where r is very small. tg is at light speed, v=c. If it is possible to slow down time,
∂tgf∂t<c (in ticks per second)
such that,
∂tgf∂t=rω
r=1ω∂tgf∂t<cω
where both ω and r are physically manageable number, consider the equation from the post "Time Travel Made Easy" again,
∂tgf∂t=1mc2(∂tgi∂tEΔh+tgi∂EΔh∂t)+∂tgi∂t ---(*)
∂tgi∂t=c
∂tgf∂t=1mc2(cEΔh+tgi∂EΔh∂t)+c
tgi∂EΔh∂t=mc2∂tgf∂t−cEΔh−mc3
tgi=2πaψnc
aψn∂EΔh∂t=12π{mc3∂tgf∂t−c2EΔh−mc4}
aψn=12π{−mc4(∂EΔh∂t)−1+mc3∂tgf∂t(∂EΔh∂t)−1−c2EΔh(∂EΔh∂t)−1}
Since,
∂EΔh∂t<0
aψn=12π{mc4(|∂EΔh∂t|)−1−mc3∂tgf∂t(|∂EΔh∂t|)−1+c2EΔh(|∂EΔh∂t|)−1}
replacing aψno=12πmc4(|∂EΔh∂t|)−1
aψn=aψno{1−1c∂tgf∂t+1mc2∂EΔh∂tt}
From (*)
∂tgf∂t=1mc2(∂tgi∂t∂EΔh∂tt+∂tgi∂tt∂EΔh∂t)+∂tgi∂t
∂tgf∂t={2tmc2∂EΔh∂t+1}∂tgi∂t --- (**)
So,
aψn=aψno{1−{2tmc2∂EΔh∂t+1}+1mc2∂EΔh∂tt}
aψn=aψno{−tmc2∂EΔh∂t}
aψn=12πc2.t
The particle travels at a forward velocity of,
vtele=12πc2
Is this speed possible? Yes, time speed has been slowed down, that implies greater than light speed space travel. However we are NOT traveling through space this way.
At time t=0−, just before t=0,
aψn,t=0=aψno{1−1c∂tgf∂t+1mc2∂EΔh∂tt0}
aψn,t=0=aψno{1−1c∂tgf∂t}
in this case, ∂tgf∂t≠0, and the teleported distance is aψnt=0. As with the case considered in the post "Teleportation", with
∂tgf∂t=constant
the particle return velocity is,
aψno1mc2∂EΔh∂tt=−12πc2.t=vtele.t
vtele=−12πc2
At t>0 however,
vtele=12πc2
It seems paradoxical that the particle can be at different locations and have different velocities just before and after t=0. The trick here is,
∂tgf∂t={2tmc2∂EΔh∂t+1}∂tgi∂t = constant
The expression ∂EΔh∂t.t is held constant such that
∂tgf∂t={2tmc2∂EΔh∂t+1}∂tgi∂t
from (**) is a constant. This means
∂EΔh∂t=At
where A is a constant.
This way, only the situation for t=0− applies and as will be discussed below t≯0.
Simple, we beam the particle through another portal that induces −∂EΔh∂t. After passing through the second portal, ∂EΔh∂t=0. Equation (*) collapses to,
∂tgf∂t=∂tgi∂t
as at t=0, EΔh=0.
And the particle does not return from x=aψn,t=0, t≯0.
Since time, tg is curled around x where itg=x as required in the post "Time Travel Made Easy". x is perpendicular to the circular plane containing tg. The portal is directional. This sort of teleportation unfortunately is restricted to line of sight. Unless we can draw a straight line between the two points, teleportation is not possible.
And the billion dollar question is, how to induce ∂EΔh∂t=At in a particle?