Time Speed Again, Two Pie Or Not Two Pie

Consider again,

Let,

$r.\Delta \theta = k.\Delta t_c$

$\omega _{ c }=\cfrac { \partial \, \theta  }{ \partial \, t_{ c } } =\cfrac{k}{r}$

$\omega =\omega _{ c }\cfrac { \partial \, t_{ c } }{ \partial \, t } =\cfrac { \partial \, \theta  }{ \partial \, t_{ c } } \cfrac { \partial \, t_{ c } }{ \partial \, t } =\cfrac{k}{r} \cfrac { \partial \, t_{ c } }{ \partial \, t }$

$\left( \omega -\cfrac { k }{ r }  \right) \cfrac { \partial \, t_{ c } }{ \partial \, t } =0$

As, $r\rightarrow\infty$,   $k\ne0$

$\omega\cfrac { \partial \, t_{ c } }{ \partial \, t } =0$

since,

$\omega\ne0$

$\cfrac { \partial \, t_{ c } }{ \partial \, t } =0$

This implies that as time stretches out, time speed is zero.

when $r\ne\infty$,

$2\pi r=kt_{\lambda}$

$2\pi rf=kt_{\lambda}f=k\cfrac { \partial \, t_{ c } }{ \partial \, t }$

$\cfrac { \partial \, t_{ c } }{ \partial \, t }=\cfrac { 2\pi rf }{ k }=\cfrac { r\omega }{ k }=\cfrac{c}{k}$

What about,

$t_{ c }=2\pi r$

$\cfrac{t_c}{t}= \cfrac { 2\pi r }{ t } =c$

$\cfrac { \partial \, t_{ c } }{ \partial \, t } =2\pi \cfrac { \partial \, r }{ \partial \, t }=c$

This is wrong because $r$ for a given time speed is fixed.  Time $t_c$, will always coil around $x$ at a fixed $r$, gvien $\cfrac{\partial\,t_c}{\partial\,t}$,

$\cfrac{\partial\,r}{\partial\,t}=0$

$r$ does not increase and the time coil does not expand as time progresses.

Time is in a helix with a phase associated with its position on its circular path along its axis of travel.  If time returns to its original phase at a distance $t_{\lambda}$ ahead, after time period $T$,

$2\pi r=kt_{\lambda}$

$\cfrac { 2\pi r }{c}=T=\cfrac{k}{c}t_{\lambda}$

$t_{\lambda}=\cfrac{c}{k}T$

so, time linear speed $v_t$ is,

$v_t=\cfrac{c}{k}$

Time speed $v_t$, is not light speed, $c$.