\(2cos^2(\cfrac{\phi}{2})=cos(\phi)+1\lt cos(\phi)+v\)
when \(v\lt1\),
\(2cos^2(\cfrac{\phi}{2})\lt cos(\phi)+v\)
because of the power of two, \(\phi\) has a period of \(\pi\) and \(\cfrac{\phi}{2}\) has a period of \(\cfrac{\pi}{2}\)
Given, \(v\lt1\), \(\cfrac{\phi}{2}\lt \cfrac{\pi}{2}\) and let \(\theta=\cfrac{\phi}{2}\)
From the post "Moving Moving...Frames" dated 25 Nov 2017,
\(\theta\lt \cfrac{\pi}{2}\)
The \(x\) axis is marked by the zeros of \(cos^2(\theta)\), and \(x=cos(\phi)+v\) the zeros of \(cos(\phi)+1\) and \(cos^2(\theta)\) are the same. Since \(\theta\lt \cfrac{\pi}{2}\), it takes at least two \(\theta\)s for \(x\) to be along \(VO\) and \(x\) stands still. Two more \(\theta\)s and \(x\) is in the reverse direction \(OH\). \(x\) is a vector, it is pointing left. \(v\) varies with each turn of \(\theta\).
In this case \(\theta\) or \(\cfrac{\phi}{2}\) is changed by adding \(v\), a moving reference frame with velocity \(v\). If \(\theta\) is changed directly, as in the case of a pulsing light by changing its flash frequency, then \(\theta\) is not restricted to be less then \(\cfrac{\pi}{2}\). As \(\theta\) is increased from zero, \(x\) stands still when \(\theta=\cfrac{\pi}{2}\), and then seems to reverse at \(\theta=\pi\).
If \(x\) are marked by the blades of a fan swinging by periodically, first the blades seem to stand still and then reverses its direction, as the flash frequency increases.
This form of time travel/reversal, requires "presence" at the furthest point back in time, traveled to. This is the origin. Time reversal cannot be further back beyond this origin.
Which is just like making a video and then reversing it, except you have no participation; the reversed playback is not real.
In this case, the motion is periodic in the first place. The same motion is repeated over and over.
Just a strobe light...except for,
\(f_{\psi}=\cfrac{c}{2\pi a_\psi}=\cfrac{c}{2\pi*14.77}=3230699.3\,GHz\)
from the post "A Treasure What Worth?" dated 30 Jul 2016. \(a_{\psi}=14.77\,nm\) is the time particle exerting a placid field (post "The Placid Field" dated 30 Jul 2016). This frequency may just make it real.
Note: The direction of \(x\) is reversed from the post "Moving Moving...Frames" dated 25 Nov 2017, such that \(\theta\) is the angle the resultant vector (\(x+v\)) makes with \(x\). If the other direction of \(x\) is considered then the resultant angle will be \(\cfrac{\pi}{2}-\theta\) and the end result is the same.
Remember alien experiment where the candle flame freezes in the (time) field?
