1.3.10 The Curvature of Light

We end the chap­ter by con­sid­er­ing the cur­va­ture of the as­ymp­tot­ic light forms. Given how these curves appear flat in their para­met­ric plots, we expect to find that their cur­va­ture equals zero. To eval­u­ate the merits of this ex­pec­ta­tion, we first take the cir­cu­lar light func­tion

and eval­u­ate its polar form, by first, con­sid­er­ing the radial com­po­nent from $f(t) = r(t)e^{i\phi(t)}$

where we see the pa­ra­me­ter in time changes the func­tion's radius. Next, we cal­cu­late the in­verse tan­gent func­tion to see how the radial el­e­ment ro­tates in time, ac­com­plished by noting the iden­ti­ty

to then con­sid­er the ex­pres­sion

set­ting $\theta = 1$, in the pre­vi­ous iden­ti­ty, to then solve for $\tan(1)$

Sub­sti­tut­ing this result back into the polar form, we find

show­ing that the radial com­po­nent of $f(t)$ does not rotate in the com­plex plane, main­tain­ing a line with the angle $e^{\left( \frac{i\pi}{4} \right)}$ from the real-axis. In polar co­or­di­nates, we can define the signed cur­va­ture for a curve in the com­plex plane, as

rep­re­sent­ing the ratio of an in­fin­i­tes­i­mal change of the an­gu­lar ro­ta­tion over an in­fin­i­tes­i­mal change in the radius in time. In the case of $f(t)$, $\phi(t)$, this will equal

and since it is not time-de­pen­dent, it is a con­stant whose de­riv­a­tive is equal to zero

im­ply­ing, as ex­pect­ed, the signed cur­va­ture for $f(t)$ equals zero

Next, we con­tem­plate the cur­va­ture of the as­ymp­tot­ic para­met­ric curve $\gamma(t)$

which has the fol­low­ing first and second-order de­riv­a­tives

The in­fin­i­tes­i­mal dis­tance rep­re­sen­ta­tion with re­spect to time for our com­plex pa­ram­e­triza­tion is the two-norm

and the com­plex-valued tan­gent func­tion ${itan}\left( \phi(t) \right)$, rep­re­sents

which, by way of the in­verse tan­gent func­tion $i\phi(t)$, is equal to

We im­plic­it­ly dif­fer­en­ti­ate

find­ing $\partial t\tan\left( \phi(t) \right)$ equals

By sub­sti­tut­ing, $\tan\left( \phi(t) \right) = \left( \frac{Im\left( y^{'}(t) \right)}{x^{'}(t)} \right)$, $\frac{d\phi}{dt}$ equals

lead­ing to pa­ram­e­trized signed cur­va­ture result

We sub­sti­tute the values for the first and second-order de­riv­a­tives for $\gamma(t)$ into the signed cur­va­ture for­mu­la

and sim­pli­fy, find­ing once again that the cur­va­ture is equal to zero for the as­ymp­tot­ic para­met­ric curve in the com­plex plane

Next, we can con­sid­er the geo­met­ric prod­uct which ro­tates the unit hy­per­bo­la to the as­ymp­tote, by first, defin­ing the vec­tors $u$, $v$, as

with com­plex basis vec­tors equal to

We cal­cu­late the geo­met­ric prod­uct $uv$, as fol­lows

This result can be put into a polar form con­sis­tent with $f(t) = r(t)e^{i\phi(t)}$, as

where this func­tion's radius will vary in time, but the rate of change of the angle this radial com­po­nent makes with the real-axis is a con­stant, lead­ing to the straight­for­ward con­clu­sion, the cur­va­ture of $f(t)$ at all points is also zero, given that

We con­sid­er the final ex­am­ple of light cur­va­ture, by eval­u­at­ing a $\mathfrak{L}(t, - 1)$-type uni­fied light func­tion,

which can be sep­a­rat­ed into real and imag­i­nary com­po­nents

Next, by defin­ing the para­met­ric form $\mathfrak{L}(t, - 1)$, as

$\gamma(t)$ has the fol­low­ing first and second de­riv­a­tives

The in­fin­i­tes­i­mal dis­tance rep­re­sen­ta­tion with re­spect to time for our com­plex pa­ram­e­triza­tion is the two-norm

and using the values from $\mathfrak{L}(t, - 1)$, $\frac{ds}{dt}$ equals

The com­plex-valued tan­gent func­tion ${itan}\left( \phi(t) \right)$, rep­re­sents

which im­plies the in­verse tan­gent func­tion of ${itan}\left( \phi(t) \right)$, equals $\phi(t)$

We plot the ${itan}\left( \phi(t) \right)$ func­tion in Figure 1.3.39, which re­veals that for $t < 0,$the graph ap­pears tan­gent-like. For in­creas­ing values of $t \in \mathbb{R}^{+}$, the curve ap­pears flat due to the ex­po­nen­tial func­tion $e^{t}$ in the nu­mer­a­tor and de­nom­i­na­tor. This flat­ness im­plies very little change occurs for the ${itan}\left( \phi(t) \right)$ func­tion for large $t \in \mathbb{R}^{+}.$

Con­tin­u­ing, we use im­plic­it dif­fer­en­ti­a­tion to find

first, eval­u­at­ing for

Next, we find that $\partial t\left( \tan\left( \phi(t) \right) \right)$ equals

and by com­bin­ing both parts as a prod­uct,$\ \frac{d\phi}{dt}$ equals

Using the signed cur­va­ture for­mu­la

we take the prod­uct of the in­verse of the $\frac{ds}{dt}$ func­tion with $\frac{d\phi}{dt}$, find­ing the signed cur­va­ture

which is plot­ted in Figure 1.3.40, as com­pared to the ${itan}\left( \phi(t) \right)$ func­tion. Due to the dom­i­nance of the ex­po­nen­tial func­tion $e^{t}$ in the ex­pres­sion for $k$, the cur­va­ture rapid­ly ap­proach­es zero for large values of $t \in \mathbb{R}^{+}$. It is also seen that for neg­a­tive values of $t \in \mathbb{R}^{-}$, the func­tion rapid­ly con­verges to a cur­va­ture of one, which is con­sis­tent with the wave-like char­ac­ter­is­tics and cur­va­ture of the unit circle.

in the $\gamma(t)$ para­met­ric form, along with the ${itan}\left( \phi(t) \right)$ func­tion.

Fi­nal­ly, we verify that the cur­va­ture result is con­sis­tent with the com­plex para­met­ric signed cur­va­ture ex­pres­sion

by sub­sti­tut­ing the de­riv­a­tive values into the $k$ ex­pres­sion

find­ing the cur­va­ture re­sults are equiv­a­lent.

The analy­sis per­formed in this sec­tion on the cur­va­ture of the as­ymp­tot­ic light func­tions is con­sis­tent with the ex­pec­ta­tions that these func­tions have flat cur­va­ture. Most of the as­ymp­tot­ic light forms are ab­solute in zero cur­va­ture, while the uni­fied light func­tions, when taken as sole func­tions, rapid­ly con­verge to zero cur­va­ture, for $t \in \mathbb{R}^{+}$.