1.8.4 Relativistic Mechanics and the Unified Field

The form defin­ing the Uni­fied Field yields pow­er­ful physics right out of the gate. For in­stance, physics strug­gles to ex­plain the lo­ca­tion of the an­ti­mat­ter from the uni­verse's birth, where an­ti­mat­ter has the prop­er­ty of being op­po­site in charge to its matter part­ner. If an­ti­mat­ter and matter were to col­lide, they would mu­tu­al­ly an­ni­hi­late, pro­duc­ing in­tense energy in the process. Far-fetched ex­pla­na­tions have been pro­posed, e.g., it has been said that an asym­me­try ex­ist­ed at the moment of the uni­verse's birth, fa­vor­ing the num­bers of matter versus an­ti­mat­ter. These kinds of ex­pla­na­tions are a direct af­front to the net-zero prin­ci­ple, and, for this reason, they are in no way valid.

Ex­am­ples of matter/an­ti­mat­ter pairs are the elec­tron/positron$\ \left( e,\overline{e} \right)$, and the proton/an­tipro­ton $\left( p,\overline{p} \right)$, where we use the bar con­ven­tion over the par­ti­cle to denote an­ti­mat­ter. Math­e­mat­i­cal­ly, an­ti­mat­ter ro­tates in the op­po­site di­rec­tion on the in­fi­nite sphere of tan­gent spaces com­pris­ing the Uni­fied Field, which are the vacuum sin­gu­lar­i­ties at the Planck scale. Matter and an­ti­mat­ter share a mutual bound­ary on these in­fi­nite spheres of tan­gent spaces, but move around in op­po­site di­rec­tions, ac­count­ing for the dy­nam­ics of the vir­tu­al par­ti­cles of the vacuum. Ex­po­nen­ti­a­tion yields a radial ex­ten­sion away from the bound­ary of the sin­gu­lar­i­ty that can move inward or out­ward de­pend­ing on the mag­ni­tude of the real-valued com­po­nent. We present gen­er­al­ized math­e­mat­i­cal terms for form as $\mathfrak{f}$, and $\overline{\mathfrak{f}}$ as its anti-form part­ner. In $m$-di­men­sions, the Lie al­ge­bra for the form is de­fined as $\mathfrak{f}(z)= \mathfrak{f}(a + ib)$, and the anti-form as $\overline{\mathfrak{f}(z)}= \mathfrak{f}( - a - ib)$, where $a,b \in \mathbb{R}^{+ m \times m}$. Ex­po­nen­ti­a­tion of the Lie al­ge­bra of the form yields

and for the anti-form

and for $a \neq 0$, the radial com­po­nent of $\exp\left( \overline{\mathfrak{f}(z)} \right)$ will be less than one, while the radial com­po­nent of $\exp\left( \mathfrak{f}(z) \right)$ will be greater than one. The pair rep­re­sents the sym­met­ric split­ting of the iden­ti­ty el­e­ment, and mutual an­ni­hi­la­tion occurs if the form/anti-form pair col­lides, given the result of their prod­uct

Math­e­mat­i­cal­ly, the answer to the ques­tion of an­ti­mat­ter's where­abouts is that it is present in the Uni­fied Field; only it exists at radial dis­tances of the Plank length and below. Im­por­tant to re­al­ize is that the dis­tance from zero to the Plank length holds just as much in­for­ma­tion as the dis­tance of the Plank length to in­fin­i­ty does. De­pict­ed in the fol­low­ing figure, we show a plot of the func­tion $e^{a}$ for $a\in \mathbb{R}$, which ad­heres to the net-zero prin­ci­ple given that it is a sym­met­ric split of the zero-di­men­sion­al iden­ti­ty, uni­tary in volume and lo­cat­ed at the func­tion's ver­ti­cal in­ter­cept.

The real ex­po­nen­tial por­tion $e^{- a}$ of the anti-form ex­pres­sion $\exp{\left( \overline{\mathfrak{f}(z)} \right) =}e^{\overline{z}} = e^{- a}e^{- ib}$ is in­verse­ly pro­por­tion­al to its form coun­ter­part $e^{a}$. But it is crit­i­cal to en­vi­sion that when bound to the space­time geom­e­try of the hy­per­bo­lae, the mag­ni­tude of the radial ex­ten­sion from the origin is equal in the para­met­ric rep­re­sen­ta­tion $\left( \cosh(a),\sinh(a) \right),\ a\in \mathbb{R}$, de­pict­ed in Figure 1.8.6, for $a = \pm \sqrt{2}$. Lim­it­ing this con­cept to only matter and an­ti­mat­ter ob­scures the true nature of math­e­mat­ics at work. Ex­tend­ing the ter­mi­nol­o­gy to form and anti-form is the ap­pro­pri­ate con­cep­tu­al treat­ment, given that matter and an­ti­mat­ter are one man­i­fes­ta­tion of the multi-faceted Uni­fied Field. An­ti­mat­ter must have all the ac­cou­ter­ments that matter does, in­clud­ing the geo­met­ric space it sits in, forces, and rel­a­tivis­tic im­pli­ca­tions of its mass, to name just a few. So, the math­e­mat­i­cal frame­work which de­fines anti-form is as com­plete and flaw­less as its form coun­ter­part.

Next, we con­sid­er a form in the guise of a flow­ing point in space­time as the ex­pres­sion,

where $e^{i\omega}$ rep­re­sents the form's wave prop­er­ties, and $e^{\alpha}$ rep­re­sents the radial ex­ten­sion beyond the Plank dis­tance of the unit sin­gu­lar­i­ty's bound­ary and en­codes its rel­a­tivis­tic prop­er­ties. Lie al­ge­bras that are not axis-based, mean­ing solely real or imag­i­nary, will cease to be lo­cat­ed on the unit sin­gu­lar­i­ty's bound­ary once ex­po­nen­ti­at­ed.

We define the fol­low­ing Lie al­ge­bra, $\mathrm{A},$as

for $\alpha\in \mathbb{R}$, rep­re­sent­ing the hy­per­bol­ic Lie al­ge­bra, and $i\Omega$ as

for $\omega\in \mathbb{R}$, rep­re­sent­ing the imag­i­nary spher­i­cal Lie al­ge­bra. Given that the al­ge­bras com­mute, $\lbrack \mathrm{A},i\Omega\rbrack = 0,$the fol­low­ing ex­pres­sions are equal

which rep­re­sents the prod­uct of two gen­er­a­tors, one hy­per­bol­ic and the other com­plex spher­i­cal. The prod­uct $e^{\mathrm{A} + i\Omega}$ can be ex­pressed in terms of hy­per­bol­ic func­tions

or the cir­cu­lar trigono­met­ric func­tions

The energy-mo­men­tum re­la­tion in physics is de­fined as

which holds for a system or a body, for the mo­men­tum $p$, the speed of light $c$, the in­vari­ant mass $m_{0}$, also known as the rest mass. In space­time, all mas­sive bodies must have some motion, but with the ide­o­log­i­cal case of a mas­sive body at rest, the energy-mo­men­tum re­la­tion re­duces to the well-known energy-mass equiv­a­lence

and for mass­less par­ti­cles, the re­la­tion sim­pli­fies to

The energy-mo­men­tum re­la­tion of a mas­sive object is re­lat­ed to the Lorentz factor by the ex­pres­sion

and the scalar mo­men­tum, $\left| \mathbf{p} \right|$, is de­fined as

Using the hy­per­bol­ic Lorentz factor equiv­a­lence, we find

which yields the fol­low­ing ex­pres­sion for energy

The scalar mo­men­tum, $\left| \mathbf{p} \right|$, is re­lat­ed to the hy­per­bol­ic $\sinh(\alpha)$ func­tion as the equiv­a­lence

and the de­riva­tion using the ra­pid­i­ty term $\beta = \tanh(\alpha)$, is the fol­low­ing

Con­sid­er­ing how the two hy­per­bol­ic func­tions relate to rel­a­tivis­tic me­chan­ics leads to the fol­low­ing rel­a­tivis­tic equiv­a­lence for the ex­po­nen­tial func­tion

and solv­ing for energy, we find

We again con­sid­er the ide­al­is­tic case for a mas­sive object at the rest ideal $(v = 0)$, where the energy re­la­tion be­comes $E = m_{0}c^{2}$. After sub­sti­tut­ing this re­la­tion into the ex­pres­sion above, we find

and given that $v = 0$, we have that

Re­turn­ing to the hy­per­bol­ic gen­er­a­tor, $e^{\mathrm{A}},$from our gen­er­a­tor prod­uct, $e^{\mathrm{A} + i\Omega},$

we sub­sti­tute our rel­a­tivis­tic equiv­a­lences, yield­ing

and fac­tor­ing the $\frac{E}{\left( m_{0}c^{2} \right)}$ term out, gives us

In terms of a rel­a­tivis­tic re­la­tion­ship, our gen­er­a­tor prod­uct $e^{\mathrm{A} + i\Omega}$ be­comes

which, after mul­ti­pli­ca­tion, be­comes

Given the iden­ti­ty

our prod­uct re­la­tion in terms of the Lorentz factor is

The matrix prod­uct above is a form rep­re­sent­ing a flow­ing point in space­time and will have an in­verse anti-form coun­ter­part, rep­re­sent­ing a flow­ing anti-point in anti-space­time, found by the eval­u­a­tion of the fol­low­ing in­verse

The result above can be ex­pressed as the matrix prod­uct $e^{- \mathrm{A}}e^{- i\Omega}$, and by sub­sti­tut­ing the rel­a­tivis­tic terms into the prod­uct, we arrive at the in­verse matrix $e^{- (\mathrm{A} + i\Omega)}$

where the anti-form coun­ter­part con­tains the in­verse and hy­per­bol­i­cal­ly de­rived rel­a­tivis­tic factor

With matrix rep­re­sen­ta­tions for $e^{\mathrm{A} + i\Omega}$ and $e^{- (\mathrm{A} + i\Omega)}$ es­tab­lished, we can solve for the energy of the $e^{\mathrm{A} + i\Omega}$ gen­er­a­tor prod­uct

for which we will use the ra­pid­i­ty term no­ta­tion, $\beta = \frac{v}{c}$. The matrix por­tion from the ex­pres­sion above re­sults from the prod­uct

which we denote as the prod­uct of the $\mathrm{B}$ matrix with $e^{i\Omega}$, and to find the energy matrix $\mathbf{E}$, we can solve the fol­low­ing ex­pres­sion

The $\left( \mathrm{B}e^{i\Omega} \right)^{- 1}$ term is equal to

given that the in­verse of the Lie group $e^{i\Omega}$ is simply $e^{- i\Omega}$

and with sub­sti­tu­tion, the energy ex­pres­sion be­comes

Since

we find the rel­a­tivis­tic energy for our gen­er­a­tor prod­uct is not de­pen­dent on the $e^{i\Omega}$ gen­er­a­tor por­tion and the ex­pres­sion re­duces to

The $e^{\mathrm{A}},\mathrm{B}^{- 1}$ ma­tri­ces, using no­ta­tion for the Lorentz factor $\gamma$ and the ra­pid­i­ty term $\beta$, are equiv­a­lent to

lead­ing to the $e^{\mathrm{A}}\mathrm{B}^{- 1}$ prod­uct result

With this result, the rel­a­tivis­tic energy of our gen­er­a­tor prod­uct sim­pli­fies to

which given $m_{o}c^{2}$ is a scalar value, it can be moved to the left-hand side of the $\gamma$-matrix, ul­ti­mate­ly, we find a result con­sis­tent with our ini­tial rel­a­tivis­tic energy equiv­a­lence $E = m_{0}c^{2}\gamma$. Given this result for the rel­a­tivis­tic energy of the gen­er­a­tor prod­uct, the energy of an ide­al­ized mas­sive object system, for $v = 0$, be­comes

which is con­sis­tent with the ex­pect­ed result from rel­a­tivis­tic me­chan­ics.

To con­sid­er the rel­a­tivis­tic energy-mo­men­tum for the gen­er­a­tor prod­uct of a system com­prised of a mass­less object, we first take the ${\gamma = cosh}(\alpha)$ func­tion iden­ti­ty

and sub­sti­tute this back into our matrix energy ex­pres­sion

yield­ing

Next, we mul­ti­ply the matrix by the scalar-valued $m_{o}c^{2}$ term, after which the $m_{0}c^{2}$ terms will cancel, lead­ing to the energy matrix, $\mathbf{E},$

For a system com­prised of a mass­less object, the in­vari­ant mass will be equal to zero, $m_{0} = 0$, and solv­ing each entry in the $\mathbf{E}$ matrix above for our mass­less object system

we find the pre­vi­ous­ly de­fined result energy-mo­men­tum iden­ti­ty

im­ply­ing that the energy matrix result has the con­sis­tent form

The matrix so­lu­tion above for $\mathbf{E}$ is the result of the energy-mo­men­tum re­la­tion­ship $E = \sqrt{(pc)^{2} + \left( m_{0}c^{2} \right)^{2}}$, and taking (solely) this ap­proach ob­scures the fun­da­men­tal re­la­tion­ship that exists be­tween energy and the spher­i­cal and hy­per­bol­ic geome­tries. For this reason, we start by con­sid­er­ing the $e^{i\Omega}$ matrix

As we have seen pre­vi­ous­ly, an iso­lat­ed gen­er­a­tor-co­or­di­nate system will evolve in time due to the gen­er­a­tor op­er­at­ing on itself through its suc­ces­sive powers. We define a spher­i­cal time vari­able $\tau$ to rep­re­sent a math­e­mat­i­cal cal­cu­la­tive pro­gres­sion. Then, we can ex­press the evo­lu­tion of the cal­cu­la­tive/spher­i­cal time-de­pen­dent gen­er­a­tor system as

which will have the fol­low­ing matrix form

This matrix can also be ex­pressed in the fol­low­ing equiv­a­lent form

where $k$ is the wavenum­ber and $c$ is the speed of light. This equiv­a­lence can be seen by start­ing with the speed-of-light iden­ti­ty

where the prod­uct of the wave­length $\lambda$ and the fre­quen­cy $f$ equals $c.$The an­gu­lar fre­quen­cy, $\omega$, is re­lat­ed to fre­quen­cy as

the wavenum­ber, $k$, to the wave­length, as

and the speed of light is re­lat­ed to the an­gu­lar fre­quen­cy and the wavenum­ber as the fol­low­ing iden­ti­ty equiv­a­lence

Solv­ing this equiv­a­lence for $\omega$ and $k$, we find the fol­low­ing

which es­tab­lish­es that $e^{i\Omega\tau} = e^{i\mathrm{K}C\tau}.$

As al­ready stated, the rel­a­tivis­tic energy for a mass­less object is equal to

where the mo­men­tum can be ex­pressed in terms of the wavenum­ber as

and in terms of the an­gu­lar fre­quen­cy

lead­ing to the energy equiv­a­lence

We recall that the time-de­pen­dent Schrödinger's equa­tion we pre­sent­ed for our gen­er­a­tor was

which ex­press­es the energy equiv­a­lence be­tween the time-de­riv­a­tive for $i\hslash\psi(X,\tau)$, as equal to the second space-de­riv­a­tive of $- \frac{\hslash^{2}}{(2m)}\psi(X,\tau)$, rep­re­sent­ing a mas­sive object in space. The energy of the system of the $e^{i\Omega\tau}$ gen­er­a­tor will take the same form of the left-hand side of the time-de­pen­dent Schrödinger's equa­tion, which can be ap­pre­ci­at­ed after defin­ing our gen­er­a­tor to be the wave­func­tion $e^{i\Omega\tau} = \psi(\Omega,\tau)$

The neg­a­tive sign for the above re­sults from our choice of a pos­i­tive-valued an­gu­lar fre­quen­cy $\omega.$Cal­cu­lat­ing the time de­riv­a­tive for the wave­func­tion gives us the result

which can be easily re­arranged to

which shows that the time-de­riv­a­tive of our $\psi(\Omega,\tau)$ wave­func­tion gen­er­a­tor is equal to the energy of the mass­less system found using the rel­a­tivis­tic energy-mo­men­tum re­la­tion­ship

When the ve­loc­i­ty in space is pos­i­tive and less than the speed of light, the Lorentz factor re­lat­ed terms $\gamma$ and $\beta\gamma$ are equal to

for which the co­or­di­nates in hy­per­bol­ic space are re­lat­ed to the object's ve­loc­i­ty in space­time. Our mass­less object system rep­re­sents axial light, with a purely spher­i­cal gen­er­a­tor geom­e­try, mean­ing it does not have a ve­loc­i­ty in space­time. Solv­ing the Lorentz factor re­lat­ed terms $\gamma$ and $\beta$ for $v = 0$, we find

and taking the Lorentz factor matrix rep­re­sen­ta­tion for $e^{\mathrm{A}}$

we plug into it our null-ve­loc­i­ty Lorentz factor re­sults, find­ing the result

which is con­sis­tent with the system we are de­scrib­ing.

The energy of our mass­less system ex­am­ple can be de­fined as

which has the matrix rep­re­sen­ta­tion

and given that

we have de­rived the result for the ini­tial mass­less system energy-mo­men­tum

What we have learned so far in this sec­tion takes us to the next step. That being, to de­scribe a rel­a­tivis­tic and light-in­clu­sive equa­tion for quan­tum sys­tems. We start by defin­ing a proper hy­per­bol­ic/space vari­able $\chi$, to rep­re­sent the hy­per­bol­ic gen­er­a­tor's evo­lu­tion in space­time

and now define the wave­func­tion $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ as the fol­low­ing prod­uct of the gen­er­a­tors

which for our ex­am­ple, $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ takes the fol­low­ing matrix form

The equa­tion that we are de­scrib­ing has been in plain sight the entire time in the guise of the neg­a­tive-valued time-de­pen­dent Schrödinger's equa­tion

rep­re­sent­ing the quan­tum system in its proper con­text, an en­twined dance be­tween hy­per­bol­ic and spher­i­cal geome­tries. We define this equa­tion as the geo­met­ric-de­pen­dent Schrödinger equa­tion, then prove that the geo­met­ric-de­pen­dent Schrödinger equa­tion holds for our de­fined matrix gen­er­a­tor wave­func­tion by first solv­ing the left-hand side

Then solve the right-hand side

Making the geo­met­ric-de­pen­dent Schrödinger equa­tion con­sis­tent with rel­a­tiv­i­ty re­veals that the wavenum­ber $k$ in the Schrödinger equa­tion is the hy­per­bol­ic vari­able $\alpha$. The re­sults for both sides con­tain ma­tri­ces that are equal to our wave­func­tion $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$, all with the re­spec­tive energy terms, and the result simply re­duces to the form

Taking $\alpha$ as the wavenum­ber, the ex­pres­sion above is con­sis­tent given our pre­vi­ous de­riva­tion show­ing the fol­low­ing energy equiv­a­lence

The result of the geo­met­ric-de­pen­dent Schrödinger equa­tion rep­re­sents an energy equiv­a­lence, which holds for par­ti­cle sys­tems bound to both hy­per­bol­ic and spher­i­cal geome­tries, lead­ing to the more spe­cif­ic de­f­i­n­i­tion that it rep­re­sents a geo­met­ric energy equiv­a­lence.

The geo­met­ric-de­pen­dent energy-mo­men­tum re­la­tion­ship can be de­fined, as

which, for the $X_{1}$-ori­ent­ed light cone $\in \mathbb{R}^{+}$ has the fol­low­ing fac­tor­iza­tion

The left-hand ma­tri­ces rep­re­sent the sym­met­ric as­ymp­tot­ic light Lie gen­er­a­tor with its $X_{1}$-axis ori­ent­ed in the pos­i­tive real and imag­i­nary quad­rant $\left( \mathbb{R}^{+},Im^{+} \right)$

and the right-hand ma­tri­ces are the Lie gen­er­a­tor rep­re­sen­ta­tion of the sym­met­ric as­ymp­tot­ic light with its $X_{1}$-axis ori­ent­ed in the $\mathbb{R}^{+},Im^{-}$ quad­rant

Cal­cu­lat­ing the $\mathrm{E}^{2}$ prod­uct using these ma­tri­ces, we find

which is con­sis­tent with our de­fined geo­met­ric-de­pen­dent energy-mo­men­tum re­la­tion­ship.

The sym­met­ric as­ymp­tot­ic light Lie al­ge­bra gen­er­a­tors are de­pict­ed in Figure 1.8.7 as al­ge­bras on the unit sin­gu­lar­i­ty's sur­face $\left( \mathcal{B}_{j},\mathcal{C}_{j} \right)$, and with their moduli ex­tend­ed to $\mathcal{B}'_{j},\mathcal{C}'_{j} = \sqrt{2}\left( \ \mathcal{B}_{j},\mathcal{C}_{j} \right)$. For our energy-mo­men­tum ex­am­ple, we are using an $X_{1}$-type gen­er­a­tor, mean­ing the vec­tors ex­tend­ing to the poly­gon ver­tices at $\mathcal{B}'_{j},\mathcal{C}'_{j}$ rep­re­sent the axes of ro­ta­tion. The point at $\epsilon$ is a sig­nif­i­cant one, given that its lo­ca­tion rep­re­sents where spher­i­cal and hy­per­bol­ic geom­e­try share a common so­lu­tion, which for the $e^{\mathrm{A}\chi},e^{i\Omega\tau}$ group gen­er­a­tors, occurs when $\chi,\tau = 0$

The $\sqrt{2}$ factor is geo­met­ri­cal­ly the radial length of hy­potenuse formed by the tri­an­gle $\mathcal{O}\epsilon\mathcal{B}_{1}^{'}\ \text{or the triangle}\ \mathcal{O}\epsilon C_{1}^{'}$, and sig­ni­fies the point at which a seg­ment or­thog­o­nal to the $X_{1}$-axis will touch the light as­ymp­totes at $\mathcal{B}_{1}^{'}$ and $\mathcal{C}_{1}^{'}$. These points rep­re­sent the first in­ter­ac­tion points that as­ymp­tot­ic light has with space­time.

As pre­vi­ous­ly stated, net-zero is a char­ac­ter­is­tic that all physics must be con­sis­tent with, and the energy-mo­men­tum re­la­tion­ship is not an ex­cep­tion. For each $X_{j}$-plane, the light as­ymp­totes co­in­cid­ing with each of the four axes are so­lu­tions to the energy-mo­men­tum re­la­tion­ship, with two pos­i­tive so­lu­tions and two neg­a­tive. In our ex­am­ple case for the $X_{1}$-plane, the fol­low­ing light as­ymp­tote prod­ucts rep­re­sent the so­lu­tions to the geo­met­ric-de­pen­dent energy-mo­men­tum re­la­tion­ship**s for each of the four axes

where the sub­scripts for each $\mathrm{E}_{\left\lbrack {c_{j}X}_{1} \right\rbrack}^{2}$ denote the axis. By ex­pand­ing each so­lu­tion, we find the re­sult­ing axis-de­pen­dent geo­met­ric-de­pen­dent energy-mo­men­tum**s

which, when summed, are equal to net-zero

The neg­a­tive-valued $\mathrm{E}_{\left\lbrack {c_{j}X}_{1} \right\rbrack}^{2}$ energy-mo­men­tums are the result of the prod­uct of imag­i­nary-valued masses and mo­men­tums, which can be shown by start­ing with the neg­a­tive-valued re­sults for the imag­i­nary axes

and then solve for the energy $\mathrm{E}$

Figure 1.8.8 de­picts the geo­met­ric arrange­ment of the as­ymp­tot­ic light Lie al­ge­bra gen­er­a­tors in the $X_{1}$-plane. The points at the vector arrow tips form pairs with ad­ja­cent points on the light as­ymp­totes, rep­re­sent­ing the axes of the so­lu­tions to the geo­met­ric-de­pen­dent energy-mo­men­tum re­la­tion­ships. The dashed lines ex­tend­ing or­thog­o­nal­ly from each axis to the primed points on the light as­ymp­totes, present an in­tu­itive glimpse as to how Eu­clid­ean spaces man­i­fest in the Uni­fied Field, given the flat geom­e­try of the tri­an­gles formed with the axes where the Pythagore­an the­o­rem is ap­plic­a­ble. These primed points rep­re­sent the bal­ance points be­tween spher­i­cal and hy­per­bol­ic geom­e­try.

The ad­ja­cent light as­ymp­totes form geo­met­ric prod­ucts that yield the ori­gins of the unit hy­per­bo­lae pic­tured in red in the figure

where con­sis­tent with net-zero, we find sym­me­try in two pos­i­tive and neg­a­tive re­sult­ing pairs. The pos­i­tive-valued unit hy­per­bo­lae are ori­ent­ed in the space-like light cone, while the neg­a­tive-valued unit hy­per­bo­lae are ori­ent­ed in time-like light cone. The hy­per­bol­ic geome­tries in­her­ent in the Lie al­ge­bra gen­er­a­tors lo­cat­ed at these axis points can be ap­pre­ci­at­ed by start­ing with the fol­low­ing $\mathcal{B}_{1}\left( X_{1} \right)$ matrix Lie group gen­er­a­tor

which is equal to our geo­met­ric-de­pen­dent Schrödinger equa­tion wave­func­tion

with its vari­ables set as $\omega = \frac{\pi}{4}$, $\tau = 1$, and $\chi = 0.$The group gen­er­a­tor can be ex­pressed in the equiv­a­lent form

which shows its as­ymp­tot­ic light char­ac­ter­is­tics. $\mathcal{B}_{1}\left( X_{1} \right)$ is uni­tary given that

and sits on the bound­ary of the unit sin­gu­lar­i­ty. We can rotate the group in a clock­wise di­rec­tion to four equidis­tant points in ra­di­ans along the ge­o­des­ic path of the $X_{1}$-plane, by mul­ti­ply­ing suc­ces­sive­ly by $- i$. This leads to the fol­low­ing four dis­tinct as­ymp­tot­ic light group gen­er­a­tors

and the com­plete ge­o­des­ic has been trav­elled given that $( - i)^{4} = \epsilon$, bring­ing us back to the start­ing group, $\mathcal{B}_{1}\left( X_{1} \right)$.

The prod­uct result of the light as­ymp­tote gen­er­a­tors $\mathcal{B}_{1}\left( X_{1} \right) \times \mathcal{C}_{1}\left( X_{1} \right)$ yields the hy­per­bol­ic Lie al­ge­bra gen­er­a­tor

which is a ro­ta­tor that acts on the $X_{1}$-plane hy­per­bol­ic gen­er­a­tor group

taking the bottom-right $2 \times 2$ sub­ma­trix from its hor­i­zon­tal ori­en­ta­tion to a ver­ti­cal one in the $X_{1}$-plane

where it can be seen that $\cosh(\alpha)$ and $\sinh(\alpha)$ have been ex­changed. This result is con­sid­ered by math­e­mati­cians and physi­cists to be a linear trans­for­ma­tion con­sis­tent with the de­riv­a­tive rep­re­sen­ta­tion of the $\exp\left( \alpha\mathcal{B}_{1}\left( X_{1} \right) \times \mathcal{C}_{1}\left( X_{1} \right) \right)$ gen­er­a­tor, seen as

Math­e­mat­i­cal­ly, the co­he­sive de­riv­a­tive of the above matrix is the result of our de­fined de­riv­a­tive op­er­a­tor trans­form

which is con­sis­tent with the result of the de­riv­a­tive of the ex­po­nen­tial func­tion

When the $\mathcal{B}_{1}\left( X_{1} \right) \times \mathcal{C}_{1}\left( X_{1} \right)$ light as­ymp­tote ro­ta­tor acts on $e^{\mathrm{A}}\left( \mathcal{B}_{1}\left( X_{1} \right) \times \mathcal{C}_{1}\left( X_{1} \right) \right)$, it re­turns to its orig­i­nal po­si­tion

given that the ro­ta­tor squares to the iden­ti­ty matrix

which gives rise to an ori­ent­ed geo­met­ric con­fine­ment these hy­per­bol­ic gen­er­a­tors main­tain be­tween the pos­i­tive ver­ti­cal and hor­i­zon­tal axes in the $X_{1}$-plane. Eval­u­a­tion of the de­ter­mi­nants for the two Lie group gen­er­a­tors yields the fol­low­ing re­sults

where the first rep­re­sents the space-like geom­e­try of the unit hy­per­bo­la ori­ent­ed in the hor­i­zon­tal pos­i­tive real-axis, and the second rep­re­sents the time-like geom­e­try of the unit hy­per­bo­la ori­ent­ed in the ver­ti­cal real-axis.

As­sum­ing the pre­vi­ous ex­am­ple was form, the hy­per­bol­ic analog anti-form is the result of the prod­uct of the $- \mathcal{C}_{1}\left( X_{1} \right)$ and $- \mathcal{B}_{1}\left( X_{1} \right)$ light as­ymp­totes

which is the ro­ta­tor that acts on the hy­per­bol­ic gen­er­a­tor group in the neg­a­tive $X_{1}$-plane

taking the bottom-right $2 \times 2$ sub­ma­trix in the neg­a­tive $X_{1}$-plane from its hor­i­zon­tal ori­en­ta­tion to a ver­ti­cal one

where it can be seen that $\cosh(\alpha)$ and ${- sinh}(\alpha)$ func­tions have been ex­changed. A sim­i­lar­ly ori­ent­ed geo­met­ric anti-form con­fine­ment is also present due to the second prod­uct power equal­ing the iden­ti­ty

and these hy­per­bol­ic gen­er­a­tors main­tain lo­ca­tions be­tween the neg­a­tive ver­ti­cal and hor­i­zon­tal axes in the $X_{1}$-plane. Eval­u­a­tion of the anti-form de­ter­mi­nants for these two Lie group gen­er­a­tors yields the fol­low­ing re­sults

where the first rep­re­sents the anti-space-like geom­e­try of the unit hy­per­bo­la ori­ent­ed in the hor­i­zon­tal neg­a­tive real-axis, and the second rep­re­sents the anti-time-like geom­e­try of the unit hy­per­bo­la ori­ent­ed in the ver­ti­cal neg­a­tive real-axis.

The pro­found im­pli­ca­tion is that time flows in one di­rec­tion for form, which is due to the geo­met­ric con­fine­ment its hy­per­bol­ic gen­er­a­tors are sub­ject to on the pos­i­tive real ver­ti­cal and hor­i­zon­tal axes. Anti-form trav­els in time in the op­po­site di­rec­tion with in­verse­ly re­lat­ed mag­ni­tudes for anti-time and anti-space com­pared to the time and space of forms. Time travel for some­thing bound to the geome­tries of form and anti-form, if pos­si­ble, re­quires the trans­for­ma­tion of form to anti-form and vice versa.

Geo­met­ri­cal­ly, mul­ti­ply­ing the $\mathcal{B}_{1}\left( X_{1} \right)$ group gen­er­a­tor by $- i$, ro­tates it by $- \frac{\pi}{2}$ ra­di­ans in the $X_{1}$-plane on the unit sin­gu­lar­i­ty. This ro­ta­tion is ex­act­ly what the $- i$ term does to the wave­func­tion $\psi(X,\tau)$ on the left-hand side of the Schrödinger equa­tion

and mul­ti­ply­ing by $i$ for the Schrödinger equa­tion of the form

ro­tates $\psi(X,\tau)$ by $\frac{\pi}{2}$ ra­di­ans in the $X_{1}$-plane on the unit sin­gu­lar­i­ty. The $- \frac{\pi}{2}$ ra­di­ans ro­ta­tion ap­plies just the same for the geo­met­ric-de­pen­dent Schrödinger equa­tion

and the vari­able spe­cif­ic $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ func­tion, that is the $\mathcal{B}_{1}\left( X_{1} \right)$ gen­er­a­tor.

Using our $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ wave­func­tion, we omit the $\hslash$ and $\frac{\hslash^{2}}{(2m)}$ terms in the geo­met­ric-de­pen­dent Schrödinger equa­tion, and eval­u­ate the equal­i­ty

find­ing for the left-hand side

then the right-hand side

and since the wave­func­tion $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$ is common to both re­sults, our equal­i­ty sim­pli­fies to

We recall that $\alpha$ is the wavenum­ber, and for matter waves, this equals

and $\alpha^{2}$ is

which, by using the energy equal­i­ty $E = \hslash\omega$, it can be put in terms of $\omega$ as

There­fore, the geo­met­ric-de­pen­dent Schrödinger equa­tion for mas­sive body sys­tems is con­tin­u­al­ly solv­ing for the mass term $m$, which is the sole in­de­pen­dent vari­able given that the an­gu­lar fre­quen­cy $\omega$ and the wavenum­ber $\alpha$ are re­lat­ed by the iden­ti­ty

where $v$ is the ve­loc­i­ty term, im­ply­ing that

Solv­ing for the mass term in the $\alpha^{2}$ ex­pres­sion, we find

and by $p = \hslash\alpha$, the mass of the system is equal to

where the $\frac{\alpha}{\omega}$ term above is equal to the in­verse of the ve­loc­i­ty

which leads to the well-known mo­men­tum ex­pres­sion

For the wave­func­tion $\psi\left( \lbrack \mathrm{A},\Omega\rbrack,\lbrack\chi,\tau\rbrack \right)$, the $\alpha$ and $\omega$ vari­able terms will be equal in mag­ni­tude, and by as­sign­ing a vari­able $\theta_{\alpha} = \alpha$, and $\theta_{\omega} = \omega$ the wave­func­tion be­comes

Using $\psi\left( \lbrack\Theta\rbrack,\lbrack\chi,\tau\rbrack \right)$, we eval­u­ate the equal­i­ty

for a vari­able, $\mathrm{K}$, find­ing for the left-hand side

then the right-hand side

and the equal­i­ty sim­pli­fies to

Using sim­i­lar vari­ables to the $\mathcal{B}_{1}\left( X_{1} \right)$ matrix Lie group gen­er­a­tor, we set $\theta_{\omega},\theta_{\alpha} = \frac{\pi}{4}$, and solve for the value of the fre­quen­cy

and for the wave­length value

We re­arrange the equal­i­ty

and given that

the ex­pres­sion be­comes

The ob­vi­ous so­lu­tion for $\mathrm{K}$ is

but the other so­lu­tion that exists for light is less ap­par­ent, which is seen by con­sid­er­ing $\mathrm{K}$ as

where the equal­i­ty be­comes

which sim­pli­fies to

Above, we find an in­verse speed of light re­la­tion­ship that hold when $c = f\lambda = 1$, which is ex­act­ly the case for our light system, ev­i­denced by sub­sti­tut­ing our fre­quen­cy and wavenum­ber values into the ex­pres­sion

Per­haps this is sur­pris­ing, but it is rooted in the es­sen­tial net-zero char­ac­ter­is­tics that light must pos­sess, mean­ing a light anti-form exists that is in­verse­ly re­lat­ed to its light-form coun­ter­part, which to­geth­er embody a per­fect bal­ance.

More ev­i­dence for the ex­is­tence of this bal­ance is ap­pre­ci­at­ed by eval­u­at­ing our light-system equal­i­ty using the matter-based ex­pres­sion for our wavenum­ber

which would be blas­phe­mous to the physi­cist, but the math­e­mat­ics clear­ly states oth­er­wise.

Solv­ing for mass we find

and given that $\theta_{\alpha} = \theta_{\omega}$, mass for the light system has two so­lu­tions

These can be ex­pressed in re­la­tion to a wave­length mass, $m_{\lambda},$and a fre­quen­cy mass, $m_{\omega},$as

where $m_{f}$ and $m_{\lambda}$ are equal. This equiv­a­lence

can be arranged in two ratios: the first

and the second

which are in­verse­ly re­lat­ed to one an­oth­er, yet equal in light sys­tems, and are indeed form and anti-form matter com­po­nents of light sys­tems. Set­ting the form and anti-form matter ratios as equal, we find

rep­re­sent­ing the same so­lu­tions found for $\mathrm{K} = \frac{1}{\theta_{\omega}}$, in the ex­pres­sion

As­ymp­tot­ic light is a bal­ance be­tween in­verse­ly re­lat­ed wave­lengths and fre­quen­cies, matter and in­verse matter, oth­er­wise known as matter and an­ti­mat­ter, form and anti-form. This brings us to the sub­ject matter of the next sec­tion, which is to math­e­mat­i­cal­ly ex­plore in­vari­ant mass and derive the as­so­ci­a­tion that a unit in­vari­ant mass has with the energy-mo­men­tum re­la­tion­ship.