# Another take on the twins paradox (split)

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Another take on twin paradox:

One twin is living in a tower on Earth and the other is flying around the Earth with constant, high, speed, at the same level with the "tower twin". Both are using very accurate clocks with 2 displays, one normal and one very big.

Now, the plane with the flying twin gets very close to the tower every time it completes a full circle around the Earth and both twins are taking pictures with both plane & tower cocks in the same frame/picture (remember the normal + big displays).

The question is: would the pictures taken show time dilation only for the other twin/clock or the pictures would agree in showing time dilation only for the twin/clock on the plane (or tower)? Why?

PS Pictures are dropped at the tower base to be easily compared.

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I like this thought experiment.

DanMP mentions only 2 or 3 or 4 possible results (not sure which).

But i see a total of 9 simple possible results (am i correct).

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They would be in the same gravity well but travelling at different speeds. Time would slow more for the twin travelling furthest wrt an inertial frame, which the spinning Earth is not.

In the scenario described this could be either travelling furthest, or it could be the same for both. So I would say different possible results even for the simple cases where the tower is on the equator, and the other twin flying West or East at varying speeds for each case.

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They would be in the same gravity well but travelling at different speeds. Time would slow more for the twin travelling furthest wrt an inertial frame, which the spinning Earth is not.

In the scenario described this could be either travelling furthest, or it could be the same for both. So I would say different possible results even for the simple cases where the tower is on the equator, and the other twin flying West or East at varying speeds for each case.

Yes indeed, this a variant of Hafele Keating - https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

It avoids the gravitational potential complication, but the issue of the rotating Earth is central.

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Yes indeed, this a variant of Hafele Keating - https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

It avoids the gravitational potential complication, but the issue of the rotating Earth is central.

Exactly. This is a simplification of HK experiment. Since the influence of gravitational time dilation is eliminated, we have only speed related time dilation. And HK exp. suggests that the pictures taken by the twins would agree about time dilation. In fact, any observer, including one from Hubble, would see/capture the same difference.

So, this is an interesting case where speed related time dilation is not reciprocal. Of course, if they observe each other short after the plane leave the tower, they would see that the other clock is ticking slower.

With this trick is possible to instantly compare the clocks and to see that in fact speed related time dilation is not really reciprocal, only appears to be, because the speed of light is limited.

Another interesting aspect is that we can not apply Lorentz transformations with one twin as stationary and the other moving in relation with the first. We can only do that if we consider a preferred frame, the one with the origin in Earth's centre. Of course, the twins are not in inertial frames, but this was/is not an issue for Fizeau experiment, where Lorentz transformations are successfully applied ... So, it seems that in the real world, where we have massive objects and rotation, we do have preferred frames, contrary to the idea that we don't ...

Back to the tower / orbiting plane scenario, let the tower be on the equator and the plane flying along the equator. If the plane is flying eastwards (same direction as the rotation of the tower in Earth's centre frame), the clock on the plane would tick slower and remain behind the other. If the plane goes westwards, the tower clock would get behind, until the plane will have double the speed of the tower around the Earth, in relation with the tower, when the clocks would be in sync. For greater westward speeds, the plane clock would fall behind again.

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Exactly. This is a simplification of HK experiment. Since the influence of gravitational time dilation is eliminated, we have only speed related time dilation. And HK exp. suggests that the pictures taken by the twins would agree about time dilation. In fact, any observer, including one from Hubble, would see/capture the same difference.

So, this is an interesting case where speed related time dilation is not reciprocal. Of course, if they observe each other short after the plane leave the tower, they would see that the other clock is ticking slower.

With this trick is possible to instantly compare the clocks and to see that in fact speed related time dilation is not really reciprocal, only appears to be, because the speed of light is limited.

Another interesting aspect is that we can not apply Lorentz transformations with one twin as stationary and the other moving in relation with the first. We can only do that if we consider a preferred frame, the one with the origin in Earth's centre. Of course, the twins are not in inertial frames, but this was/is not an issue for Fizeau experiment, where Lorentz transformations are successfully applied ... So, it seems that in the real world, where we have massive objects and rotation, we do have preferred frames, contrary to the idea that we don't ...

Back to the tower / orbiting plane scenario, let the tower be on the equator and the plane flying along the equator. If the plane is flying eastwards (same direction as the rotation of the tower in Earth's centre frame), the clock on the plane would tick slower and remain behind the other. If the plane goes westwards, the tower clock would get behind, until the plane will have double the speed of the tower around the Earth, in relation with the tower, when the clocks would be in sync. For greater westward speeds, the plane clock would fall behind again.

You are making the common mistake of confusing time dilation with the accumulated difference between two clocks if they are separated and then brought together again. Time dilation is the "moment to moment" comparison of clock rates, and while it is a component in determining the final difference between two clocks separated and brought back together, it is not the same thing.

For illustration purposes, we will assume a tower sitting at the Equator and a plane flying west at just the right speed to cancel out the Earth's rotation.( this way we can treat the plane as an inertial frame.

Then from the Plane's frame, the tower clock always runs slow, and thus is always behind his when they meet up again.

From the Tower clock non-inertial frame something else happens. For it, during certain periods between the moments the clocks meet up, the plane clock runs slow (the slowest when the clocks are passing each other) and at others it will be running faster than his own (the fastest when the clocks are on the opposite sides of the Earth.

The rate at any given moment is determined by the relative speed between the two, which remains constant and the magnitude of the centripetal acceleration of the tower clock needed to maintain the tower clock in its circular path (which also remains constant)combined with the distance between the two clocks as measured along the acceleration vector.

when the two clocks are passing each other the distance along the acceleration vector is 0, and you only account for the relative speed. (At this moment, time dilation is purely reciprocal). When the clocks are on the opposite sides of the Earth, then the distance is its greatest, and this overcomes the speed difference.

The periods during which the plane clock runs fast accumulate more time than it loses when it runs slow and the net time accumulation between meetings of the clocks is less as measured from the Tower clock.

This is just the twin paradox with one of the twins constantly accelerating along a long circular path instead than going out at a constant speed, doing a turn around acceleration, and then returning at a constant speed.

Edited by Janus
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With this trick is possible to instantly compare the clocks and to see that in fact speed related time dilation is not really reciprocal, only appears to be, because the speed of light is limited.

The H-K experiment is not reciprocal because the two planes are not at the same speed when measured against an inertial frame of reference.

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I don't understand much of these explanations.

How about this special case of the OP. The tower-twin is on Earth's equator & the tower is so hi that he is in circular free orbit. The plane-twin is also in circular free orbit, "flying" the other way. This setup fits inside the OP -- but the answer is not the same as any given here to date. Why???

A slight difference to the OP.

In my special case the pix could not be dropped to the tower base -- if dropped they would collect at the tower top (ok i think).

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This setup fits inside the OP -- but the answer is not the same as any given here to date. Why???

It's not? What do you think the answer is?

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swansont.

I think the answer in my special case of identical opposite free-fall (inertial) orbits is that the twin's 2 pix would show equal times (eg 9:00 & 9:00 in both pix), ie showing zero relative time dilation.

Whereas i think that the/a previous answer was that the 2 pix (for the case of the original OP) would show equal relative time dilations.

Although, it seems to me that this wordage is ambiguous. It could mean that the 2 pix showed identical times for both clocks (ie say tower-clock=9:01 & plane-clock=9:00 in both pix) -- or it could mean that the 2 pix showed different times but equal dilations (ie tower-clock=9:01 & plane-clock=9:00 taken by tower-twin, & tower-clock=9:00 & plane-clock=9:01 taken by plane-twin).

I am thinking that my answer in my special case might be ok, & different answers in other cases might also be ok, the (perhaps only) problem being that the original OP wrongly (i think) insinuated that only one correct answer was possible (in accordance with SR & GR).

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swansont.

I think the answer in my special case of identical opposite free-fall (inertial) orbits is that the twin's 2 pix would show equal times (eg 9:00 & 9:00 in both pix), ie showing zero relative time dilation.

Show your work. Let's see your calcuations.

Whereas i think that the/a previous answer was that the 2 pix (for the case of the original OP) would show equal relative time dilations.

There have been several posts that say that this is not the case, and mention why.

Although, it seems to me that this wordage is ambiguous. It could mean that the 2 pix showed identical times for both clocks (ie say tower-clock=9:01 & plane-clock=9:00 in both pix) -- or it could mean that the 2 pix showed different times but equal dilations (ie tower-clock=9:01 & plane-clock=9:00 taken by tower-twin, & tower-clock=9:00 & plane-clock=9:01 taken by plane-twin).

I am thinking that my answer in my special case might be ok, & different answers in other cases might also be ok, the (perhaps only) problem being that the original OP wrongly (i think) insinuated that only one answer was possible (in accordance with SR & GR).

The pictures will not disagree with each other. Events are not relative; something that happens in one frame must happen in all frames.

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You are making the common mistake of confusing time dilation with the accumulated difference between two clocks if they are separated and then brought together again. Time dilation is the "moment to moment" comparison of clock rates, and while it is a component in determining the final difference between two clocks separated and brought back together, it is not the same thing.

No, according to Wikipedia:

time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses.

The instances when the plane is passing the tower are, in this scenario, the "two" events and the differences in the elapsed time can be seen/calculated from the pictures taken (by the way, nobody answered the OP question: the pictures would agree or not?).

Even if time dilation is what you are saying, I'm more interested in analyzing the real differences in clocks rates, not what the observers are seeing "moment to moment", because their perception is altered by the finite speed of light, so it isn't relevant.

For illustration purposes, we will assume a tower sitting at the Equator and a plane flying west at just the right speed to cancel out the Earth's rotation.( this way we can treat the plane as an inertial frame.

This is the preferred frame I wrote about ... We do have preferred frames in the real world ...

(The frame is not really inertial. The Earth rotates around the Sun, and the Sun around the galactic centre and so on ...)

From this preferred frame we can see (like here, "the case of a flexible loop of optical fiber moving like a conveyor belt with some arbitrary shape") the problem as one-dimensional ... and calculate the time differences using Lorentz transformations.

That's why I wrote:

Back to the tower / orbiting plane scenario, let the tower be on the equator and the plane flying along the equator. If the plane is flying eastwards (same direction as the rotation of the tower in Earth's centre frame), the clock on the plane would tick slower and remain behind the other. If the plane goes westwards, the tower clock would get behind, until the plane will have double the speed of the tower around the Earth, in relation with the tower, when the clocks would be in sync. For greater westward speeds, the plane clock would fall behind again.

This is just the twin paradox with one of the twins constantly accelerating along a long circular path instead than going out at a constant speed, doing a turn around acceleration, and then returning at a constant speed.

Yes. That's why I posted this scenario in The twin paradox and other variants. topic.

Another interesting idea is that the rate of an accelerated clock doesn't depend on its acceleration

an accelerating clock will count out its time in such a way that at any one moment, its timing has slowed by a factor (γ) that depends only on its current speed; its acceleration has no effect at all.

The pictures will not disagree with each other. Events are not relative; something that happens in one frame must happen in all frames.

Good answer! Thank you.

Edited by DanMP
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Yes, i see, any 2 pix of the one event taken at close range at the same place & time will of course always agree (silly me).

Ok, so my 9 possible outcomes in #2 are in fact only 3, eg (+1 & +1) or (0 & 0) or (-1 & -1).

For my special case i arrive at (0 & 0) by symmetry (no calcs needed).

Assuming of course that my special case is possible (ie that scenario might be impossible even for a thought experiment).

And, as an aside, i suspect that the twins can synchronise clocks, if not on the first pass, then at least on subsequent passes.

And, furthermore, i suspect that this applies to all such scenarios in the OP, not just to my special (symmetrical) case, if the relative time dilation gain per pass is constant,

eg if the pairs of pix each show 0, 0, 0, 0............ this shows 0 relative time dilation per pass, &

eg if the pairs of pix each show +1, +2, +3, +4............ this shows +1 relative time dilation per pass.

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Wow - this brings up some interesting situations. Let me throw out a couple of scenarios (moderators please let me know if either or both of these should be in a new thread).

1) What if A and B are in separate space vehicles, in orbit (so they're both strictly inertial), but moving in different directions (opposite, or orthogonal), but still pass by each other twice per orbit. Assume they can photograph each other's clock as mentioned above. Either seems free to claim the role of "stationary." Now, it seems clear to me that no actual time dilation will occur - they're both in entirely parallel situations, so I assume their clocks will agree every time they meet. But both of them would expect the other's clock to be running slower.

2) I'm really getting into deep waters for me, but I did some reading about space time topology recently, and it seemed like they were saying that the universe could be closed like the surface of a sphere. So if you kept going long enough you'd "go around" the universe. So if that's true, we could use it in the two-twin situation. A and B match clocks, and A takes off and circuits the universe. I guess when she gets home she'll be younger? That avoids the need for a turn-around. There's also the more precisely parallel version of that where they both take off in opposite directions.

Really looking forward to replies on these - fascinating!

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Yes, i see, any 2 pix of the one event taken at close range at the same place & time will of course always agree (silly me).

Ok, so my 9 possible outcomes in #2 are in fact only 3, eg (+1 & +1) or (0 & 0) or (-1 & -1).

For my special case i arrive at (0 & 0) by symmetry (no calcs needed).

Assuming of course that my special case is possible (ie that scenario might be impossible even for a thought experiment).

And, as an aside, i suspect that the twins can synchronise clocks, if not on the first pass, then at least on subsequent passes.

And, furthermore, i suspect that this applies to all such scenarios in the OP, not just to my special (symmetrical) case, if the relative time dilation gain per pass is constant,

eg if the pairs of pix each show 0, 0, 0, 0............ this shows 0 relative time dilation per pass, &

eg if the pairs of pix each show +1, +2, +3, +4............ this shows +1 relative time dilation per pass.

I agree...though they only agree at that point...they tend toward disagreement as they diverge and back toward agreement as they converge

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[re kipingram].

I hadn't twigged that 2 identical circular orbits must always cross twice, not matter the angle.

And that oribitors if meeting once must meet twice per orbit, no matter the direction.

[re J C MacSwell].

I don't see any tending toward disagreement (& then tending back).

Although u would i think get such tendings if including real-world moon & sun effects etc, & perhaps Earthly spin -- but then the whole circular orbit thing explodes.

Unless there is something else i haven't thought of that gives little (perfectly balanced, twice per orbit, plus & minus) tendings.

Would your tendings involve change of speed??, or change of shape of orbit??? (or both????).

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