Confirm: Smooth Twin Paradox Intuition

[DrGreg]

And as for the twin who was at rest in the global inertial coordinates of Minkowski space-time: would he/she now be following an orbit (in Born coordinates) of something like $u = \frac{1}{\sqrt{1 - \omega^{2}r^{2}}}\partial_{t} - \frac{\omega r}{\sqrt{1 - \omega^{2}r^{2}}}\frac{1}{r}\partial_{\phi}$ i.e. the traveling twin who is now at rest in Born coordinates would see the "stay at home" twin moving along an arc of the same (allowable) radius but in the opposite azimuthal direction, as represented in Born coordinates?

I have to confess, on re-reading post #18, I'm not clear what the diagram there means. At first glance I had assumed both twins were at rest relative to the rotating disk, but in that case I don't understand what the "0 c" and "0.8c" would mean. Is B supposed to be moving "radially" (or rather "spirally"?)

The simplest scenario is simply to have one twin on the periphery of the disk and the other at its centre. Then both are at rest relative to the disk, so we just need to compare proper time at r=0 and r=r₀ (both for constant phi and z).

For a twin at rest in the inertial frame, but not at the disk centre, yes I think what you said is correct.

 

[WannabeNewton]

Oh ok, I didn't even catch that diagram. I was picturing something completely different: a "stay at home" twin and a second twin equipped with a rocket engine who would go out into space and at some point fire the rocket so as to traverse a semicircular arc and turn off the rocket to then head back the same path, towards the "stay at home" twin. But you are correct that the physical scenario in the diagram would be much simpler because we won't have to somehow "smoothly patch together" different metrics (i.e. for the transition from rocket off to rocket on and then back off) to describe the entire round trip if we wanted to go the metric tensor route.

In the scenario given in the diagram, it seems the traveling twin is, from the start, traveling in a circular path such that the "stay at home" twin is at the center of this circular path (which is the origin of the Born coordinates). The only problem I have with this is that the Born metric in Born coordinates is only defined for $0 < r < \frac{1}{\omega}$ so how would we even describe the worldline of the central twin in Born coordinates?

 

[Dale]

the Born metric in Born coordinates is only defined for $0 < r < \frac{1}{\omega}$

Hmm, that doesn't seem right to me. It should be $0<r<\infty$

 

[WannabeNewton]

Hmm, that doesn't seem right to me. It should be $0<r<\inf$

Those are indeed the bounds for the Minkowski metric in cylindrical coordinates but when we simultaneously boost to the momentarily comoving inertial frames of a circling observer at all events on his/her worldline, and construct the Born metric in Born coordinates, we have to restrict ourselves to the open subset $0 < r < \frac{1}{\omega}$ of Minkowski space-time so that the boost speed doesn't exceed the speed of light i.e. $v = \omega r < 1$ where $v$ is the boost speed.

 

[Dale]

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

 

[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.