1.8.5 The Unit Invariant Mass and the Flowing Point Ideals

Pre­vi­ous­ly in the chap­ter, we con­sid­ered the ide­al­is­tic ex­am­ple of an in­vari­ant mass with zero ve­loc­i­ty in space, where the energy-mo­men­tum re­la­tion­ship of this system is re­duced to $E = m_{0}c^{2}$, by way of the fol­low­ing

In spe­cial rel­a­tiv­i­ty and physics, an in­vari­ant mass is a quan­ti­ty that is the same for all ob­servers in all rest frames. Math­e­mat­i­cal­ly, an in­vari­ant mass re­sults from the point of in­ter­sec­tion be­tween spher­i­cal and hy­per­bol­ic geome­tries lo­cat­ed on the sin­gu­lar­i­ty's bound­ary. In the case of an in­vari­ant mass that is pur­port­ed to be in real space, the quan­ti­ty is static and not moving, im­ply­ing that the mass has no ki­net­ic energy. The tem­per­a­ture of a system is di­rect­ly re­lat­ed to the ki­net­ic energy of the ob­jects com­pos­ing it, and a system with zero ki­net­ic energy is, there­fore, a system at ab­solute zero in tem­per­a­ture.

Ab­solute zero is an ideal state when bound to space­time geom­e­try, given the in­fi­nite amount of work re­quired to cool the system to it. In physics, the Heisen­berg un­cer­tain­ty prin­ci­ple is the rea­son­ing behind the im­pos­si­bil­i­ty of ab­solute zero in space­time. By the prin­ci­ple in a system that ap­proach­es ab­solute zero, we are clos­ing in on a known pre­ci­sion for the mo­men­tum of its quan­tum par­ti­cles, which im­plies we are ap­proach­ing a com­plete lack in our abil­i­ty to know the po­si­tions of these quan­tum par­ti­cles. The Heisen­berg un­cer­tain­ty prin­ci­ple re­sults from the Fouri­er trans­form's math­e­mat­i­cal frame­work, where in­her­ent to its analy­sis, a func­tion and its Fouri­er trans­form cannot be lo­cal­ized in si­mul­tane­ity.

In space­time, zero ve­loc­i­ty and ab­solute zero are ideals. They do not exist, but math­e­mat­i­cal­ly, the ideals exist in Fouri­er trans­forms and the Uni­fied Field dy­nam­ics, where a trans­form can be lo­cal­ized en­tire­ly. At the same time, its func­tion is al­to­geth­er non-lo­cal­ized. A Lie al­ge­bra on the bound­ary of the in­fi­nite sphere of tan­gent spaces is wholly lo­cal­ized, while its matrix ex­po­nen­tial is in­fi­nite and non-lo­cal­ized. A gen­er­a­tor em­bod­ies the mo­men­tum frame­work, and the co­or­di­nate vec­tors it acts upon rep­re­sent the po­si­tions in the Uni­fied Field. The in­vari­ant mass with zero ve­loc­i­ty in space exists math­e­mat­i­cal­ly on the sin­gu­lar­i­ty's bound­ary, the bound­ary of the in­fi­nite sphere of tan­gent spaces.

To begin to con­sid­er the unit in­vari­ant mass, we first recall the hy­per­bol­ic Lorentz factor as it re­lates to energy-mo­men­tum

and solve this for the in­vari­ant mass

The in­ter­sec­tion be­tween spher­i­cal and hy­per­bol­ic geome­tries occurs at the point where the $\cosh(\alpha)$ func­tion has a joint so­lu­tion with the $\cos(\omega)$ func­tion, this being for $\alpha = 0$ and $\omega = 2\pi n$, for $\in \mathbb{Z}$, and we stay mind­ful of the equiv­a­lences for the fol­low­ing iden­ti­ties

In the con­text of the unit in­vari­ant mass, $m_{0}(\epsilon),$the common point of in­ter­sec­tion will be where the origin of the unit hy­per­bo­la meets the com­plex unit circle at the point $1$ on the real-axis, and this point is also the iden­ti­ty, $\epsilon,$for each.

In the two-di­men­sion­al space of $\mathbb{C}$, the al­ge­bra­ic uni­tary light so­lu­tion to $c^{2} = 1$ is the result of the prod­uct of the fol­low­ing light as­ymp­tote al­ge­bras

and is lo­cat­ed at the same iden­ti­ty point, $\epsilon$, on the sin­gu­lar­i­ty's bound­ary. Plug­ging these uni­tary values into our in­vari­ant mass ex­pres­sion, we have

and this result rep­re­sents the co­a­les­cence of the as­ymp­tot­ic light al­ge­bras with that of the unit hy­per­bo­la's geom­e­try at the iden­ti­ty.

The co­a­les­cence point is on the sin­gu­lar­i­ty's bound­ary, and as such, we use the fol­low­ing ex­pres­sion for energy, $E$,

where $\hslash$ rep­re­sents the re­duced Planck's con­stant, and $\omega$ the an­gu­lar fre­quen­cy.

The an­gu­lar fre­quen­cy in the energy ex­pres­sion for $m_{0}(\epsilon)$, is the result of the so­lu­tion for $e^{i\omega} = \epsilon,$oc­cur­ring at the iden­ti­ty when $\omega = 2\pi n$, which we sub­sti­tute into the energy ex­pres­sion

and given that

$m_{0}(\epsilon)$ is equal to

Planck's con­stant, $h$, is a fun­da­men­tal physics con­stant and, in SI units, is equal to the very small quan­ti­ty

in units of $J \bullet Hz^{- 1},$rep­re­sent­ing SI units of energy over units of fre­quen­cy, equal to one cycle per second. The energy of a photon is de­fined as the prod­uct of Planck's con­stant with the photon's fre­quen­cy

and the con­stant $h$ plays a fun­da­men­tal role in mass-energy equiv­a­lence, re­lat­ing fre­quen­cy and mass. The reader should note that the two energy ex­pres­sions are equiv­a­lent

ap­pre­ci­at­ed by the fact that $\omega = 2\pi f$, and for the $m_{0}(\epsilon)$ case, we see that $f = 1$, given $\omega = 2\pi$. Geo­met­ri­cal­ly, Planck's con­stant is the radius of the unit sin­gu­lar­i­ty and, in the sense of on­to­log­i­cal math­e­mat­ics, is equal to unity. In clas­si­cal me­chan­ics, the energy ex­pres­sion $hf$ is anal­o­gous to the energy/work con­cept of torque, where $h$, Planck's con­stant, is the lever arm, and $f$, the fre­quen­cy, is the force acting on it.

All the terms in the so­lu­tion to $m_{0}(\epsilon)$, the unit in­vari­ant mass, are now joined at the co­a­les­cence point, the iden­ti­ty, which is the zero-di­men­sion­al point of pri­mor­dial light. The unit in­vari­ant mass rep­re­sents the math­e­mat­i­cal form the Uni­fied Field con­fers the mass char­ac­ter­is­tics upon and de­notes its small­est pos­si­ble value, ul­ti­mate­ly defin­ing the mass of the elec­tron and positron.

The figure above rep­re­sents the geo­met­ric co­a­les­cence and math­e­mat­i­cal basis for what the unit in­vari­ant mass, $m_{0}(\epsilon)$, de­fines, which, like all as­pects of the Uni­fied Field, is a direct result of the split­ting of pri­mor­dial light of the zero-point, the iden­ti­ty. Every­thing out­lined re­gard­ing the unit in­vari­ant mass and its math­e­mat­i­cal un­der­pin­nings ap­plies to our higher-di­men­sion­al gen­er­a­tor forms. This is ap­pre­ci­at­ed by start­ing with the fol­low­ing matrix rep­re­sen­ta­tion of our gen­er­a­tor $e^{\mathrm{A} + i\Omega}$

which, after in­clud­ing our proper spher­i­cal/time and hy­per­bol­ic/space vari­ables, we can define as the $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ wave­func­tion

The uni­tary as­ymp­tot­ic light gen­er­a­tors we denote as $\mathcal{c}_{1}$ and $\mathcal{c}_{7}$, which have the matrix rep­re­sen­ta­tions

We recall that the energy-mo­men­tum re­la­tion­ship is

and the geo­met­ric-de­pen­dent Schrödinger equa­tion de­rived in the last sec­tion is

which we found to be equal to

where the $\alpha^{2}$ term rep­re­sents the wavenum­ber squared, $k^{2}$. From the right-hand side of the equa­tion above, we take the ex­pres­sion that mul­ti­plies the wave­func­tion, $\frac{\hslash^{2}\alpha^{2}}{2m_{0}},$and given that, for matter waves, the wavenum­ber is equal to

we can ex­press it as the fol­low­ing mo­men­tum re­la­tion­ship

In the above, the $\hslash^{2}$ terms cancel, and after sub­sti­tut­ing $p = \sqrt{2m_{0}E}$, the ex­pres­sion be­comes

We use the matter wave energy equiv­a­lence $E = m_{0}c^{2}$, and cancel the like terms to find

where we have ar­rived at the $m_{0}c^{2}$ from the energy-mo­men­tum re­la­tion $\mathrm{E} = m_{0}c^{2}\gamma$, and the geo­met­ric-de­pen­dent Schrödinger equa­tion is found to equal

Using the wave func­tion form

$\mathrm{E} = m_{0}c^{2}\gamma$ can be ex­pressed using the geo­met­ric-de­pen­dent Schrödinger equa­tion as

where $c^{2} = \mathcal{c}_{1} \bullet \mathcal{c}_{7}$ rep­re­sents the prod­uct of the light as­ymp­totes. The $m_{0}\left( \mathcal{c}_{1} \bullet \mathcal{c}_{7} \right)$ on the right-hand side, when ex­pand­ed in matrix form, is

which simply equals

and given that $\gamma(\chi) = \cosh(\alpha\chi)$, the right-hand side in its en­tire­ty, $m_{0}c^{2}\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right),$is equal to

For the right-hand side, we set the proper hy­per­bol­ic/space value to one, $\chi = 1$, and sub­sti­tute the same values for our vari­ables, $\alpha = 0$, $\omega = 2\pi$. Given that $\beta\gamma = \sinh(\alpha)$, when $\alpha = 0$, $\beta\gamma = 0$, and the right-hand side sim­pli­fies to

For the left-hand side, we sub­sti­tute $\hslash = \frac{1}{2\pi}$ and once again $\chi = 1,\alpha = 0$, $\omega = 2\pi$, to find it sim­pli­fies to the iden­ti­ty matrix

and the so­lu­tion to the geo­met­ric-de­pen­dent Schrödinger equa­tion with our uni­tary choice of vari­ables

is equal to

rep­re­sent­ing the unit in­vari­ant mass $m_{0} = \epsilon.$