I posted this on my old forum in case someone there can answer my questions on circular motion and SR.
I was introduced to the new wrinkle of circular motion that no one can answer my questions to. There are two variants to this circular motion scenario:
One is the relative velocity and resulting time dilation of the circumference of a centrifuge to its center. This is similar to the HafelKeating Experiment (HKX) if there was a guy at the center of the Earth conducting it. The other is the HKX where two planes leave an airport at the north pole in opposite longitudinal orbits around the Earth neglecting gravitational effects.
Now on this forum we discussed closing speed as not being relative velocity. An example of closing speed would be orbits parallel to yours on Earth where the orbiting satellite maintains the same distance from you. There is no linear vector between you and the satellite so the relative velocity between the two of you should always be zero. This should be the same relative velocity between a rotational centrifuge and its center. But here is some irrefutable math from John Rennie on the PSX who proves that assertion not true.
"Suppose you're whirling about a pivot with velocity v at a radius and I'm watching you from the pivot. I'm going to measure your position using polar coordinates (t,r,θ,ϕ), and in polar coordinates the line interval is given by (I'm leaving c in the equation this time):
ds2=−c2dt2+dr2+r2(dθ2+sin2θdϕ2)ds2=−c2dt2+dr2+r2(dθ2+sin2θdϕ2)
Note that this is just the flat space, i.e. Minkowski metric, in polar coordinates. We're using the flat space metric because there are no masses around to curve spacetime (we'll assume you and I have been on a diet
. We can choose our axes so you are rotating in the plane θ=π/2θ=π/2, and you're moving at constant radius so both dr and dθ are zero. The metric simplifies to:
ds2=−c2dt2+r2dϕ2ds2=−c2dt2+r2dϕ2
We can simplify this further because in my frame you're moving at velocity v so dϕ is given by:
dϕ=vrdtdϕ=vrdt
and therefore:
ds2=−c2dt2+v2dt2=(v2−c2)dt2ds2=−c2dt2+v2dt2=(v2−c2)dt2
In your frame you're at rest, so ds2=−c2dt′2ds2=−c2dt′2, and equating this to my value for ds2 gives:
−c2dt′2=(v2−c2)dt2−c2dt′2=(v2−c2)dt2
or:
dt′2=(1−v2c2)dt2dt′2=(1−v2c2)dt2
or:
dt′=dtγ
which you should immediately recognise as the usual expression for time dilation in SR. Note that the centripetal force/acceleration does not appear in this expression. The time dilation is just due to our relative velocities and not to your acceleration towards the pivot."
One of the blowhards on my new forum who doesn't understand that reciprocal time dilation is not the same as permanent age difference caused by the twin paradox, insists that for every revolution, the time difference between the center and the guy on the circumference is accumulating. Basically the centrifuge has become a Jules Verne time machine.
So I drew a Minkowski diagram (Md) to support reciprocal time dilation between the two but it fell apart when I tried to stop the centrifuge to compare clocks to establish permanent age difference between the two. The fact remained no matter how many orbits the centrifuge performed, when stopped, the only separation between them was always just the length of the radius. In normal linear twin paradox, the total age difference greatly increases the farther the separation between them when a change in velocity is made. So I asked the blowhard to draw me an Md that supported his theory. Of course these philosophers have no way to back up their opinions with math so he left in a huff calling me a mentally ill liar. I got an even worse reaction on the PSX to this question and I'm not allowed to ask any more questions.
There is just one more question. The only difference between the HKX and centrifuge scenarios is the participants begin separated in the centrifuge and begin together in the HKX. Otherwise they could be both modelled as circular motion. Except the HKX is free to expand into ever larger and more pointed elliptical motion which would at some point be the same as the linear twin paradox scenario which ends in permanent age difference, not reciprocal time dilation like the centrifuge scenario. So I also asked for an Md for the circular HKX which I assume is not an example of reciprocal time dilation but is always an example of the round trip twin paradox. Of course, no one understands this question either.