1.3.9 The Asymptotic Light Functions

In this sec­tion, math­e­mat­i­cal cur­va­ture is con­sid­ered and then ap­plied in the con­text of math­e­mat­i­cal physics. In par­tic­u­lar, the hy­per­bol­ic as­ymp­tote serves as the bound­ary for the para­met­ric curve $\gamma(t) = \cosh(t),\sinh(t)$, which ap­proach­es the as­ymp­tote per the limit

In physics, this as­ymp­tote is a bound­ary sec­tion of the rel­a­tivis­tic light cone, which acts to sep­a­rate time from space. The light cone is the basis for Ein­stein-Minkows­ki space­time, which di­vides causal­i­ty into a frame­work of the future and the past, re­lat­ing to a tem­po­ral present. The light cone bound­ary sec­tions the geom­e­try into space­like, time­like, and light­like spaces. Figure 1.3.22 shows the time axis run­ning from past to present and then to future. Space is de­pict­ed as the area out­side the light cone, the time-space as in the cone, and the present/now as the center of the cone. We note that the space­like region is sit­u­at­ed on the hor­i­zon­tal plane pass­ing by the origin, the light­like is on the cone, and the time-like is in the volume space inside the light­cone. In sec­tion 1.1.26, we de­fined the $D_{3\mathbb{c}}$ di­he­dral group, whose basis re­volves around an ex­cep­tion­al number, that being$\ \sqrt{i}$, which has mul­ti­ple ex­pres­sions, ev­i­denced by the equiv­a­lent forms

It is also es­sen­tial to con­sid­er that the neg­a­tive forms also suf­fice for $\sqrt{i}$,

and more pre­cise­ly, $\sqrt{i}$ is equal to

where the light cone sep­a­rates past, present, future, and space.

Its coun­ter­part, $\sqrt{- i}$, can be stated in a sim­i­lar way,

for which, all four of these num­bers are lo­cat­ed at points on the bound­ary of the com­plex unit circle through which the hy­per­bol­ic as­ymp­totes em­anate. Figure 1.3.23 rep­re­sents how these num­bers are ori­ent­ed on the com­plex unit circle, demon­strat­ed as the points where the red as­ymp­totes meet the circle's bound­ary. The values that are bound­ary spe­cif­ic are de­pict­ed inside the com­plex unit circle, and the ex­po­nen­tial map­pings for the hor­i­zon­tal­ly ori­ent­ed unit hy­per­bo­la and circle are shown as the $f(t)$ func­tions.

The as­ymp­totes rep­re­sent a bal­ance be­tween the sphere's pos­i­tive cur­va­ture and the hy­per­bo­la's neg­a­tive cur­va­ture. The ques­tion be­comes, what func­tions serve as the as­ymp­totes in com­plex space? We will use the im­plied com­plex geom­e­try and the geo­met­ric prod­uct to find out. We start by taking the com­plex unit circle Lie group iso­mor­phism,

and ex­press this in vector format,

where the $\cos(t)$ and $\sin(t)$ func­tions rep­re­sent vari­able-de­pen­dent scalar por­tions of the vector. Next, we define a vector, $v = \left( \frac{e_{1}}{\sqrt{2}},\frac{e_{2}}{\sqrt{2}} \right)$, to rep­re­sent the as­ymp­tot­ic Lie al­ge­bra, $e^{\left( \frac{i\pi}{4} \right)}$, then con­sid­er the geo­met­ric prod­uct be­tween the two vec­tors

By cal­cu­lat­ing the geo­met­ric prod­uct, we find the dot prod­uct equals

and the wedge prod­uct

By the iso­mor­phism $i = I$, the sum of the prod­ucts rep­re­sent­ing $uv$ is

a result of equal in mag­ni­tude con­tri­bu­tions from the dot and wedge prod­ucts. In Figure 1.3.24, the vec­tors that form this geo­met­ric prod­uct appear par­al­lel. De­spite this par­al­lel ap­pear­ance, in­cor­po­rat­ed is the nested bivec­tor $I$ into the $u$-vector, which acts to main­tain a con­tri­bu­tion from the wedge prod­uct in the result. Like the cosine and sine func­tion ex­am­ple in the last sec­tion, we can dis­trib­ute the bases con­tain­ing the bivec­tor $I$, to rep­re­sent the vector $u$

which will result in the same geo­met­ric prod­uct once cal­cu­lat­ed. Next, we arrange the result into real and imag­i­nary com­po­nents, defin­ing a two-di­men­sion­al para­met­ric vector $f(t)$ in the com­plex plane, as

The result is seen by plot­ting the func­tion $f(t)$ in Figure 1.3.25, re­veal­ing a flat and di­ag­o­nal­ly ori­ent­ed curve con­sis­tent with the unit hy­per­bo­la's as­ymp­tote. This as­ymp­tot­ic con­sis­ten­cy is ap­pre­ci­at­ed given the flat curve in­ter­sects the com­plex unit circle at the points $e^{\left( \frac{i\pi}{4} \right)}$ and $e^{\left( \frac{- 3i\pi}{4} \right)}.$The geo­met­ric in­tu­ition behind this ex­am­ple can be shown when we con­sid­er the com­plex plane, for $\pm isin(t)$ func­tions ori­ent­ed about the ver­ti­cal axis, and for $\pm cos(t)$ func­tions on the hor­i­zon­tal axis, as seen in Figure 1.3.26.

When ro­tat­ing the $\frac{e^{- it}}{\sqrt{2}}$ vector in the com­plex plane by $\frac{i\pi}{2}$ ra­di­ans, it will align the vector func­tions. This is the same thing as adding the sta­tion­ary vector$\frac{e^{it}}{\sqrt{2}}$ , to the ro­tat­ed vector $\frac{ie^{- it}}{\sqrt{2}}$ , rep­re­sent­ed by the ex­pres­sion

which is the same result as the geo­met­ric prod­uct cal­cu­la­tion. For $t\in \mathbb{R}$, the com­plex con­ju­gate of the func­tion $f(t)$, is

and using the iso­mor­phism, we set $I = i$, for the $f(t)$, $f(t)^{*}$ func­tions, and pro­ceed to eval­u­ate their geo­met­ric prod­uct

As these vec­tors are or­thog­o­nal­ly re­lat­ed, we find the dot prod­uct equals zero

and the wedge prod­uct

where ul­ti­mate­ly, the geo­met­ric prod­uct is solely due to the con­tri­bu­tion from the wedge prod­uct

By way of the iden­ti­ty

we find that the geo­met­ric prod­uct $f(t)f(t)^{*}$ pro­duces a real-valued sine wave that is always $\geq 0$, given that the $+ 1$ in the ex­pres­sion shifts the wave­form up by one on the ver­ti­cal axis. To find the other as­ymp­tot­ic curve func­tion, we re­verse the vector order in the geo­met­ric prod­uct that yield­ed the $f(t)$ func­tion, to $vu$'

where we dis­trib­uted the bivec­tor, $I$, in the bases of the orig­i­nal $u$-vector, be­com­ing, $u'$,

The dot prod­uct cal­cu­lates to

and the wedge prod­uct

lead­ing to the $vu'$result

After ar­rang­ing it into a vector, we find $vu^{'}$ is iso­mor­phic to the com­plex con­ju­gate form $f(t)^{*}$

In Figure 1.3.27, we plot the $f(t),f(t)^{*}$ func­tions in the com­plex plane with the unit hy­per­bo­la, where it can read­i­ly be seen these func­tions are the as­ymp­tote func­tions to the unit hy­per­bo­la. Given that

we can take the $f(t)$ func­tion to define the para­met­ric func­tion $\gamma(t)$, and its com­plex con­ju­gate $\gamma(t)^{*}$, as fol­lows

radius, $r = \sqrt{2}$ (dashed), for $\gamma(t) = \frac{1}{\sqrt{2}}\left( \cos(t) + \sin(t) \right) + \frac{i}{\sqrt{2}}\left( \cos(t) + \sin(t) \right).$

In Figure 1.3.28, the func­tions $\gamma(t)$ and $\gamma(t)^{*}$ are plot­ted, re­veal­ing their flat curves extend beyond the unit circle to a max­i­mum mod­u­lus equal to $\sqrt{2}$, for $n\in \mathbb{N}$, oc­cur­ring when $\gamma(t)$ equals

This ex­ten­sion beyond the unit circle bound­ary is due to the per­pet­u­al pres­ence of equal mag­ni­tude real and imag­i­nary values. This should be clear since the same trigono­met­ric func­tion rep­re­sents the real and imag­i­nary di­men­sions of $\gamma(t).$We now define the time-based func­tion $f(\tau)$, by using this com­pact form

which moves through time $t$ in the com­plex plane. The real por­tion of the wave func­tion is the $\sin\left( t + \frac{\pi}{4} \right)$ com­po­nent, and the imag­i­nary com­po­nent ${isin}\left( t + \frac{\pi}{4} \right)$ is its or­thog­o­nal­ly ori­ent­ed coun­ter­part.

In Figure 1.3.29, we plot the func­tion $f(\tau)$, which can be seen to move to the right in time, having or­thog­o­nal­ly ori­ent­ed wave com­po­nents os­cil­lat­ing in the real and imag­i­nary di­men­sions. The $f(\tau)$ func­tion has pe­ri­od­ic roots in the time di­men­sion, oc­cur­ring when $t = - \frac{\pi}{4} \pm \pi n$ $,$and wave com­po­nents demon­strat­ing peaks and troughs for $t = \frac{\pi}{4} \pm \pi n$. For the reader fa­mil­iar with elec­tro­mag­net­ic waves, these or­thog­o­nal waves are per­fect­ly con­sis­tent with the char­ac­ter­is­tics of light. Also cap­tured in the di­a­gram is the in­tu­ition behind the wave-par­ti­cle du­al­i­ty of light, mean­ing the phe­nom­e­nol­o­gy of light ap­pears to be wave-like and par­ti­cle-like. The par­ti­cle-like char­ac­ter­is­tics of light man­i­fest at each pe­ri­od­ic root along the time di­men­sion for $t = - \frac{\pi}{4} \pm \pi n$, oc­cur­ring twice per wave period as each wave com­po­nent hits the time axis in unison. While the wave char­ac­ter­is­tic, in­her­ent to its form, is de­fined by two or­thog­o­nal­ly ori­ent­ed waves os­cil­lat­ing through time.

with or­thog­o­nal­ly ori­ent­ed waves os­cil­lat­ing in re­spec­tive real and imag­i­nary di­men­sions.

Elec­tro­mag­net­ic waves also have spa­tial forms that travel through spa­tial dis­tances in time, and we can define the com­plex plane wave with spa­tial and tem­po­ral com­po­nents as fol­lows

We then ex­press $e^{i( - kx + \omega t)}$, as the vector $u$

and as before, the vector $v$, is de­fined

We pro­ceed to cal­cu­late the geo­met­ric prod­uct be­tween the two vec­tors

find­ing the dot prod­uct por­tion equals

and the wedge prod­uct por­tion equals

lead­ing to the fol­low­ing result for $uv$

which sim­pli­fies to the form

Using this result, we define the func­tion $f(x,t)$ to rep­re­sent a wave trav­el­ing through the spa­tial and tem­po­ral di­men­sions, as

Maxwell's equa­tions de­scribe elec­tro­mag­net­ism, and are de­fined by the fol­low­ing four equa­tions in their dif­fer­en­tial forms:

where $\overset{\rightharpoonup}{E}$ rep­re­sents the elec­tric field, and $\overset{\rightharpoonup}{B}$ is the mag­net­ic field. The first two equa­tions are dot prod­ucts, while the second two are cross-prod­ucts. For the static equa­tions, Gauss's law states that the elec­tric flux pass­ing through a closed sur­face is pro­por­tion­al to the total charge on its sur­face, and the mag­net­ic flux pass­ing through a closed sur­face is always equal to zero. While for the dy­nam­ic equa­tions, Fara­day's law states that a chang­ing mag­net­ic field in time pro­duces a mag­net­ic field, and Ampere's law says a chang­ing elec­tric field in time cre­ates a mag­net­ic field. The equa­tions in free space with­out sources can be rep­re­sent­ed as

where $\rho$ and $\overset{\rightharpoonup}{J}$ are now equal to zero.

Next, we define the $\overset{\rightharpoonup}{E}(x,t)$ and $\overset{\rightharpoonup}{B}(x,t)$ fields to be equal to the $f(x,t)$-wave func­tion,

and con­sid­er a lin­ear­ly po­lar­ized wave trav­el­ing at the speed of light in the $x$-di­rec­tion, with the elec­tric field $\overset{\rightharpoonup}{E}$ con­strained to the $y$ di­rec­tion, and the mag­net­ic field $\overset{\rightharpoonup}{B}$ con­strained to the $z$-di­rec­tion (Figure 1.3.30).

We set the fol­low­ing basis vec­tors for the three di­men­sions

which lend them­selves to the fol­low­ing vector field rep­re­sen­ta­tions

By eval­u­at­ing the di­ver­gence for both fields

we find they equal zero, agree­ing with Gauss' law and the mag­net­ic law from the equa­tions in free space. Next, we eval­u­ate Fara­day's law for our field func­tions, to find

noting the speed of light is equal to the fol­low­ing wavenum­ber-an­gu­lar fre­quen­cy re­la­tion­ship

The time de­riv­a­tive of the mag­net­ic field is

and we equate the mag­ni­tudes of both sides of Fara­day's law, and sim­pli­fy

for which the dif­fer­en­tial equa­tion holds. Next, we eval­u­ate Ampere's law for our trav­el­ing wave

and the time de­riv­a­tive of the elec­tric field

We equate the mag­ni­tudes of both sides of Ampere's law

to find this dif­fer­en­tial equa­tion holds as well. The fact that the func­tion $f(x,t)$ is con­sis­tent with Maxwell's equa­tions, shows that these as­ymp­tot­ic num­bers and func­tions are light-based, specif­i­cal­ly elec­tro­mag­net­ic waves. The treat­ment of Maxwell's equa­tions was cur­so­ry in this ex­am­ple, but we will return to this in more depth in the forth­com­ing volume con­cern­ing the dy­nam­ics of light.

The uni­fied light func­tions are the final and ar­guably most im­pres­sive as­ymp­tot­ic light func­tions we con­sid­er in this sec­tion. The term uni­fied, means that the spher­i­cal and hy­per­bol­ic geome­tries have been brought into align­ment for the func­tions.

The de­fined vector $u$

rep­re­sents the $e_{1},e_{2}$-plane, for the $e^{t}$,$\ e^{It}$ ex­po­nen­tial func­tions ori­ent­ed by their bases. We can rotate to any point on the func­tion plane using the fol­low­ing geo­met­ric prod­uct,

where $\left\| \left( ae_{1} + be_{2} \right) \right\| = 1,$rep­re­sents a point on the bound­ary of the unit circle in the $e_{1},e_{2}$-plane.

In Figure 1.3.31, we depict the $e_{1},e_{2}$-plane with hy­per­bol­ic and cir­cu­lar ex­po­nen­tial func­tions ori­ent­ed along the pos­i­tive and neg­a­tive $e_{1}$,$e_{2}$-axes, adding red vec­tors to rep­re­sent the axes of ro­ta­tion for the $e_{1},e_{2}$-plane seg­ments, using $\left( \pm \frac{e}{\sqrt{2}}, \pm \frac{ie}{\sqrt{2}} \right)$, as the ro­ta­tion values. It turns out that ro­tat­ing the $e_{1},e_{2}$-plane about these as­ymp­tot­ic axes, re­sults in the light func­tions we are con­sid­er­ing. By set­ting $v_{1} = \left( ae_{1} + be_{2} \right) = \frac{1}{\sqrt{2}}\left( e_{1} + e_{2} \right),$we trans­form the vector $u$, as fol­lows

for which, we have arranged the result by real and imag­i­nary valued com­po­nents. Real and imag­i­nary hy­per­bol­ic and wave-like el­e­ments are noted in this result, which is read­i­ly seen in the com­plex plane by the para­met­ric plot in Figure 1.3.32. The sym­met­ric bal­ance be­tween the real and imag­i­nary hy­per­bol­ic el­e­ments, re­sults in a de­fin­i­tive­ly flat geom­e­try $\gamma_{h}(t),$seen in the plot, while the wave el­e­ments com­prise the unit circle $\gamma_{s}(t)$ geom­e­try.

The vector $uv_{1}$ is com­posed of scalar por­tions and bivec­tors por­tions, form­ing a func­tion that is a hy­bridiza­tion of the unit hy­per­bo­la with the unit circle. We con­tin­ue the process, ro­tat­ing around the plane in a clock­wise manner to all the as­ymp­tot­ic points and axes, by using the fol­low­ing prod­ucts

where we find re­sults that are axis ori­ent­ed for $uv_{j}$ with even $j$-values, and func­tions that are as­ymp­tote ori­ent­ed for odd $j$-values. The reader can orient them­selves to the re­sult­ing vector arrange­ments by view­ing Figure 1.3.33, from the per­spec­tive of the $e_{1},e_{2}$-plane, where the odd as­ymp­tot­ic vector func­tions are de­pict­ed in red, and the axis vector func­tions in black.

The square plane seg­ments should stand out to the reader in Figure 1.3.33, whose sig­nif­i­cance is that the ro­ta­tion of light occurs by moving ori­ent­ed plane seg­ments, which are the bivec­tor forms of the geo­met­ric prod­uct. This dif­fers from ro­tat­ing a point or a vector in the com­plex two-di­men­sion­al plane, where the scalars and the $e_{1},e_{2}$ bases are suf­fi­cient to de­scribe the process. In the case of the as­ymp­tot­ic light func­tions, ro­ta­tions occur in a higher-di­men­sion­al man­i­fold, ev­i­denced by the dy­nam­ics of bivec­tor ro­ta­tions.

Next, we take a closer look at the $uv_{1}$ light func­tion by plot­ting $uv_{1}$ in Figure 1.3.34, its imag­i­nary and real com­po­nents

which con­veys that these light wave­forms are a hy­bridiza­tion of wave and radial el­e­ments.

and the para­met­ric plot, $\gamma(t) = \left( Re(uv),i \bullet Im(uv) \right)$.

The cir­cu­lar trigono­met­ric func­tions impart wave char­ac­ter­is­tics, while the hy­per­bol­ic func­tions impart radial ex­ten­sion. For $t < 0$, the ex­po­nen­tial wave por­tion of each func­tion dom­i­nates, while for $t > 0$, the ex­po­nen­tial radius por­tion be­comes om­nipresent. This qual­i­ty of the light func­tions clar­i­fies what occurs when the macroscopy of clas­si­cal physics be­comes quan­tum physics at the mi­cro­scop­ic level. De­spite the dom­i­nance of the radial ex­ten­sion of the ex­po­nen­tial func­tion in the mor­phol­o­gy to the right of the origin, the wave-like char­ac­ter­is­tics remain present in the form. On the other hand, to the left of the origin, it can also be said that de­spite the mor­pho­log­i­cal wave dom­i­nance, the radial ex­ten­sion from the ex­po­nen­tial func­tion re­mains present, albeit in an at­ten­u­at­ed form. The func­tion plots show this as wave mor­phol­o­gy dom­i­nance for $t < 0$, trans­form­ing to hy­per­bol­ic dom­i­nance for $t > 0$, after pass­ing through $\sqrt{2}$ in that process.

The $v_{j}$-ro­ta­tion vec­tors are not ar­bi­trary, being all number forms re­lat­ed to $\sqrt{i}$ , as men­tioned at the onset of the sec­tion. The number $e^{\frac{i\pi}{4}} = \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\ \right),$is close­ly re­lat­ed to the vector $v_{1} = \left( \frac{e_{1}}{\sqrt{2}} + \frac{e_{2}}{\sqrt{2}} \right).$What should not be over­looked is that the hand­ed­ness of the geo­met­ric prod­uct can result in ro­ta­tions in the op­po­site di­rec­tion. So, the ini­tial prod­uct eval­u­at­ed was $uv_{1}$, with $v_{1}$ on the right side, which ef­fec­tive­ly ro­tates $u$ in the in­verse di­rec­tion of the re­lat­ed ro­ta­tion $e^{\left( \frac{i\pi}{4} \right)}$, which is then ef­fec­tive­ly a ro­ta­tion of $u$ by$\ e^{\left( - \frac{i\pi}{4} \right)}.$

The $uv_{j}$ func­tions form the as­ymp­tot­ic cones having the fol­low­ing matrix rep­re­sen­ta­tions:

The top row of each cone matrix, de­not­ed $c_{j},$can take the para­met­ric func­tion form

and the bottom row, the form

The ori­en­ta­tion of the cone seg­ments can be seen in Figure 1.3.35, where the axes $e_{1}, - e_{2}, - e_{1},e_{2}$ are the re­spec­tive cone axes for $c_{1},c_{2},c_{3},c_{4}.$The cone ma­tri­ces all have de­ter­mi­nants equal to zero, which occurs since the vector arrange­ment mimics the dot prod­uct. Using the first cone matrix as an ex­am­ple, we ex­press the $uv_{1}$,$uv_{3}$ vec­tors, as

We now cal­cu­late the dot prod­uct of the $uv_{1}$,$uv_{3}$ vector com­po­nents

where the result equals zero, and this is con­sis­tent with the two or­thog­o­nal vec­tors. The geo­met­ric prod­uct $uv_{1}uv_{3}$ is there­fore, equal to the wedge prod­uct $uv_{1} \land uv_{3}$

where the wedge prod­uct is equal to the $uv_{2}$ vector squared. Now, we con­sid­er the vec­tors $C_{1}$ and $C_{1}^{'}$, which rep­re­sent the columns of the Cone 1 group

for which, the values for each vector rep­re­sent the column vec­tors from the $Cone\ 1$ matrix. Next, we add the two vec­tors to­geth­er

and the $e_{2}$ basis por­tions cancel, lead­ing to a vector sum equal to the $uv_{2}$ geo­met­ric prod­uct scaled by $\sqrt{2}$, ori­ent­ing with the $e_{1}$ axis

It is easily ver­i­fied that sim­i­lar vector sums for the other cones hold. Ob­tained as fol­lows:

Geo­met­ri­cal­ly, these sums rep­re­sent the di­ag­o­nals of the ori­ent­ed par­al­lel­o­grams, formed by the pair of vec­tors. We demon­strate this for the case of the $C_{1} + C_{1}^{'}$ vector sum, in the figure below.

the di­ag­o­nal of the ori­ent­ed par­al­lel­o­gram formed by the vec­tors.

The im­por­tance of these vector ad­di­tion re­sults is that $\sqrt{2}$ plays a cen­tral role in rel­a­tiv­i­ty and the per­cep­tion of the speed of light. It will be con­sid­ered in more detail toward the end of the book, but suf­fice it to say, $\sqrt{2}$ rep­re­sents the di­ag­o­nal length of the com­plex unit square, which plays a sig­nif­i­cant role as the con­stant for the com­plex unit cube's shear trans­for­ma­tions and main­tains this in­vari­ant area rep­re­sent­ing the speed of light.

The light cones and light axes result from a dy­nam­ic process of cir­cu­lar and hy­per­bol­ic geome­tries moving past one an­oth­er. These geome­tries should be con­sid­ered out of phase in the com­plex angle space, which was al­lud­ed to ear­li­er in the chap­ter and elu­ci­dat­ed by the simple ex­am­ple $e^{it} = e^{Bt}$, which holds for $B = e^{\left( \frac{i\pi}{2} \right)}.$The light as­ymp­totes have or­thog­o­nal re­la­tion­ships to the re­spec­tive as­ymp­totes ori­ent­ed a dis­tance $e^{\left( \pm \frac{i\pi}{2} \right)}$ away, as do the light axes. This phase dif­fer­ence, $e^{\left( \pm \frac{i\pi}{2} \right)}$, de­fines the or­thog­o­nal­i­ty in the com­plex plane for the light func­tions. The hy­per­bol­ic and cir­cu­lar planes can be viewed as ro­tat­ing past one an­oth­er in op­po­site di­rec­tions. This is shown in Figure 1.3.37, where the chosen rep­re­sen­ta­tion has the red tan­gent vec­tors of the $e^{it}$ cir­cu­lar geom­e­try with neg­a­tive/clock­wise ori­en­ta­tion, and the black tan­gent vec­tors of the $e^{t}$ hy­per­bol­ic geom­e­try with pos­i­tive/coun­ter­clock­wise ori­en­ta­tion. When the light func­tions are viewed in this manner, we can define the gen­er­al­ized uni­fied light func­tions:

De­f­i­n­i­tion (1.3.345) – The gen­er­al­ized spa­tial-tem­po­ral uni­fied light func­tions $\mathfrak{L}\left( \mathbf{t}, - \mathbf{\tau} \right)$, $\mathfrak{L}\left( \mathbf{t},\mathbf{\tau} \right)$ are de­fined, as

where the time-phased, $i\omega$, will be a mul­ti­ple of the as­ymp­tot­ic angle, $\pm \frac{i\pi}{4}$ ra­di­ans, the func­tions may also take on the spa­tial forms

Using the $\mathfrak{L}(t, - \tau)$ func­tion, we set $\theta = \frac{\pi}{4}$, and con­sid­er in­te­ger values for $\tau$, start­ing at $\tau = 0$ , then watch how the light system evolves

Ap­par­ent from the above, is the light system ro­tates in the clock­wise di­rec­tion with re­spect to the uni­fied light func­tion values for the as­ymp­totes and axes. The rep­re­sen­ta­tion of the light func­tions in Figure 1.3.37 is linked to the con­cept of torque in physics. Torque is typ­i­cal­ly de­fined as the mag­ni­tude of a tan­gen­tial force com­po­nent times the dis­tance, im­ply­ing the work done per unit angle. A torque vector $\mathbf{\tau}\mathbf{}$ is rep­re­sent­ed, as

for $\mathbf{r} \times \mathbf{F}$, the cross prod­uct be­tween a po­si­tion vector $\mathbf{r}$, orig­i­nat­ing at a point where torque is being mea­sured to the point that a force is being ap­plied, and a vector $\mathbf{F}$, rep­re­sent­ing the ap­plied force. Given the pres­ence of the cross prod­uct, it pro­duces a vector t­­hat is or­thog­o­nal to both $\mathbf{r}$ and $\mathbf{F}$, with an ori­en­ta­tion that fol­lows the right-hand rule. With the torque vector com­po­nents de­scribed, torque can be de­fined, as

for $\theta$, the angle be­tween the force vector $\mathbf{F}\mathbf{}$ and the po­si­tion vector $\mathbf{r}$. We can con­sid­er torque in terms of geo­met­ric al­ge­bra by con­sid­er­ing the po­si­tion vector $\mathbf{r}$ in the para­met­ric form,

for which, the unit bivec­tor in the plane is the imag­i­nary number $i$, and noting the iso­mor­phism $i \cong e_{1} \land e_{2} = e_{1}e_{2} = I$, lead­ing to the ex­pres­sion

In physics, the de­f­i­n­i­tion of work is the amount of energy trans­fer as­so­ci­at­ed with the move­ment of an object over some dis­tance, which is ac­com­plished by an ex­ter­nal force. For a system com­posed of a ro­tat­ing body about a fixed axis, we can define the work done $dW$ by a tan­gen­tial force $F_{\tan}$

The in­fin­i­tes­i­mal change in dis­tance can be ex­pressed in the radian mea­sure $ds = \mathbf{r}d\theta$, for $\mathbf{r}$, the radius vector, and $d\theta$, the in­fin­i­tes­i­mal dis­place­ment angle. We can rewrite the ex­pres­sion for work, as

for $F_{\tan}\mathbf{r}$, the torque$\ \tau$, re­sult­ing from the ex­ter­nal and tan­gen­tial force $F_{\tan}$, lead­ing to the iden­ti­ty

which can be solved for torque

In vector no­ta­tion, the ex­ter­nal force is ex­pressed as $\mathbf{F}$, which has a tan­gen­tial com­po­nent force acting on a ro­tat­ing body, as­sumed to be con­stant. We now state torque in terms of our vec­tors, as

where $\frac{dr}{d\theta}$ is the change of the $\mathbf{r}$ vector over an in­fin­i­tes­i­mal dis­place­ment angle

This en­ables us to ex­press torque using geo­met­ric al­ge­bra, as

and by ex­pand­ing the po­si­tion vector, we find

which we use to take the dot prod­uct with $\mathbf{F}$

In the above, we used the per­pen­dic­u­lar basis vector $e_{3}$ in the force vector $F_{\bot}e_{3}$, to be con­sis­tent with the cross prod­uct's ori­ent­ed normal vector. No­tably, the result is con­sis­tent with the wedge prod­uct, given that

and this wedge prod­uct $\mathbf{r} \land \mathbf{F}$ rep­re­sents the torque of a system with­out $\mathbf{r}$ and $\mathbf{F}$ ex­pressed as mag­ni­tudes per the tra­di­tion­al de­f­i­n­i­tion. This leads us to define the torque vector in geo­met­ric al­ge­bra no­ta­tion, as

Using the bivec­tors native to geo­met­ric al­ge­bra, we do away with the need for a normal axis of ro­ta­tion, which re­sults from the cross-prod­uct. This turns out to be more con­sis­tent with the frame­work of the com­plex plane and its ro­ta­tions, given that a normal axis to the plane of ro­ta­tion is not math­e­mat­i­cal­ly re­quired in the for­mal­ism. With torque de­fined in the con­text of geo­met­ric al­ge­bra, we apply it to the light func­tions de­scribed in this sec­tion. We con­sid­er the light system com­posed of two torques, by first, defin­ing two po­si­tion vec­tors

where we ab­stract that a po­si­tion in the geo­met­ric space de­fines a time-de­pen­dent ori­en­ta­tion of the com­plex unit circle and unit hy­per­bo­la, co­in­cid­ing with the pre­vi­ous­ly de­rived $uv_{2}$, $uv_{8}$ geo­met­ric prod­ucts. The ap­plied forces on each of the po­si­tion vec­tors are

having both mag­ni­tude and ori­en­ta­tion. Next, we cal­cu­late the in­di­vid­ual torque com­po­nents

which are de­pict­ed in Figure 1.3.38, as the ori­ent­ed bivec­tors $\mathbf{\mathrm{T}}_{1}\circlearrowright$ , $\mathbf{\mathrm{T}}_{2}\circlearrowleft$. These torque bivec­tors can be seen as equal in area but op­po­site in ori­en­ta­tion.

The net torque of the func­tion system is found by adding the torque com­po­nents, which eval­u­ate to

noting a result which is con­sis­tent with our pre­vi­ous­ly de­rived, $uv_{1}$, geo­met­ric prod­uct.

So, the ori­ent­ed torque bivec­tor result de­pends on the func­tion's dis­tance in com­plex space, which varies by the value of the time vari­able $t$. In­tu­itive­ly, this should make sense, given that the $\mathbf{r}$-vector rep­re­sents an ori­ent­ed dis­tance in the com­plex plane for which, a tan­gen­tial force acts on the form as­so­ci­at­ed with its vector head. Put in simple terms, $\mathbf{\mathrm{T}}_{1} + \mathbf{\mathrm{T}}_{2}$ is a torque func­tion with re­sults that orient in the com­plex plane. It should be noted, the $\mathbf{r}_{1}$, $\mathbf{r}_{2}$, dis­tance func­tion vec­tors have a de­fault torque that is par­tic­u­lar to their forms, which stems from the pres­ence of the com­plex unit circle in each. For in­stance, we take the $- {Ie}^{It}$ por­tion from the $\mathbf{\mathrm{T}}_{1}$ torque cal­cu­la­tion, to con­sid­er the form

in which, the sim­pli­fi­ca­tion shows that $e_{2}$ is the ori­en­ta­tion basis of the func­tion. The ex­po­nen­tial func­tion $e^{It}$ does have an an­gu­lar fre­quen­cy, which is simply uni­tary $\omega = 1$, and set­ting $\omega = \sin\left( \frac{\pi}{2} \right)$, the ori­ent­ed com­plex unit circle be­comes

So, a torque occurs in the ex­po­nent of the func­tion with an or­thog­o­nal­ly ori­ent­ed force, $\mathbf{F} = {e_{2}\sin}\left( \frac{\pi}{2} \right),$which is wedged with the unit radius basis vector $e_{1}$. The ex­po­nent por­tion of the com­plex unit circle can now be seen in terms of its torque re­la­tion­ship, as

which amounts to a nested torque lo­cat­ed in the ex­po­nent, rep­re­sent­ing work done over a dis­tance in time, man­i­fest­ed as the radian dis­tance in the tan­gent space. The ex­po­nen­tial func­tion $e^{t}$ has no an­gu­lar ro­ta­tion com­po­nent, and there­fore, does not have an innate torque like the com­plex unit circle. The reader should be aware that if we have $\mathbf{r}_{j}$-vec­tors, as fol­lows

being the same ab­solute dis­tance from the origin, when the same-valued forces are ap­plied

the in­di­vid­ual torque com­po­nents are found to be

This leads to a net-torque force that is equal to zero, seen by eval­u­at­ing the sum of torque com­po­nents

So, we can think of a net-zero torque as the geo­met­ric prod­uct be­tween two func­tion vec­tors, ori­ent­ed in an or­thog­o­nal basis in re­la­tion to one an­oth­er. For two or­thog­o­nal func­tion vec­tors ${ue}_{1}\bot ve_{2}$, with a chosen ori­en­ta­tion ${ue}_{1} \land e_{2} = v$ and ${ve}_{2} \land e_{1} = u,$the torque will be equiv­a­lent to the geo­met­ric prod­uct

If the net-torque

has ro­ta­tion forces $\mathbf{F}_{\mathbf{1}}$ and $\mathbf{F}_{2}$, that are equal value, this leads to a wedge prod­uct equal to zero

given we are wedg­ing a par­al­lel scaled vector along its basis vector. By set­ting, $\mathbf{F}_{1}\mathbf{=}\mathbf{F}_{2}$, by as­so­cia­tiv­i­ty, we have the iden­ti­ty

Eval­u­at­ing the geo­met­ric prod­uct $\left( ue_{1} + ve_{2} \right)\left( \mathbf{F}_{1}e_{2}\mathbf{+}\mathbf{F}_{2}e_{1} \right)$, we find

where the dot prod­uct por­tion is zero, given that $\mathbf{F}_{1}$ acts on $ue_{1}$ per­pen­dic­u­lar­ly, as does $F_{2}$ on $ve_{2}\mathbf{.}$ By eval­u­at­ing the net-torque $\mathbf{\mathrm{T}}= \mathbf{r} \land \mathbf{F}$, we find

which is equiv­a­lent to the geo­met­ric prod­uct of or­thog­o­nal vec­tors. We can fur­ther state by the de­fined or­thog­o­nal ori­en­ta­tion of the vec­tors $u,v$, that

which im­plies the geo­met­ric prod­uct equals

This primer on torque shows how it can be ap­plied to the frame­work of the geo­met­ric prod­uct. In­stead of being a scalar value, its mean­ing has been ex­tend­ed to that of a func­tion ori­ent­ed in the com­plex plane. We can math­e­mat­i­cal­ly con­cep­tu­al­ize the forces acting on the cir­cu­lar and hy­per­bol­ic ex­po­nen­tial func­tions in con­text to the uni­fied light func­tions. To show this, we take the tem­po­ral uni­fied light func­tion as the ex­am­ple

where we have em­pha­sized in bold, the $- i\theta\tau$ por­tion in the ex­po­nents. This rep­re­sents the forces acting on the func­tion com­po­nents $e^{i\omega t}$, $e^{\omega t}$, each serv­ing to create torque in the com­plex func­tion­al plane. This is ap­pre­ci­at­ed by defin­ing the func­tion­al dis­tance vector $\mathbf{r}$, as

and the force vector $\mathbf{F}$, is

Note, we have changed the sign of the ex­po­nent for $e^{- i\theta\tau}$, given the torque equa­tion has the force on the right-hand side. Using the ex­pres­sion $\mathbf{\mathrm{T}}= \mathbf{r} \land \mathbf{F}$, we cal­cu­late the func­tion­al geo­met­ric torque

find­ing the $\mathbf{F}_{1} = {Im(e}^{i\theta\tau})$ force acts on the func­tion $\left( e^{\omega t} - {Ie}^{I\omega t} \right)$, and $\mathbf{F}_{2} = Re\left( e^{i\theta\tau} \right)$ acts on the func­tion $\left( {Ie}^{\omega t} + e^{I\omega t} \right)$. We dis­trib­ute the re­spec­tive bivec­tors into the func­tions, find­ing that the net geo­met­ric torque equals

rep­re­sent­ing a sum of prod­ucts, with the real force com­po­nent matched to the $e_{1}$-axis func­tion­al and the imag­i­nary force com­po­nent matched to the $e_{2}$-axis func­tion­al. We set a pref­er­ence to the $e_{2}$-axis func­tion $\left( {Ie}^{\omega t}{\ + \ e}^{I\omega t} \right)$, to denote the geo­met­ric torque as

and given that

$\mathbf{\mathrm{T}}$ is equiv­a­lent to

which rep­re­sents the geo­met­ric al­ge­bra form of the uni­fied light equa­tions.

This sec­tion has served as our in­tro­duc­tion to light's geo­met­ric and dy­nam­i­cal prop­er­ties in the com­plex plane. The meth­ods re­flect­ed here will soon be ap­plied when we con­sid­er the higher-di­men­sion­al rep­re­sen­ta­tions of form.