Deja Vous Change

In view of the previous post "Like Doppler" dated 15 Jan 2018, the expression,

$\Delta t*v_s=\cfrac{v_{boom}}{c}*v_s=S$ --- (*)

from the post "Light Speed Back In Time" also dated 15 Jan 2018, might change to,

$\Delta t*v_s=\left(1-\cfrac{v_{boom}}{c}\right)*v_s=S$ --- (**)

The former expression (*) suggests an absolute change in $t$ and the latter expression (**) might suggest a relative change in $t$ with respect to the photon that remains at light speed $c$.  But consider this;

if

$1-\cfrac{v_{boom}}{c}=\cfrac{\Delta}{\text{refer.}}$

$\cfrac{v_{boom}}{c}=\Delta$

then,

$\text{refer.}=\cfrac{\cfrac{v_{boom}}{c}}{1-\cfrac{v_{boom}}{c}}=\cfrac{v_{boom}}{c-v_{boom}}$

Which reminds of the expressions,

$E=mc\int ^{c}_{v_{boom}}{ dv }$

quoted in the post "Like Doppler", originally from the post "No Poetry for Einstein" dated 06 Apr 2014;  and,

$E=mc\int ^{c}_{c-v_{boom}}{ dv }$

This implies that $\text{refer.}$ is with reference to the a photon at speed $c-v_{boom}$ instead of $c$.  ie the situation when $v_{boom}$ flips sign.  Given $v_{boom}$, numerically, it is possible that,

$c-v_{boom}$  and $c+v_{boom}$

that the torus collapse to a photon going in the opposite direction.

Expression (*) should be used, where $c$ is the reference.