Dubbelosix Posted June 22, 2021 Report Share Posted June 22, 2021 (edited) As some of you may remember, I did investigate the Dirac spinors in the style of torsion, but I realised, I hadn't actually spoke about the Dirac equation itself and how gravity and then torsion would enter this brilliant and beautiful equation. The four equations which summed up the previous investigation was ∇x Γ = - 1/2c² ⋅ ∂Ω/∂t Ω = 1/2mc² ⋅(dU/dt) B x v = 1/2emc² ⋅ 1/r ⋅ dU(r)/dt ⋅ Jv e(B x v) = 1/2mc² ⋅ 1/r ⋅ dU(r)/dr ⋅ Jv The first equation is my take on how the gravitational field is related to the torsion, remember... torsion should be taken as important, since it encompasses the full Poincare group of spacetime symmetries, which means its a phenom that should occur in nature, and as my arguments have progressed, its something that may be important on the scale of particles, where acceleration and the tightly bound curvilinear trajectories are analogous to large gravitational contributions. Acceleration will be a major feature of our second treatise. The second equation, is how I define torsion explicitly, related to a central potential in which a particle can travel round a fixed axis. The third equation described the coupling of gravity and magnetism through a cross product of velocity, so that gravimagnetism arises from motion in the gravitational field, analogous to how a particle can experience an electromagnetic field as it has motion through it. In fact, an early pet theory of a gravitational mass was explained by me as a possible reason to why certain particles experience an inertial mass as opposed to some all pervading Higgs field, but such an alternative model would certainly be difficult for an academia to swallow, so I dropped the idea a while back. The fourth equation unified the third, with that of an analogous Lorentz gravimagnetic force equation, but as always in my own professing, we must be clear that gravity is not a real force from the first principles of relativity. So, what can we now say about a particle, that is following a curved path, and how do we translate these ideas into gravity carefully? In order for a particles trajectory to be following a tightly curved path in space it must have a maximum of gravity that is translatable from GR, the maximum acceleration it can have will be approximate, or possible equal to (though this does not include the idea of black holes, which is the true upper limit using a gravitational classical upper bound) as, a ~ mc³/ħ acceleration did appear in one of the derivations i made, which was itself modelled from the work of Sciama, whose own paper was also on gravimagnetism, titled the "origin of inertia" and we implemented it into the spin orbit equation while defining a torsion directly from acceleration in the rotating frame a = Ω x v Lets not toy too much as I really want to move on to the Dirac equation. There are many types of connections in mathematical physics, in fact, its one of the most complicated aspects of physics, I can't remember now off the top of my head how many there are, but its over a dozen. Whats unique about the connection though, is that they don't just come in different components, they are often written under different dimensions. For instance, in our work above, we defined the gravitational field as having units of an inverse length ∇x Γ = - 1/2c² ⋅ ∂Ω/∂t These dimensions are most commonly found in literature, mostly because two connections define the Ricci curvature R, with dimensions of inverse length squared. Its also possible however, to define it under the j its of acceleration, in fact, because gravity and acceleration are phenomena that are coupled and for that matter, unified as the same thing under relativity, you might argue it is more accurate to measure it in these dimensions. If we were to create a gravitational field with these dimensions the next task would be to define how that itself is related to torsion. It can't be done in any ad hoc way, we need an argument to explain why the torsion would be related to it, just like how we related the gravitational field to torsion by a curve round some central potential. Keeping the torsion Ω as an artifact of motion, then we know from basic physics that (a change in velocity) divided by (a change in time) is the acceleration, so by redefining the connection as acceleration, we can now accutely redefine the torsion with dimensions of (a change in) velocity, so in a theoretical sense, the argument holds merit by such a remodelling. Such an equation would look like Γ = ΔΩ/Δt In this case, the gravitational field is taken in the rotating frame of reference, so that it can be related torsion. Now we want to write the Dirac equation in terms of a gravitational field. In can remember roughly how to do this from earlier essays a few years back when investigating how spin 1/2 particles would be written when interacting with some external gravitating mass, it is in the most simplest form [pc ⋅ α + (mc² + mΓ ℓ ) β] = 0 where α and β are the usual Dirac matrices, following the unique properties of the Clifford algebra, where ℓ is some affine length. We recall now that the gravitational field, is the ratio of the torsion with time, where again the numerator is measured as a velocity and the denominator with time. Since it is an acceleration, the maximal acceleration of the torsion field must be ΔΩ/Δt ~ mc³/ħ Edited June 23, 2021 by Dubbelosix Quote Link to comment Share on other sites More sharing options...
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