1.5.9 Subtangent Vectors

John Still­well¹ has a nice ap­proach to in­tu­itive­ly de­scribe the tan­gent vec­tors of a Lie group as "ve­loc­i­ty vec­tors", for points moving smooth­ly through the iden­ti­ty. We take this ve­loc­i­ty vector de­scrip­tion, and expand upon it by in­clud­ing sub­tan­gent vec­tors, which can be de­scribed as "ac­cel­er­a­tion vec­tors" for points moving smooth­ly through the iden­ti­ty. This notion of "ac­cel­er­a­tion" can be thought of as a pulling to­wards the origin of the groups lo­cat­ed on the sur­face of the sphere of tan­gent spaces, and this pull is a man­i­fes­ta­tion of the space of angles. To con­sid­er the re­la­tion­ship be­tween the tan­gent and sub­tan­gent spaces and how they relate, we refer to el­e­ments of the sub­tan­gent space by $u\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, and the tan­gent space $t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$,

The matrix group gen­er­a­tors are de­rived by way of the ex­po­nen­tial map­ping, $\exp\left( t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right) \right) \rightarrow e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)}.$Two vari­ables, $u,t$, now exist for the tan­gent space $t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)$, in ad­di­tion to the angle ar­gu­ment $\theta_{j}$. Matrix el­e­ment de­riv­a­tives with re­spect to $t,u,$co­in­cide with the re­sults from the prior the­o­rem, where $\frac{d}{dt}$ of any matrix entry $\in e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)}$, for $t = 0$, rep­re­sents the ex­po­nen­ti­at­ed sub­tan­gent space, which is equiv­a­lent to the tan­gent space el­e­ment

Using the matrix group gen­er­a­tor-type $\mathcal{X}_{1}^{\infty} \in S_{3\mathbb{C}}^{\infty}$, we have the group gen­er­a­tor

Taking the de­riv­a­tive with re­spect to $t$ of $e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)}$, and for $t = 0$, we find

We con­tin­ue by eval­u­at­ing the de­riv­a­tive $\frac{d}{du}\left( \frac{d}{dt}e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)} \right)$ , for $u = 0$

find­ing that $\frac{d^{2}}{dudt}\left( e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)} \right)$, rep­re­sents the ex­pect­ed sub­tan­gent el­e­ment $\mathcal{T}\left( \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( \mathcal{X}_{1} \right) \right)$

Other de­riv­a­tives con­sid­ered are:

of which the first, $\frac{d^{2}}{dt^{2}}\left( e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)} \right)$ is con­sis­tent with the matrix group ex­po­nen­tial $e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)}$ being phase-shift­ed by $2u\left( i\theta_{j} \right)$ sub­tan­gent ra­di­ans on the sphere of tan­gent spaces, while the second and third ex­am­ples, $\frac{d}{du}\left( e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)} \right)$, $\frac{d^{2}}{du^{2}}\left( e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)} \right)$, are in­ter­est­ing by way of their ap­pli­ca­tion to group cir­cu­lar motion. As the radius is con­stant for the sphere of tan­gent spaces, we have that $\frac{d}{dR},\frac{d^{2}}{dR^{2}} = 0$, for $R = 1$. We define, $z = Re^{te^{u\left( i\theta_{j} \right)}}$, having the ve­loc­i­ty $v = \ \dot{z}$

for $\omega_{\mathfrak{T}} = \ t\theta_{j}e^{u(i\theta_{j})}$, the an­gu­lar ve­loc­i­ty of the tan­gent space, and $z$, the ex­po­nen­tial of the tan­gent space $Re^{te^{u\left( i\theta_{j} \right)}}$. We can also define ${\dot{\theta}}_{\mathfrak{T}}(u) = {t\theta}_{j}e^{u(i\theta_{j})}$, lead­ing to

Since ac­cel­er­a­tion is the de­riv­a­tive of ve­loc­i­ty, $a = \dot{v} = \frac{d^{2}z}{du^{2}}$, it im­plies that for $a$, we have

From the ve­loc­i­ty ex­pres­sion, $\omega_{\mathfrak{T}} = t\theta_{j}e^{u\left( i\theta_{j} \right)}$, ${\dot{\omega}}_{\mathfrak{T}}$ will equal

and the ve­loc­i­ty squared $\omega_{\mathfrak{T}}^{2}$ is

Plug­ging the ${\dot{\omega}}_{\mathfrak{T}},\omega_{\mathfrak{T}}^{2}$ terms into the ex­pres­sion for ac­cel­er­a­tion, the result match­es the second de­riv­a­tive $\frac{d^{2}}{du^{2}}\left( e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)} \right)$, for R = 1

which is the math­e­mat­i­cal for­mal­ism rep­re­sent­ing the matrix group ac­cel­er­a­tion for $S_{3\mathbb{C}}^{\infty}.$We are mind­ful that the matrix el­e­ment cen­tripetal ac­cel­er­a­tion term $- \omega_{\mathfrak{T}}^{2}$ is ra­di­al­ly ori­ent­ed inward and is phase-shift­ed $2u\left( i\theta_{j} \right)$ ra­di­ans com­pared to $e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right)}$. The tan­gen­tial com­po­nent matrix el­e­ment $i{\dot{\omega}}_{\mathfrak{T}}$ is also ra­di­al­ly ori­ent­ed inward, $i{\dot{\omega}}_{\mathfrak{T}} = - t\theta_{j}^{2}e^{u\left( i\theta_{j} \right)}$, which dif­fers from the analog in cir­cu­lar me­chan­ics on the com­plex unit circle, and $i{\dot{\omega}}_{\mathfrak{T}}$ is only $u\left( i\theta_{j} \right)$ phase-shift­ed com­pared to$\ \omega_{\mathfrak{T}}^{2}$. To rep­re­sent the fa­mil­iar re­la­tion­ships be­tween ve­loc­i­ty and ac­cel­er­a­tion in cir­cu­lar motion, we us the vari­able $\tau$ to rep­re­sent both vari­ables $u,t\in \mathbb{R}$, for $e^{\left( {te}^{i{u\theta}_{j}} \right)}$

and solve for $\tau$

The ve­loc­i­ty com­po­nent can now be found by cal­cu­lat­ing $\frac{d}{d\tau}\left( e^{i\tau\theta_{j}} \right)$

and since ac­cel­er­a­tion is $a = \dot{v}$, we take the de­riv­a­tive $\frac{dv}{d\tau}$ to find the fa­mil­iar ac­cel­er­a­tion form

An im­por­tant con­sid­er­a­tion re­volves around how to define time for the matrix groups, tan­gent spaces, and sub­tan­gent spaces. The matrix ex­po­nen­tial $e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)}$, con­sid­ered ear­li­er in the sec­tion, had the vari­able $t$, which was in­sert­ed into the ex­po­nen­tial func­tion's ar­gu­ment, form­ing a prod­uct with the tan­gent space t$\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)$. When viewed in the matrix group rep­re­sen­ta­tion, $e^{t\mathfrak{T}\left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty},\left( X_{1} \right) \right)}$ equals

for $t$, present in the ex­po­nen­tial and trigono­met­ric ar­gu­ments. Typ­i­cal­ly, a time vari­able $t$, when mul­ti­plied by ve­loc­i­ty $vt$, gives us a dis­tance trav­eled in time, where the mag­ni­tude of the dis­tance de­pends on $t$. From the stand­point of group theory, a dis­tance trav­eled cor­re­lates to the mul­ti­pli­ca­tion op­er­a­tion, play­ing out as a group making a prod­uct with itself, the prod­ucts be­tween two or more dis­tinct groups, and the group matrix prod­ucts with co­or­di­nate vec­tors. The mul­ti­pli­ca­tion op­er­a­tion ad­vances the dy­nam­ics of any system com­prised of group gen­er­a­tors and its co­in­cid­ing vector el­e­ments. Time should be seen as math­e­mat­i­cal­ly cal­cu­la­tive in nature, pro­ceed­ing in the matrix group sense by enu­mer­a­tive mul­ti­plica­tive op­er­a­tions. To demon­strate, we rewrite $e^{t\mathfrak{T}\left( \left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right),\left( X_{1} \right) \right)}$ in the equiv­a­lent form

This form rep­re­sents the means for which the simple system $e^{\mathfrak{T}\left( \left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right),\left( X_{1} \right) \right)^{t}}$ pro­ceeds in cal­cu­la­tive time, oc­cur­ring it­er­a­tive­ly for $t$, rep­re­sent­ing the enu­mer­a­tive value for which, $e^{\mathfrak{T}\left( \left( u \times \mathfrak{s}_{3\mathbb{C}}^{\infty} \right),\left( X_{1} \right) \right)}$ has mul­ti­plied itself. This con­cept is im­por­tant since it gets to bottom of how time should be con­sid­ered in terms of its math­e­mat­i­cal con­struct. As it per­tains to the in­fin­i­ty sym­me­try group $S_{3\mathbb{C}}^{\infty}$, cal­cu­la­tive time moves for­ward for prod­ucts com­prised of pos­i­tive angle ar­gu­ments, and back­ward for neg­a­tive. Re­gard­ing a for­ward and back­ward dis­tinc­tion for the di­rec­tion of cal­cu­la­tive time, it is in­ter­est­ing to ponder the re­ver­sal of cal­cu­la­tive time's arrow, for $- t$, in the fol­low­ing forms

1. Still­well, John, "Naive Lie Theory", Springer 2008