2π.tct=c
2π.tc=ct
2π∂tc∂t=c
∂tc∂t=c2π
In a similar way and considering the direction of these dimensions,
∂tT∂t=−ic2π
∂tg∂t=ic2π
And
∂x∂t=∂x∂tc∂tc∂t+∂x∂tg∂tg∂t+∂x∂tT∂tT∂t --- (*)
∂x∂t=c2π{∂x∂tc+i∂x∂tg−i∂x∂tT}
If all time dimensions are equivalent,
∂x∂tc=∂x∂tg=∂x∂tT
∂x∂t=c2π∂x∂tc
We also have
(∂x∂t)2=(∂x∂tc)2+(∂x∂tg)2+(∂x∂tT)2
∂x∂t=√3∂x∂tc
This is possible only if,
∂x∂t=∂x∂tc=0
At this point we can swap x for tc, since they are both dummy variables and conclude by symmetry that when ∂x∂t=c2π, ∂tc∂t=0
However, for the fun of it,
When x wraps around tc,
2π.xt=c
2π.x=ct
2π∂x∂t=c
and,
2π∂x∂tc=c∂t∂tc
So,
∂x∂t=∂x∂tc∂tc∂t --- (***)
Since,
∂x∂tc=∂x∂tg∂tg∂tc+∂x∂tT∂tT∂tc
As changes in x and tc does not effect tg and tT, and as far as x is concern tg and tT are equivalent, !?!
ie.
∂x∂tc∂tc∂t=c2π --- (**)
and
∂x∂tc∂tc∂tg=2∂x∂tg
From which we formulate,
∂x∂tc∂tc∂tg∂tg∂t=2∂x∂tg∂tg∂t --- (1)
Similarly,
∂x∂tc∂tc∂tT∂tT∂t=2∂x∂tT∂tT∂t --- (2)
Therefore (1)+(2),
∂x∂tc{∂tc∂tg∂tg∂t+∂tc∂tT∂tT∂t}=2{∂x∂tg∂tg∂t+∂x∂tT∂tT∂t}
Substitute (**) into the above,
c2π{∂tc∂tg∂tg∂t+∂tc∂tT∂tT∂t}=2∂tc∂t{∂x∂tg∂tg∂t+∂x∂tT∂tT∂t}
But,
∂x∂t=∂x∂tc∂tc∂t+∂x∂tg∂tg∂t+∂x∂tT∂tT∂t
So,
c2π{∂tc∂tg∂tg∂t+∂tc∂tT∂tT∂t}=2∂tc∂t{∂x∂t−∂x∂tc∂tc∂t}
But from (***),
∂x∂t=∂x∂tc∂tc∂t
So,
c2π{∂tc∂tg∂tg∂t+∂tc∂tT∂tT∂t}=0
But,
∂tc∂t=∂tc∂tg∂tg∂t+∂tc∂tT∂tT∂t
Thus,
c2π∂tc∂t=0
∂tc∂t=0
Blasphemy, I know. The point is,
when ∂x∂t=c2π, ∂tc∂t=0 and when ∂tc∂t=c2π, ∂x∂t=0 are consistent.
The factor of 12π appearing before c is the result of circular motion around the orthogonal axis. A particle traveling in circular motion in space about the time axis is not traveling in time, so time speed equals zero. Similarly, the same particle traveling in circular motion in time about the space axis has zero space speed.
This relationship can be rewritten as,
v2+v2t=(c2π)2 --- (+)
where v is the particle velocity in space, vt is the particle velocity in time and c a constant. Pythagoras' Theorem applies because v is perpendicular to vt. Obviously expression (+) satisfies the boundary value conditions.
More importantly, the time speed limit occurs at c2π not at c, because the particle goes into circular motion.