Confirm: Smooth Twin Paradox Intuition

[WannabeNewton]

There is no reason to do that. You just get that there are no timelike worldlines at rest outside that critical radius.

I'm not sure I follow DaleSpam. Perhaps the wiki article linked by Dr.Greg might put forth a common ground: http://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart
The Born chart is restricted to the aforementioned open subset in this article as well.

Regardless, there is still the issue (at least as far as I can tell) that the chosen coordinates are not defined at $r = 0$ which is where the "stay at home" twin stays put in the scenario depicted by that diagram.

 

[Dale]

I'm not sure I follow DaleSpam. Perhaps the wiki article linked by Dr.Greg might put forth a common ground: http://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart
The Born chart is restricted to the aforementioned open subset in this article as well.

Yes, I saw that, but it seems like an unnecessary restriction. Nothing pathological occurs there.

Regardless, there is still the issue (at least as far as I can tell) that the chosen coordinates are not defined at $r = 0$ which is where the "stay at home" twin stays put in the scenario depicted by that diagram.

Yes, that is the usual restriction for polar coordinate systems. As far as I know, you can't avoid that problem except by changing coordinates.

 

[WannabeNewton]

Yes, that is the usual restriction for polar coordinate systems. As far as I know, you can't avoid that problem except by changing coordinates.

Right so wouldn't the Born coordinates, as written down in that wiki article, be a bad choice of coordinates for the diagram scenario in which the traveling twin is traversing a circle about the central "stay at home" twin, if our goal is to write down a time-like vector field for each twin representing their respective 4-velocities?

 

[Dale]

Agreed. The coordinates I showed don't have that problem.

 

[WannabeNewton]

Agreed. The coordinates I showed don't have that problem.

Ah I actually missed that part of your post, I apologize. Yes in cartesian coordinates things look rather nice, with regards to the above issue, for the "stay at home" twin at the center. Looks like I can just play around with that form of the metric once I finish my late night TV shows Cheers DaleSpam!

 

[tom.stoer]

Sure. Start with a standard inertial frame ...

Thanks, it's simply the orbit. Now I rememer that I posted this a couple of month ago in another thread ;-)

Actually the metric coefficients are time-independent but space-dependent, which accounts for the time dilation.

Yes, my expectation was wrong.

But the fourth term has no such interpretation that I am aware of. It is dependent on velocity, not speed, so it is not like the usual time dilation in a static gravitational field nor is it like the usual SR time dilation. You could consider it to be a gravitational time dilation due to the "gravity" of the Coriolis force, but that seems stretching the analogy a little too far for me.

I think Coriolis force plus harmonic oscillator potential are a perfect analogy.

$d\tau^2/dt^2=1- v^2 - 2\omega\,r_\perp \times v+ \omega^2 r_\perp^2$

This is the "time-dependency" I expected. However it resides in the Coriolis force due to the off-diagonal dx*dt and dy*dt terms in the line element, not in the metric coefficients itself.