How does the twin paradox work in a circular orbit?

[phinds]

There have been a couple of posts over the last few months that posit a relativistic-speed path in a circle around the Earth and I want to make sure I correctly understand the ramifications. It's the twin paradox in a circle. SO ... here's a scenario that I think will solidify it for me:

This is a thought experiment, not something that would be practical, but it is remotely within the realm of physical possibility.

Rocket achieves Earth orbit and then with a massive waste of fuel accelerates in the circular orbit to very low relativistic speeds (say .001c), and then decelerates in the orbit and returns to earth.

The person on the craft would be slightly younger than the one on earth, yes? I don't care if it's fractions of a second or what, just the absolute fact.

That is, the end result is exactly the same as if the craft had accelerated away from Earth and then it had turned around and came back (which is in essence what it IS doing).

 

[Cosmo Novice]

Thats entirely correct as I understand it.

My understanding is time dilation is directly related to curvature, curvature is intrinsically linked to gravity and geometry: acceleration can be used to "create" curvature and therefore time dilation.

Any corrections to my understanding are welcome.

 

[ThomasT]

There have been a couple of posts over the last few months that posit a relativistic-speed path in a circle around the Earth and I want to make sure I correctly understand the ramifications. It's the twin paradox in a circle. SO ... here's a scenario that I think will solidify it for me:

This is a thought experiment, not something that would be practical, but it is remotely within the realm of physical possibility.

Rocket achieves Earth orbit and then with a massive waste of fuel accelerates in the circular orbit to very low relativistic speeds (say .001c), and then decelerates in the orbit and returns to earth.

The person on the craft would be slightly younger than the one on earth, yes? I don't care if it's fractions of a second or what, just the absolute fact.

That is, the end result is exactly the same as if the craft had accelerated away from Earth and then it had turned around and came back (which is in essence what it IS doing).

Well, first, I think your question is more appropriate for the SR-GR forum than for the Cosmology forum.

Second, you can answer this by yourself by doing some simple calculations.

 

[phinds]

Well, first, I think your question is more appropriate for the SR-GR forum than for the Cosmology forum.

Second, you can answer this by yourself by doing some simple calculations.

I suspect that if I could do the calculation, I would not have needed to ask the question.

 

[ThomasT]

I suspect that if I could do the calculation, I would not have needed to ask the question.

I've read lots of your posts, and from those I believe that you're quite capable of doing the calculations that will answer your question.

EDIT: Though you might have to look something up. Everybody has to do this from time to time.

 

[phinds]

Oh, I might be able to do the math, but first I'd have to know what math to DO, and I don't.

 

[Drakkith]

Phinds, that would only be correct if the velocity is high enough to offset the reduced effect of Earth's gravity on time dilation. GPS satellites are slower than Earth based clocks by 7 microseconds per day due to time dilation caused by their velocity, however they are FASTER by 45 microseconds per day due to being further away from Earth. Combined the two effects cause satellite clocks to be faster by 38 microseconds per day. AKA they age faster by 38 microseconds per day than those of us on Earth do.

 

[Drakkith]

Oh, I might be able to do the math, but first I'd have to know what math to DO, and I don't.

Does this help? http://en.wikipedia.org/wiki/Time_dilation#Time_dilation_due_to_gravitation_and_motion_together

 

[DaveC426913]

Phinds, that would only be correct if the velocity is high enough to offset the reduced effect of Earth's gravity on time dilation.

OK so move the experiment to a location around a nominal mass asteroid. Or an imaginary point in space.

 

[Drakkith]

OK so move the experiment to a location around a nominal mass asteroid. Or an imaginary point in space.

Exactly!

 

[ThomasT]

OK so move the experiment to a location around a nominal mass asteroid. Or an imaginary point in space.

Whether wrt the Earth or an asteroid or an imaginary point in space, I'm pretty sure, without doing the calculations because, yeah, I'm lazy, that a traveller moving at .001c away from and then back to the Earth or an asteroid or an imaginary point in space will have aged less than a person on either the Earth or an asteroid or occupying an imaginary point in space, respectively.

My point to Phinds was that by doing a few actual calculations, then he can answer lots of these sorts of questions for himself, more or less intuitively.

As to the deeper question of what, physically, causes differential aging. Who knows? Superficially, it suggests that what we refer to as empty space isn't empty. And, as Cosmo Novice suggested, acceleration/deceleration seems to be affecting (producing changes in) the periods/frequencies of oscillators that are accelerating/decelerating.

 

[DaveC426913]

My point to Phinds was that by doing a few actual calculations, then he can answer lots of these sorts of questions for himself, more or less intuitively.

I think he addressed this adequately by pointing out that he does not know which equations he would apply and how, even if he is able to do the calculations once he has them.

 

[George Jones]

The analysis of circular motion does contain some surprises.

Consider a spherical planet of uniform density and five clocks (changing notation slightly):

clock A is thrown straight up from the surface and returns to the surface;
clock B is dropped from rest through a tunnel that goes through the centre of the planet;
Clock C remains on the surface;
clock D remains at the centre of the planet;
clock E orbits the body right at the surface.

Assume that A is thrown at the same time that B is dropped, and that the initial velocity of A is such that A and B arrive simultaneously back at the starting point. The times elapsed on the clocks A, B, and C between when they are all are together at the start and when they are all together at the end satisfy $t_A > t_C > t_B$.

Since A and B are freely falling and C is accelerated, it might be expected that $t_A > t_C$ and $t_B > t_C$, so $t_C > t_B$ seems strange.

Assume that clock E is coincident with clocks A, B, and C when A and B start out. As Fredrik has noted, unless the density of the planet has a specific value, E will not be coincident with with A, B, and C when A and B arrive back, but E will be coincident again with C at some other event. The elapsed times between coincidence events of E and C satisfy $T_C > T_E$. Again, since E is freely falling and C is accelerated, this seems strange.

 

[phinds]

Phinds, that would only be correct if the velocity is high enough to offset the reduced effect of Earth's gravity on time dilation. GPS satellites are slower than Earth based clocks by 7 microseconds per day due to time dilation caused by their velocity, however they are FASTER by 45 microseconds per day due to being further away from Earth. Combined the two effects cause satellite clocks to be faster by 38 microseconds per day. AKA they age faster by 38 microseconds per day than those of us on Earth do.

THanks for reminding me of that. I WAS aware of it, but didn't think about it in this instance. I often don't connect the dots between various facts that I am aware of. Probably comes from (1) only doing this stuff casually, (2) not remembering all of them all of the time, and (3) ... uh ... I forget.

 

[phinds]

Does this help? http://en.wikipedia.org/wiki/Time_dilation#Time_dilation_due_to_gravitation_and_motion_together

Thanks for that. I'll check it out. I usually do my own research on things like this, but I was feeling lazy when I started this thread.

 

[ThomasT]

I think he addressed this adequately by pointing out that he does not know which equations he would apply and how, even if he is able to do the calculations once he has them.

I guarantee you that he has the ability to find the proper equations and do the calculations on his own. He was just being a bit lazy, and I called him on it. That's all.

While I admire your defending him, I don't see any problem with requiring him to do a bit of homework.

If you want some examples of extreme laziness, then just look at some of my posts.

Anyway, bottom line, we all, including now Phinds I think, know the answer to his question. The really deep, intriguing question concerns, at least for me, the physical nature of relativistic differential aging.

EDIT: And, thanks to George Jones for presenting a nice complication.

 

[ThomasT]

I often don't connect the dots between various facts that I am aware of.

Me either.

Probably comes from (1) only doing this stuff casually, (2) not remembering all of them all of the time, and (3) ... uh ... I forget.

I think you nailed it. I'm pretty sure that I've forgotten a lot (most?) of the physics I learned many years ago, but I can't be sure.

Anyway, keep asking questions. I think we both find this stuff fascinating, and are thankful for the science advisors and mentors who donate their time to help us better understand what physics can tell us about our world.

And, I apologize if I in any way offended you. My replies weren't meant to do that. As I mentioned, I've read many of your posts, and from those I think you are a thoughtful and intelligent person.