years 2014-2016 - THEORY OF EVERYTHING

posted Mar 17, 2016, 11:46 PM by David Jaroš   [ updated Mar 31, 2016, 11:22 AM ]
I didn't do much in 2014 on my independent research topics, as I was busy days and nights with my work. Atmosphere in the company was still stormy, and many people lost  their jobs. I was fully dedicated to my work of software engineer. I also enjoyed my new wife, as I got married in march. But you can't stop ideas they are coming all the time! Therefore late in 2014 and mainly in summer 2015 I really made a breakthrough. In the autumn 2014, something reminded me that in 2013, just before I left Prague, in a flash of intuition I wrote on paper very simple quaternion equation. As I suspected that equation,  contained all  electromagnetism on one single line, and had barely 3 symbols besides equal and minus signs.  Here is the summary in it's  full beauty:

On the left side of equation is biquaternion eight intensity, which will be described later. On right side is minus biquaternion gradient of biquaternion vector potential.  Definition follows: 



The  biquaternion eight intensity  E8   is generalized version of  electrical intensity, known from Maxwell equations. The subscript "8" I added later,  to emphasize, that this intensity has up to 8  real dimensions (or four complex). As expected this simple equation could be expanded to more complex expression, which shows relation to electromagnetic quantities E,B (these are written as purely imaginary quaternions with real components). Purely imaginary quaternion means, that it contains only vector part, therefore neither E or B has scalar component in the following equation.


Immediately I have also realized, that the mysterious indwelling  sub-expression, that I started call G,  is indeed equal to expression  from  Lorenz gauge condition (LGC), but  it's not equal to 0 here anymore! Instead G it's on the left side of equation (and thus can have any value): 

I knew  already in Prague, that this magical and simple expression represents complete theory of electromagnetic field. This includes both field components  B and E as well as all Maxwell equations.  Some quantities like  charge density or vectors of current densities are a bit obscured in this equation, but they are inherently still there. E.g. if we  calculate quaternion Laplacian (note there are even 2 versions or components  of this  operator which differ in sign at space components), we clearly see current densities Ji. To see this immediately, either knowledge of wave  equation for vector potential is needed (which have been derived based on LGC assumption e.g. G=0, or we can see this also in other way. Let's  also set G=0  i.e. LGC, for which this wave  equation was derived, and then we apply  appropriate  quaternion gradient (also has 2 variants which differ in sign at space components ) also on E8. Because of LGC  precondition,  E8 has now zero scalar component, and is  a purely  imaginary quaternion according to quaternion  terminology. By expression of components of quaternion gradient of E8 we can  immediately see that the real part of the  equation express  rot(B)  vector operator i.e. Ji. The similar is valid for imaginary part of  E8, i.e. electrical  intensity. When G=0 and quaternion gradient is applied on E8, we get  div(E), and on right side free charge density divided by c. (Note: but there is also one more element related to divergence of vector potential - this should be checked, why is there this extra component?).

From the above explanation, it is clear, that even though obscured, the current density and charge density, is there  inherently in the one line equation of electromagnetism.

 Why I call the new element G? Well simply G stands for gradient or gauge. Gradient because it can be generated by 4-dimensional gradient of vector 4-potential. Space  components have much higher influence, in this case. But we can call G also gauge, and in electromagnetism generally this gauge can be any (see e.g. Lorentz gauge, Coulomb gauge or Weyl gauge on wikipedia). But there are infinitely many possibilities how to select gauge. And any gauge can be converted to Lorentz gauge (or LGC or G=0).

This is very similar to selection of inertial frame in classical Newton dynamics or even theory of relativity. Therefore I started to hypothesize, that this similarity is not random,  but G corresponds to acceleration or in other words to GRAVITY!!! I other words, if G=0 (LGC) there is no acceleration or gravity. If G<>0 then there is some acceleration or gravity in the frame of reference.  This connection I anticipated even earlier, but I got closer to that conclusion only on the beginning of year 2016 when I was in  Richmond.

 Here I'm going to show also another indication to this conclusion. Let's start with energy calculation of generalized electromagnetic field.
 I have calculated the energy density & flow W as:

Note that except of 3 standard electromagnetic elements, which are exactly equal to elements from standard electromagnetic theory (magnetic energy, electrical energy and  flowing electromagnetic energy i.e. Poynting vector) there are also other two elements containing G. First element  has scalar and the second one vector character. Please note, that about formula is valid only for real G, and in this case there is no magneto-scalar energy. For complex G, you must use full formula, which can be found in Theory of everything.