made up drivel.

Good Lord, please don't!

 

This is another of your delusions. I'm not a moderator at that site and I stopped posting there about 9 months ago (apart from once, I think).

 

I've called it, the Dirac Poincare Spinor equation, another one I considered for a later equation, for later work has abate if the Schrodinger Wigner Localisation. The latter name will become clear when I get to writing it up. The reason why I chose the first is because of the melding of the the spinor correction terns for the torsion making it part of the full Poincare group.

 

Thank you, but I really wasn't for sympathy, I was just explaining my motivations fir returning.

 

The following model I will construct will be used to describe the previous model we invented to create a semi classical wave equation, without directly inviting any monopole of the gravitational field (graviton) just as there is no monopole of the magnetic force, so it's interesting to note that gravitomagnetic is a theory about linear gravity couplings at weak and larger perturbation pseudoforce unification with spin. As I said on multiple occasions, gravity isn't a real force but it can be melded into the language of wave mechanics and can follow quantum mechanical rules. While the wave spreads out in space, it does not not necessarily follow from the previous model that we can talk about curvilinear geometric properties.

When I spoke about the relativitistic correction to transition the Dirac spinor verbein connection, there are already noteworthy corrections to speak about, one as a good example is how Einstein added a correction term to spacetime Euclidean geometry,

∇ = (∂ + Γ)

It is also important to know how ot arises in general relativity for the Ricci curvature. It appears like

R = ∂ Γ + ΓΓ

because it has those essential space derivatives associated to the gradient with how geometry, more specifically curvature spreads through space with dimensions of inverse length squared . But this identity of the curvature, lacks sone essential ingredients which are important. While doing my essays for a boivector theory for gravity again, to get the full relationship, you can simply expand

(∂ + Γ)(∂ + Γ)

with appropriate indices, and from it you find the parallel transport from ordinary concepts of curvature and the geodesics that matter couples to. From it you find the antisymmetric part involving torsion. Again. In bivector theory, this part arises even more naturally than what you might expect I'm GR. I think ipersonally, bivector theory is more intuitive in this matter as it avoids unnecessary debates as to whether torsion should vanish as a symmetry of ordinary general relativity. When I imposed the gamma Pauli spin to it, we see it preserves with it the generally accepted laws of Poincare spacetime symmetries. In other words, anything preserving the Poincare symmetries, we should in principle expect it to be a real facet of nature. Expanding it we get a set of commutators which describes an uncertainty relationship between space and tend (this relationship is even supported by experiments)

<∇(1),∇(2)>

= (∂ (1)+ Γ)(2)(∂ (2)+ Γ(1))

=-[∂ (1),Γ(2)]

+[∂ (2),Γ(1)]

+-[Γ(1),Γ(2)]

And cutting a longer set of calculations down, the result is three terms, the first two representing curvature and the final commutator being the non vanishing torsion part. Since commutators belong in the phase space, the torsion may very well be important for quantum interactions. Commutators are not special alone in quantum theory, tjey also appear on classical theory, it's only that commutators in the quantum theory appear to be rife in the Hilbert space. Now, I was a bit ambitious hoping to get all this done tonight but we'll see. In bivector theory, when I constructed a gravitational analogue I found that the torsion is naturally nonvanishing as it is fundamental to the bivector mathematical basis of the theory.

 

Now, we should investigate this sin space under the bivector model I invented for analogy to the one which already exists for the Em field. The bivector theorems I published were as essays sent to the gravity research foundation.

∇D =∂⋅D + iσ⋅(∇∧D)

Where ∧is the wedge product. Thisrsms it can be rewritten as

∇D =∂⋅D + iσ⋅(Γ × D)

Where this time × is the cross product and without any silly agents about whether torsion vanishes in general relatobity, we understand it is a crucial non vanishing component when gravity I'd looked at through the spectacles of bivector. This is because the last term on the previous equations defined a crucial relationship with torsion as

-(Γ × D) = ∂Ω/∂t

And the energy required to make an object spin I'm curved torsion background requires

K = L × Ω

which is the angular momentum cross product with torsion.