# Twin paradox with a twist

## Recommended Posts

Posted (edited)

We all know the twin paradox with the travelling twin returning younger than the stay at home twin. The acceleration in order to return is considered important for the outcome. Now, let's add a new acceleration, let's consider the travelling twin revolving around an axis, in order to feel as on Earth surface (same g force).

If we have 1 clock on the Earth twin location (#1) and 2 on the ship, one on the axis of rotation (#2) and the other (#3) where the traveling twin is staying, which clock is recording the least time between departure and arrival and why?

In order to see if only the (added) acceleration is important, let's add another, bigger, "wheel" on the starship, with a different rotation speed, but with the same centripetal/centrifugal acceleration, g. The 4th clock, on the rim of the bigger wheel, would record the same time as the clock #3, on the rim of the smaller wheel? If not, why not?

Edited by DanMP
##### Share on other sites

A rotation will slow the clock down; this has been independently measured

citations 82-84 in https://en.m.wikipedia.org/wiki/Tests_of_general_relativity

You can analyze these as equivalent to gravitational redshifts with acceleration v^2/r

##### Share on other sites

2 hours ago, DanMP said:

let's add another, bigger, "wheel" on the starship, with a different rotation speed, but with the same centripetal/centrifugal acceleration, g. The 4th clock, on the rim of the bigger wheel, would record the same time as the clock #3, on the rim of the smaller wheel? If not, why not?

No. The centripetal force is Fc = mv^2/r, so greater r means greater v, which means a larger Lorentz factor relative to the centre of the wheel.

Would that give the same answer as a gravitational redshift analysis assuming negligible spaceship mass?

##### Share on other sites

6 hours ago, DanMP said:

... The acceleration in order to return is considered important for the outcome. ...

Are you claiming the acceleration is a direct cause of the differential aging in the twins' paradox?

##### Share on other sites

6 hours ago, DanMP said:

In order to see if only the (added) acceleration is important, let's add another, bigger, "wheel" on the starship, with a different rotation speed, but with the same centripetal/centrifugal acceleration, g

The time dilation is not simply a function of g; it’s the gravitational potential that’s important. for constant g, the dilation is given by gh/c^2. The distance matters.

As md65536 points out, a larger wheel with the same g will have a larger dilation. v^2 is bigger. Or, if you want to view it via the acceleration, ah is bigger.

##### Share on other sites

22 hours ago, swansont said:

A rotation will slow the clock down

Yes, of course.

19 hours ago, md65536 said:

The centripetal force is Fc = mv^2/r, so greater r means greater v, which means a larger Lorentz factor relative to the centre of the wheel.

That was/is my line of thinking as well.

19 hours ago, md65536 said:

Would that give the same answer as a gravitational redshift analysis assuming negligible spaceship mass?

I don't know.

16 hours ago, pzkpfw said:

Are you claiming the acceleration is a direct cause of the differential aging in the twins' paradox?

No, on the contrary, I think that only the speed of the travelling twin is the cause, but the acceleration is important to establish which twin is travelling and also his path in spacetime, if I understood correctly.

15 hours ago, swansont said:

As md65536 points out, a larger wheel with the same g will have a larger dilation. v^2 is bigger. Or, if you want to view it via the acceleration, ah is bigger.

What do you mean with "ah is bigger"?

##### Share on other sites

32 minutes ago, DanMP said:

What do you mean with "ah is bigger"?

The potential term has variables of acceleration * distance (a*h); the product is larger

Even though the acceleration is the same, the position has changed. It is the equivalent to being deeper in a potential well

##### Share on other sites

10 minutes ago, swansont said:

The potential term has variables of acceleration * distance (a*h); the product is larger

Even though the acceleration is the same, the position has changed. It is the equivalent to being deeper in a potential well

Ok, thank you.

##### Share on other sites

To give the twin scenario more “twist”, you can allow spacetime to not be flat (GR twin scenario). For even more twist, allow spacetimes that are topologically non-trivial

##### Share on other sites

Posted (edited)
On 4/27/2024 at 8:46 AM, Markus Hanke said:

To give the twin scenario more “twist”, you can allow spacetime to not be flat (GR twin scenario).

Ok, let's consider a modified Hafele-Keating experiment:

The stay-at-home twin would be in a tower, at the equator, and the traveling-twin would fly around the Earth, over the equator, at the same altitude as the stay-at-home twin (in order to have the same gravitational time dilation).

The Earth is not spinning and flat, with mirrors placed all along the equator. The twins would see each other in mirrors, exactly as described in the classical, linear, experiment.

Now, at the opposite side of the Earth (from the tower), the travelling twin turns the telescope from backwards to forwards, and by this action he is changing his perception from moving away to moving towards the tower twin. In this case there is no acceleration (change in velocity), no actual turning back, but the difference in ageing would be the same as he turns back with instant change in velocity and direction, like in a relay version.

To me, this is yet another indication that:

On 4/26/2024 at 4:00 PM, DanMP said:

only the speed of the travelling twin is the cause

for the differential aging in the twins' paradox, not the acceleration, nor the frame change.

Edited by DanMP
##### Share on other sites

1 hour ago, DanMP said:

Ok, let's consider a modified Hafele-Keating experiment:

The stay-at-home twin would be in a tower, at the equator, and the traveling-twin would fly around the Earth, over the equator, at the same altitude as the stay-at-home twin (in order to have the same gravitational time dilation).

The Earth is not spinning and flat, with mirrors placed all along the equator. The twins would see each other in mirrors, exactly as described in the classical, linear, experiment.

Now, at the opposite side of the Earth (from the tower), the travelling twin turns the telescope from backwards to forwards, and by this action he is changing his perception from moving away to moving towards the tower twin. In this case there is no acceleration (change in velocity), no actual turning back, but the difference in ageing would be the same as he turns back with instant change in velocity and direction, like in a relay version.

To me, this is yet another indication that:

for the differential aging in the twins' paradox, not the acceleration, nor the frame change.

The moving twin is accelerating. You can’t move around a circle without accelerating.

##### Share on other sites

17 hours ago, swansont said:

The moving twin is accelerating. You can’t move around a circle without accelerating.

Yes, but I never heard that centripetal acceleration is playing a significant role in explaining the time differences in Hafele-Keating experiment. If you did, please elaborate.

##### Share on other sites

1 hour ago, DanMP said:

Yes, but I never heard that centripetal acceleration is playing a significant role in explaining the time differences in Hafele-Keating experiment. If you did, please elaborate.

Gravity provides the centripetal acceleration. They were at some location in a potential well, and the explanation used that, but gravity and acceleration are equivalent.

##### Share on other sites

2 hours ago, swansont said:

They were at some location in a potential well, and the explanation used that, but gravity and acceleration are equivalent.

I don't quite understand. Can you provide a link to the explanation of Hafele-Keating experiment you are mentioning?

##### Share on other sites

1 hour ago, DanMP said:

I don't quite understand. Can you provide a link to the explanation of Hafele-Keating experiment you are mentioning?

I don’t understand. You can’t find a link to the experiment that you brought up?

An object moving in a circle is in an accelerating frame of reference. That doesn’t change just because you’re flying in a plane; gravity is supplying a (part of) the centripetal force of any object on the earth. It’s not mentioned in any explanation, because you have the GR framework already in place, but if you read their paper you’ll note that they point out that the earth is not an inertial frame - that’s why the east- and west-bound planes don’t have the same time dilation. The earth’s surface can’t be treated as being at rest. They do the analysis from an inertial frame, and the east-bound plane is moving faster, which is why its clocks slowed down. The west-bound clocks are moving slower, and sped up.

In your scenario, the clocks on planes are still in an accelerated frame. They will be the ones that slow down; there is no symmetry between inertial frames.

##### Share on other sites

17 hours ago, swansont said:

You can’t find a link to the experiment that you brought up?

No, I can't find a link to the explanation you mentioned:

21 hours ago, swansont said:

They were at some location in a potential well, and the explanation used that, but gravity and acceleration are equivalent.

the one you brought up when I said:

23 hours ago, DanMP said:

I never heard that centripetal acceleration is playing a significant role in explaining the time differences in Hafele-Keating experiment. If you did, please elaborate.

17 hours ago, swansont said:

They do the analysis from an inertial frame, and the east-bound plane is moving faster, which is why its clocks slowed down. The west-bound clocks are moving slower, and sped up

This confirms what I said: the speed is the relevant cause for the differential aging in the twins' paradox, not the acceleration, nor the frame change, as the classical, linear scenario, suggests.

In my modified Hafele-Keating experiment, it makes no difference if the plane continues forward or is turning back. Only the speed/velocity matters. And, by using the mirrors, the twins observe each other as in the classical, linear, scenario, with a relay (the relay scenario is the one where a third twin/clock is introduced, in order to skip the turnaround maneuvers).

17 hours ago, swansont said:

In your scenario, the clocks on planes are still in an accelerated frame. They will be the ones that slow down; there is no symmetry between inertial frames.

What inertial frames? Only the tower twin is in an inertial frame. And in the classical Hafele-Keating experiment the tower twin is also in an accelerated frame ...

By the way, as far as I know, Lorenz transformations are between inertial frames ... but it turns out that they can be applied successfully between the non-rotating Earth frame and twins accelerated frames.

This reminds me of the clock postulate

##### Share on other sites

3 hours ago, DanMP said:

This confirms what I said: the speed is the relevant cause for the differential aging in the twins' paradox, not the acceleration, nor the frame change, as the classical, linear scenario, suggests.

Which twin is relevant to what?

##### Share on other sites

4 hours ago, DanMP said:

This confirms what I said: the speed is the relevant cause for the differential aging in the twins' paradox, not the acceleration, nor the frame change, as the classical, linear scenario, suggests.

I wasn’t challenging that

4 hours ago, DanMP said:

In my modified Hafele-Keating experiment, it makes no difference if the plane continues forward or is turning back. Only the speed/velocity matters. And, by using the mirrors, the twins observe each other as in the classical, linear, scenario, with a relay (the relay scenario is the one where a third twin/clock is introduced, in order to skip the turnaround maneuvers).

My point was the clocks in the accelerated frame will be the ones to slow down.

4 hours ago, DanMP said:

What inertial frames? Only the tower twin is in an inertial frame.

I said there was no symmetry between inertial frames, which is because there aren’t multiple inertial frames. But here you acknowledge that the planes are not in inertial frames, so what’s the problem - this is what I was pointing out! There is a frame change, continual in this case, which is why you remove the expectation of symmetrical observations of time dilation.

The frame change is crucial to remove that symmetry.

4 hours ago, DanMP said:

And in the classical Hafele-Keating experiment the tower twin is also in an accelerated frame ...

In the H-K experiment the analysis was done from the view of an inertial frame, rather than from the frame of the surface of the earth

##### Share on other sites

On 4/29/2024 at 10:31 AM, DanMP said:

Now, at the opposite side of the Earth (from the tower), the travelling twin turns the telescope from backwards to forwards, and by this action he is changing his perception from moving away to moving towards the tower twin. In this case there is no acceleration (change in velocity), no actual turning back, but the difference in ageing would be the same as he turns back with instant change in velocity and direction, like in a relay version.

To me, this is yet another indication that:

On 4/26/2024 at 9:00 AM, DanMP said:

only the speed of the travelling twin is the cause

for the differential aging in the twins' paradox, not the acceleration, nor the frame change.

Yes, when the orbiting twin turns his gaze around, he will appear to be approaching instead of receding. So the redshifted view of the tower clock will change to blue shift, the difference being purely Doppler effect in both directions.

The dilation is due to speed, and speed isn't affected by where anybody is looking, so the dilation is unchanged at the far side of the planet.

Yes, there is acceleration, but all of it orthogonal to motion, so since the 'twins' are at the same potential, the dilation is constant for the entire orbit.

It is objective. The orbiting twin will be younger when the meet again, just like the one that goes out and back to the distant star.

To do this in special relativity, the planet can have no mass, and the 'orbiting' twin would need to curve his path via say a string tied to the center of Earth to get him to curve is path like that. Rockets also works, but the engineer in me hates to waste fuel when there's a better way.

##### Share on other sites

Posted (edited)
20 hours ago, swansont said:

My point was the clocks in the accelerated frame will be the ones to slow down.

OK, I got it.

20 hours ago, swansont said:

There is a frame change, continual in this case, which is why you remove the expectation of symmetrical observations of time dilation.

The frame change is crucial to remove that symmetry.

Now I get your point about the importance of (centripetal) acceleration. Clever.

20 hours ago, swansont said:

In the H-K experiment the analysis was done from the view of an inertial frame, rather than from the frame of the surface of the earth

Correct. Still, if frame-dragging is present (and significant), the clocks at rest in the dragged frame would be the fastest ... while rotating (with respect to distant stars) and having(?) centripetal acceleration ...

1 hour ago, Halc said:

Yes, when the orbiting twin turns his gaze around, he will appear to be approaching instead of receding. So the redshifted view of the tower clock will change to blue shift, the difference being purely Doppler effect in both directions.

The dilation is due to speed, and speed isn't affected by where anybody is looking, so the dilation is unchanged at the far side of the planet.

Yes, there is acceleration, but all of it orthogonal to motion, so since the 'twins' are at the same potential, the dilation is constant for the entire orbit.

It is objective. The orbiting twin will be younger when the meet again, just like the one that goes out and back to the distant star.

To do this in special relativity, the planet can have no mass, and the 'orbiting' twin would need to curve his path via say a string tied to the center of Earth to get him to curve is path like that. Rockets also works, but the engineer in me hates to waste fuel when there's a better way.

You understood exactly what I intended to convey. Thank you!

Edited by DanMP

## Create an account

Register a new account