Yesterday Once More Once More

It is possible to start with the assumption that time speed is a terminal velocity in free space related to space density by ${v}_{t}^{2}=F-D{ d }_{ s }(x)$ where $F, D$ are constants and ${ d }_{ s }(x)$ is a decreasing exponential with a value of ${d}_{n}$ the normal density of free space at $x\rightarrow\infty$.

F-De^(-x/S) Inverse Proportion Profile

${v}_{t}^{2}=F-D{ d }_{ s }(x)$

Dividing throughout by ${c}^{2}$,

${\gamma (x)}^{2} = \cfrac{{v}_{t}^{2}}{{c}^{2}}= \cfrac{F}{{c}^{2}}-\cfrac{D}{{c}^{2}}{ d }_{ s }(x)$

Since $x \rightarrow \infty$, ${d}_{s}(x)\rightarrow{d}_{n}$ and $\gamma (x)\rightarrow 1$

$1= \cfrac{F}{{c}^{2}}-\cfrac{D}{{c}^{2}}{ d }_{ n }$

In a black hole, space density is ${d}_{B}$ and ${v}_{t}$ = 0, so $\gamma (x)$ = 0

$0 = \cfrac{F}{{c}^{2}}-\cfrac{D}{{c}^{2}}{ d }_{ B }$

Solving for $\cfrac{F}{{c}^{2}}$ and $\cfrac{D}{{c}^{2}}$

$1= \cfrac{D}{{c}^{2}}{ d }_{ B }-\cfrac{D}{{c}^{2}}{ d }_{ n }$

$\cfrac{D}{{c}^{2}}= \cfrac{1}{{d}_{B}-{d}_{n}}$

$\cfrac{F}{{c}^{2}}=  \cfrac{{ d }_{ B }}{{d}_{B}-{d}_{n}}$

${\gamma (x)}^{2} =  \cfrac{{ d }_{ B }}{{d}_{B}-{d}_{n}}- \cfrac{{ d }_{ s }(x)}{{d}_{B}-{d}_{n}}$

${\gamma (x)}^{2} = 1+ \cfrac{{ d }_{ n }}{{d}_{B}-{d}_{n}} - \cfrac{{ d }_{ s }(x)}{{d}_{B}-{d}_{n}}$

As before,

${\gamma (x)}^{2} = 1- \cfrac{{ d }_{ s }(x)-{d}_{n}}{{d}_{B}-{d}_{n}}$

and so,

${\gamma (x)}^{2} = 1- \cfrac{{ d }_{ s }(x)-{d}_{n}}{{d}_{e}-{d}_{n}} \cfrac{{ d }_{ e}-{d}_{n}}{{d}_{B}-{d}_{n}}$

where $\cfrac{{ d }_{ e}-{d}_{n}}{{d}_{B}-{d}_{n}}$ is a constant.

Which bring clearly the assumption into light.  Unfortunately, it is in a black hole.  There is no reason why time speed should be a terminal velocity in free space.  If time is somehow related to photon and that photon is at terminal speed in free space, it will be consistent with the fact that photon is in a helical path through space and not a sinusoidal path confined in a one dimensional plane.  The latter requires an awkward force changing the direction of the photon, whereas the helical path suggest losses along the way in the medium of some density, that eventually the photon stops.  Not unless of course photons are magnetic monopoles, propelled by inherited magnetic and electrical properties of free space.

Time however, has never been proven to be related to light speed at all.  The fact that photons do not catch up with you does not mean time stopped.  If you are travelling away from a clock face at light speed, the clock face does not update, does not mean time stopped.