\(v_s=\dot{x}=c\)
\((\dot{x}-v_s).F=0\)
zero. This is the limiting case, when the power output from the particle is tangential to the surface of the sphere, perpendicular to a radial line. It is as if power is being shaved from the surface of the sphere by an observer standing in relative speed \(\dot{x}-v_s\) just above the surface.
Power output slows and remains around the particle could explain the increase in energy detected around spinning plasma. Such power is directly proportional to the spin velocity \(v_s\) of the plasma, on the equatorial plane of the particle,
\(P_{rad}=(\dot{x}-v_s).F\)
By conservation of energy,
\(P_{\bot}=v_sF\)
which is consistent with the limiting case of \(v_s\to c\),
\(P_{bot}\to P=c.F\)
with \(\dot{x}=c\)
where the power output is tangential to a radial line. Furthermore,
\(\overset {\rightarrow }{ \cfrac{ P_{\bot} }{v_s}}+\overset {\rightarrow}{\cfrac{P_{rad}}{(c-v_s)}}=\overset{\rightarrow }{\cfrac{P}{c}}\)
where the power vector \(\overset{\rightarrow }{\cfrac{P}{\dot{x}}}\) takes the direction of \(\dot{x}\), in view of the fact that,
\(F.\dot{x}=\cfrac{d\,KE}{dt}\)
\(F=\cfrac{1}{\dot{x}}\cfrac{d\,KE}{dt}\)
\(F\) is not a force, but a power vector.
But,
\(\overset {\rightarrow }{ \cfrac{ P_{\bot} }{v_s}}=\overset {\rightarrow}{\cfrac{P_{rad}}{(c-v_s)}}=\overset{\rightarrow }{\cfrac{P}{c}}=F\)
\(\left|\overset {\rightarrow }{ F}\right|+\left|\overset {\rightarrow }{ F}\right|=\sqrt{2}\left|\overset {\rightarrow }{ F}\right|\)
since the total output power is a constant, we have instead,
\(\cfrac{1}{\sqrt{2}}\left|\overset {\rightarrow }{ F}\right|+\cfrac{1}{\sqrt{2}}\left|\overset {\rightarrow }{ F}\right|=\left|\overset {\rightarrow }{ F}\right|\)
both expressions suggest that there can be a reduction by a factor of \(\frac{1}{\sqrt{2}}\) as the power vectors sum vectorially, in the time dimension. Where did this lost energy go? Into the space dimension from which we harness. So, as we manipulate mass in the time dimension \(\frac{1}{c}\), by manipulating light speed \(\dot{x}=c\) in the space dimension, we can gain energy in the space dimension. However, \(c\) is a constant, as the example illustrates, \(c\) is manipulated by adding an orthogonal component to \(c\). In this case the spin velocity \(v_s\).
The maximum lost of power in the time dimension is at \(\cfrac{1}{\sqrt{2}}P\) when \(v_s=\cfrac{c}{2}\), and so the maximum power available to the space dimension is \(\left(1-\frac{1}{\sqrt{2}}\right)P\) when \(v_s=\cfrac{c}{2}\).
If there is ever a bullshit alert this will surely sets it off loudly.
Good night.
