# Is the Twin Paradox Real or Just Apparent in Relativity?

• ananthu
In summary, Friends, in this conversation, the concepts of time dilation, the twin paradox, and the increase of mass with velocity according to the theory of relativity are discussed. It is explained that the different passage of time is real and the aging of cells is also affected by it. The term "apparent" is not applicable in this scenario as each twin's conclusion about the other's age is accurate. As for the increase of mass with velocity, it is clarified that it is frame dependent and not a fundamental property, and that photons have no mass and cannot travel at less than the speed of light.
ananthu
Friends,
I will be happy if anybody throws light on the following concepts:

(1). In the case of time dilation it is said that a clock in a moving frame appears to go slow to an observer in a resting frame.It leads to the famous twin paradox in which 'A' who spends some time in a spaceship and returns to the Earth appears younger to 'B' who is his twin and stays on Earth at the time A leaves earth. With respect to B , say, 20 years had passed on the Earth but for A, say, only one year has passed in his ship. Here I don't understand one point.
If the theory of relativity says that A will appear 20 years younger than B, is it just an apparent one or real one? During the journey in the spaceship had the body cells of A really slowed down in its metabolism of aging? This point is highly confusing.

(2). According the theory of relativity, the mass of a body increases with the increase in its velocity. In that case, since a photon behaves like a particle, does it acquire mass because of its velocity?

(3). When the velocity of a body increases its mass increases while its length should decrease. Let us suppose that a rod approaches the velocity of light ( not become equal to it ). Then while its mass approaches infinity its length should approach zero! How is it possible for a body to shrink to an almost zero size but at the same time possesses nearly infinite mass?

I know the theory of relativity is a highly a complex one to comprehend. Still if anybody has simple explanations for my above doubts I will be very happy.

1) the different passage of time is real. Every observer will agree on the elapsed time on the clocks when they meet up. The ageing and all time dependent processes will reflect this.

2) The so called 'mass increase' is frame dependent, so it can't be affecting any fundamental properties like inertial mass. It is better to think of the inertial mass remaining constant, but the applied force appears to get smaller from moving frames.
It is sometimes written F=(m/Y)a, where Y is gamma, but it should be YF=ma.

3) nothing shrinks, it just appears so from a moving frame.

All these relativistic effects are seen when one frame measures things in another frame. So 'reaching the speed of light' has no meaning, unless you say what frame (observer) this velocity is relative to. There's no absolute velocity.

ananthu said:
In the case of time dilation it is said that a clock in a moving frame appears to go slow to an observer in a resting frame.

Initially, neither frame is "the resting frame", and neither is "the moving frame". In SR, all inertial motion is relative. It makes no sense to ask, "Which inertial twin is REALLY moving?".

During the initial leg of the traveler's journey, EACH twin is moving inertially (no acceleration), and each twin's velocity, relative to the other, is constant. During that phase, there is NO difference in the state of motion of the two twins.

If the theory of relativity says that A will appear 20 years younger than B, is it just an apparent one or real one?

The term "apparent" is frequently used in describing the traveling twin problem, and it shouldn't be ... it leads to serious misunderstandings. During that first leg of the trip, the two twins are COMPLETELY equivalent. Each can make elementary measurements and simple, first-principle calculations about the current age of the other twin. Each will conclude that the other twin is ageing more slowly. And each of them is correct in their conclusion. Each of their conclusions is as real as anything can be ... it is NOT some kind of illusion.

Mike Fontenot

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ananthu said:
Friends,
I will be happy if anybody throws light on the following concepts:

(1). In the case of time dilation it is said that a clock in a moving frame appears to go slow to an observer in a resting frame.It leads to the famous twin paradox in which 'A' who spends some time in a spaceship and returns to the Earth appears younger to 'B' who is his twin and stays on Earth at the time A leaves earth. With respect to B , say, 20 years had passed on the Earth but for A, say, only one year has passed in his ship. Here I don't understand one point.
If the theory of relativity says that A will appear 20 years younger than B, is it just an apparent one or real one? During the journey in the spaceship had the body cells of A really slowed down in its metabolism of aging? This point is highly confusing.

I know the theory of relativity is a highly a complex one to comprehend. Still if anybody has simple explanations for my above doubts I will be very happy.

Regarding 2 and 3, my knowledge does not reach those skies, but i can tell you a thing about the first one.
You see every atom itself has a fully synchronized atomic clock in it, therefore the cells in a body are synced with the frame that you are in, so as time passes slower for you seen by the observer in a rest frame,the same happens for every cell in your body, therefore you will experience slower rate of growth as you have entered a slower shell of time.

Ofc i might be wrong, my knowledge is as much as i understand from relativity books :D

ananthu said:
Friends,
I will be happy if anybody throws light on the following concepts

(2). According the theory of relativity, the mass of a body increases with the increase in its velocity. In that case, since a photon behaves like a particle, does it acquire mass because of its velocity?

A photon has no mass, and cannot travel at less than C, or greater than C. I think relativistic mass is gained as a percentage of its original mass, and any percentage of 0 is still 0, but this might not even be wrong. As light cannot accelerate, however, and is already at C, its zero mass would remain constant.

Light slowing down as it travels through a medium has more to do with photon/electron interaction where an electron absorbs a photon, and then later emits it, which intrinsically slows "light" down, but not the velocity of the photon itself, as "it" takes several trips between electrons, being delayed not by its velocity, but by its absorption into electrons.

If either reference frame is equal, isn't each twin aging slower by each other's perspective? Why then, when one twin returns is one of them is dead and the other one young? I know about the doppler effect of flying back, where events on Earth would speed up drastically by his perception, but this indicates that time dilation is an illusion, and we know it isn't.

JJRittenhouse said:
[...]
If either reference frame is equal, isn't each twin aging slower by each other's perspective? Why then, when one twin returns is one of them is dead and the other one young?
[...]

Here's a repeat of a previous posting of mine, addressing the same question:
___________________________________________________

During the constant-speed legs of the trip, BOTH twins conclude that the other twin is ageing slower. But when the trip is over, they both agree that the stay-at-home twin is older. How is that possible?

It's possible because, during the turnaround, the traveler will conclude that the home twin quickly ages, with very little ageing of the traveler. The home twin concludes that neither of them ages much during the turnaround. When you add up all these segments of ageing, you get the result that the home twin is older (and both twins exactly agree on that).

Years ago, I derived a simple equation (called the "CADO" equation) that explicitly gives the ageing of the home twin during accelerations by the traveler (according to the traveler). The equation is especially easy to use for idealized traveling twin problems with instantaneous speed changes. But it also works for finite accelerations. I've got a detailed example with +-1g accelerations on my webpage:

http://home.comcast.net/~mlfasf

And I've published a paper giving the derivation of the CADO equation:

"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.
_______________________________________________________

Here's a brief description of my "CADO" equation:
__________________________________________________ __

Years ago, I derived a simple equation that relates the current ages of the twins, ACCORDING TO EACH TWIN. Over the years, I have found it to be very useful. To save writing, I write "the current age of a distant object", where the "distant object" is the stay-at-home twin, as the "CADO". The CADO has a value for each age t of the traveling twin, written CADO(t). The traveler and the stay-at-home twin come to DIFFERENT conclusions about CADO(t), at any given age t of the traveler. Denote the traveler's conclusion as CADO_T(t), and the stay-at-home twin's conclusion as CADO_H(t). (And in both cases, remember that CADO(t) is the age of the home twin, and t is the age of the traveler).

My simple equation says that

where

L is their current distance apart, in lightyears,
according to the home twin,

and

v is their current relative speed, in lightyears/year,
according to the home twin. v is positive
when the twins are moving apart.

(Although the dependence is not shown explicitely in the above equation, the quantities L and v are themselves functions of t, the age of the traveler).

The factor (c*c) has value 1 for these units, and is needed only to make the dimensionality correct. That factor can be ignored when doing calculations.

The equation explicitly shows how the home twin's age will change abruptly (according to the traveler, not the home twin), whenever the relative speed changes abruptly.

For example, suppose the home twin believes that she is 40 when the traveler is 20, immediately before he turns around. So CADO_H(20-) = 40. (Denote his age immediately before the turnaround as t = 20-, and immediately after the turnaround as t = 20+.)

Suppose they are 30 ly apart (according to the home twin), and that their relative speed is +0.9 ly/y (i.e., 0.9c), when the traveler's age is 20-. Then the traveler will conclude that the home twin is

CADO_T(20-) = 40 - 0.9*30 = 13

years old immediately before his turnaround. Immediately after his turnaround (assumed here to occur in zero time), their relative speed is -0.9 ly/y. The home twin concludes that their distance apart doesn't change during the turnaround: it's still 30 ly. And the home twin concludes that neither of them ages during the turnaround, so that CADO_H(20+) is still 40.

But according to the traveler,

CADO_T(20+) = 40 - (-0.9)*30 = 67,

so he concludes that his twin ages 54 years during his instantaneous turnaround.

The equation works for arbitrary accelerations, not just the idealized instantaneous speed change assumed above. I've got an example with +-1g accelerations on my web page:

http://home.comcast.net/~mlfasf

The derivation of the equation is given in my paper

"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.
________________________________________________________

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JJRittenhouse said:

If either reference frame is equal, isn't each twin aging slower by each other's perspective?
That question is the twin paradox! But the resolution of the paradox is that only inertial frames are on equal footing, whereas in order for the two twins to move apart and later reunite, one of them must accelerate to turn around at some point, so that twin didn't remain at rest in a single inertial frame. Acceleration, unlike speed, is an absolute in SR--if you're accelerating, you know you're accelerating because you feel G-forces. A good resource on the twin paradox:

Mike_Fontenot said:
It's possible because, during the turnaround, the traveler will conclude that the home twin quickly ages, with very little ageing of the traveler. The home twin concludes that neither of them ages much during the turnaround. When you add up all these segments of ageing, you get the result that the home twin is older (and both twins exactly agree on that).
As I pointed out earlier, this is only true if you choose a particular type of non-inertial frame for the non-inertial twin, and there is no fundamental physical sense in which that frame uniquely defines the "perspective" or "measurements" of the non-inertial twin.

JesseM said:
As I pointed out earlier, this is only true if you choose a particular type of non-inertial frame for the non-inertial twin, and there is no fundamental physical sense in which that frame uniquely defines the "perspective" or "measurements" of the non-inertial twin.

I disagree. I show in my paper that the ONLY reference frame which doesn't contradict the accelerating observer's own elementary measurements, and first-principle calculations, is the reference frame I describe in that paper.

Mike Fontenot

Mike_Fontenot said:
I disagree. I show in my paper that the ONLY reference frame which doesn't contradict the accelerating observer's own elementary measurements, and first-principle calculations, is the reference frame I describe in that paper.

Mike Fontenot
Well, you never responded to my questions in post #130 about how you can define terms like "elementary measurements" and "first-principle calculations" in a non-circular way which doesn't just presuppose that the non-inertial observer must at every moment have the same definitions of simultaneity as in their instantaneously comoving inertial frame. Since your claims are nonstandard and your paper isn't available online (and is from a journal that http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHESEM000019000001000006000001&idtype=cvips&gifs=yes&ref=no ), I think you should explain the detailed basis for your argument, either here or in the Independent Research forum, before promoting the conclusions on threads like this one.

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Mike_Fontenot said:
I disagree. I show in my paper that the ONLY reference frame which doesn't contradict the accelerating observer's own elementary measurements, and first-principle calculations, is the reference frame I describe in that paper.

Mike Fontenot

To my mind, all coordinate systems are a matter of convention and utility. To get the physics of what each twin sees, you could simply compute what happens if the Earth twin sends signals every second (for example) along its world line and vice versa; and work out when (along each twin's world line) each twin receives the others signals. This would tell you, physically, how each twin perceives the other, and it can all be done whatever coordinates you find convenient (all determinations must agree).

JesseM said:
Well, you never responded to my questions in post #130 about how you can define terms like "elementary measurements" and "first-principle calculations" in a non-circular way which doesn't just presuppose that the non-inertial observer must at every moment the same definitions of simultaneity as in their instantaneously comoving inertial frame. Since your claims are nonstandard and your paper isn't available online (and is from a journal that http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHESEM000019000001000006000001&idtype=cvips&gifs=yes&ref=no ), I think you should explain the detailed basis for your argument, either here or in the Independent Research forum, before promoting the conclusions on threads like this one.

JesseM,

I've often wondered about what are reasonable approaches to simultaneity for non-inertial observers, or observers in highly curved geometries. One I've played with as follows, and I wonder if you could say anything about its pros and cons:

For some distant event, imagine a pulse of known brightness was emitted, ignoring any issues of dust etc. Some observer receives it at time t1 on their world line. From the received brightness, they compute a distance based on standard inverse square assumption. Divide this imputed distance by c, and they assume the event was simultaneous to an event (t1 - d/c) on their world line.

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It is not the acceleration that causes the difference in the ages of the twins, it's the difference in the length of time that one twin spends coasting than the other twin that causes a difference in their ages.

Consider this variant (which I have read on this forum) to show that this is true:

Both identical twins start out at rest with respect to each other at the same age in the same (approximate) location. They both accelerate identically for some period of time. They then coast. During this interval of time, they will again be at rest with respect to each other and at the same age as each other and in the same (approximate) location as each other. Then one of the twins accelerates in the opposite direction as before until traveling back toward the starting point. Now both twins are coasting but in opposite directions. Then, at the appropriate time, the first twin decelerates to come to rest at the starting point. The second twin is still coasting away from the first twin and their common starting point. Later on, the second twin undergoes exactly the same acceleration in the opposite direction that the first twin experienced to bring him back home. He coasts until the approprate time comes for him to decelerate in exactly the same manner as the first twin to bring him to rest at the starting point. Now both twins will be at rest in their initial starting points but the second twin that spent more time coasting will be younger.

Note that in this variant, both twins experience exactly the same accelerations and deceleration so this cannot explain the difference in age. Rather, it's the difference in time that each twin spends coasting between the accelerations and deceleration that produces the different ages. The accelerations cause different aging rates in the twins but until they spend time at those different aging rates, their ages will remain the same.

PAllen said:
JesseM,

I've often wondered about what are reasonable approaches to simultaneity for non-inertial observers, or observers in highly curved geometries. One I've played with as follows, and I wonder if you could say anything about its pros and cons:

For some distant event, imagine a pulse of known brightness was emitted, ignoring any issues of dust etc. Some observer receives it at time t1 on their world line. From the received brightness, they compute a distance based on standard inverse square assumption. Divide this imputed distance by c, and they assume the event was simultaneous to an event (t1 - d/c) on their world line.
Well, in GR I know you can use any simultaneity convention you like thanks to the principle of diffeomorphism invariance (see this article), so I think the choice of coordinate system is usually based on symmetries in the spacetime (like spherical symmetry for a Schwarzschild spacetime, or surfaces of simultaneity where the matter density is perfectly uniform in a FLRW spacetime) which mean that the equations for the metric will take a particularly simple form in certain types of coordinate systems. So although you could use your convention, doing so might make the expression for the metric in that coordinate system very complicated and hard to deal with. But I'm not very knowledgeable about the detailed mathematics of GR so you might be better off asking some of the other posters here with more expertise.

ghwellsjr said:
It is not the acceleration that causes the difference in the ages of the twins, it's the difference in the length of time that one twin spends coasting than the other twin that causes a difference in their ages.

Consider this variant (which I have read on this forum) to show that this is true:

Both identical twins start out at rest with respect to each other at the same age in the same (approximate) location. They both accelerate identically for some period of time. They then coast. During this interval of time, they will again be at rest with respect to each other and at the same age as each other and in the same (approximate) location as each other. Then one of the twins accelerates in the opposite direction as before until traveling back toward the starting point. Now both twins are coasting but in opposite directions. Then, at the appropriate time, the first twin decelerates to come to rest at the starting point. The second twin is still coasting away from the first twin and their common starting point. Later on, the second twin undergoes exactly the same acceleration in the opposite direction that the first twin experienced to bring him back home. He coasts until the approprate time comes for him to decelerate in exactly the same manner as the first twin to bring him to rest at the starting point. Now both twins will be at rest in their initial starting points but the second twin that spent more time coasting will be younger.

Note that in this variant, both twins experience exactly the same accelerations and deceleration so this cannot explain the difference in age. Rather, it's the difference in time that each twin spends coasting between the accelerations and deceleration that produces the different ages. The accelerations cause different aging rates in the twins but until they spend time at those different aging rates, their ages will remain the same.

I don't think it is meaningful to talk about what part of a path responsible for its interval. Its a global property of the path and all parts contribute. Let's transfer the problem to Euclidean plane geometry. The hypotenuse of the triangle is shorter than the sum of the legs. Which part of the legs are responsible for the difference? Meaningless question.

ghwellsjr said:
The accelerations cause different aging rates in the twins but until they spend time at those different aging rates, their ages will remain the same.

I don't think I understand what you mean. How can you not spend time at an aging rate? How does spending time at a differing aging rate keep their ages the same? With respect to what? Each other, or themselves?

PAllen said:
I don't think it is meaningful to talk about what part of a path responsible for its interval. Its a global property of the path and all parts contribute. Let's transfer the problem to Euclidean plane geometry. The hypotenuse of the triangle is shorter than the sum of the legs. Which part of the legs are responsible for the difference? Meaningless question.

..wish I'd read this first, that makes sense.

JJRittenhouse said:
I don't think I understand what you mean. How can you not spend time at an aging rate? How does spending time at a differing aging rate keep their ages the same? With respect to what? Each other, or themselves?

Everyone is spending time at some aging rate and I didn't say spending time at different aging rates keeps their ages the same. I said the exact opposite. I said they had to spend time at different aging rates in order for their ages to become different.

With respect to the initial starting point which is also the final ending point. This turns out to be with respect to each other and to themselves comparing their initial and final states.

PAllen said:
I don't think it is meaningful to talk about what part of a path responsible for its interval. Its a global property of the path and all parts contribute. Let's transfer the problem to Euclidean plane geometry. The hypotenuse of the triangle is shorter than the sum of the legs. Which part of the legs are responsible for the difference? Meaningless question.
I am not sure I follow you here.

In flat spacetime suppose we have two events E1 and E2 and two worldlines W1 and W2 between them, if W1 has acceleration and W2 not then it is always true that between these events W1 ages less than W2.

Passionflower said:
I am not sure I follow you here.

In flat spacetime suppose we have two events E1 and E2 and two worldlines W1 and W2 between them, if W1 has acceleration and W2 not then it is always true that between these events W1 ages less than W2.

I thought I was perfectly clear. There were earlier posts asking about 'what part' of the W1 path is responsible for the 'less aging'. The fact that it is non-inertial guarantees this, but I claim it is ludicrous to attribute the time difference to a potentially infinitesimal period of acceleration, or to 'coasting after acceleration'.

The analogy to a triangle is literally equivalent. Because the two legs of a triangle do not constitute a geodesic, their length is greater than the hypotenuse. Which part of the legs are the extra length? I claim this is completely meaningless. All parts contribute to the extra length, and the extra length 'exists' because the path is not straight. I think all further answers have no real meaning.

PAllen said:
I thought I was perfectly clear. There were earlier posts asking about 'what part' of the W1 path is responsible for the 'less aging'. The fact that it is non-inertial guarantees this, but I claim it is ludicrous to attribute the time difference to a potentially infinitesimal period of acceleration, or to 'coasting after acceleration'.
What do you think it is ludicrous?

At each point on W1's path we can connect a geodesic back to E1 and this geodesic will show a greater elapsed time. Both the acceleration period and the coasting period contribute to the time differential between W1 and W2.

Passionflower said:
What do you think it is ludicrous?

At each point on W1's path we can connect a geodesic back to E1 and this geodesic will show a greater elapsed time. Both the acceleration period and the coasting period contribute to the time differential between W1 and W2.

Which part of the legs of the triangle are 'extra' compared to the hypotenuse? If you want to say all are, that's ok, I guess, but I would view it as meaningless. Equivalent to asking which part of a quart of water is 'extra' compared to a pint of water.

Passionflower said:
What do you think it is ludicrous?

At each point on W1's path we can connect a geodesic back to E1 and this geodesic will show a greater elapsed time. Both the acceleration period and the coasting period contribute to the time differential between W1 and W2.
What does connecting a geodesic back to E1 have to do with it? The "contribution" of each section of W1's path to W1's total aging is just the proper time W1 experiences along that segment, no?

Passionflower said:
What do you think it is ludicrous?

At each point on W1's path we can connect a geodesic back to E1 and this geodesic will show a greater elapsed time. Both the acceleration period and the coasting period contribute to the time differential between W1 and W2.

We can also start from E2 working back or from somewhere in the middle working out. Again, the triangle analogy is exact. There is no meaning to any of these procedures.

PAllen said:
Which part of the legs of the triangle are 'extra' compared to the hypotenuse? If you want to say all are, that's ok, I guess, but I would view it as meaningless. Equivalent to asking which part of a quart of water is 'extra' compared to a pint of water.
I tried to understand what you say here.

I attached a small diagram, could you describe what is exactly meaningless?

The curved line is the accelerating one while the straight line is the geodesic.

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• worldline.jpg
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Passionflower said:
I tried to understand what you say here.

I attached a small diagram, could you describe what is exactly meaningless?

The curved line is the accelerating one while the straight line is the geodesic.

What is meaningless to claim that some part of the non-inertial path is what is responsible for the less aging. What is responsible is that it isn't inertial. Along the path, a second is a second, no part is more responsible than any other except by some arbitrary convention.

The analogy to a triangle is to ask which part of the legs is extra. It can't be answered. The legs are longer because the path isn't straight. There is no meaningful way to say the some part of the legs are extra.

PAllen said:
What is meaningless to claim that some part of the non-inertial path is what is responsible for the less aging. What is responsible is that it isn't inertial. Along the path, a second is a second, no part is more responsible than any other except by some arbitrary convention.
The differential aging depends on the total diversion from the geodesic path.

If three people go from event E1 to E2 and one goes inertially, one accelerates a little bit and the third one accelerates the most than it is always true that the third one ages less.

Passionflower said:
The differential aging depends on the total diversion from the geodesic path.

If three people go from event E1 to E2 and one goes inertially, one accelerates a little bit and the third one accelerates the most than it is always true that the third one ages less.
How are you quantifying accelerating a little vs. accelerating more? For example, B and C could both move inertially away from A at the same time, then B could accelerate back toward A earlier, then accelerate in the opposite direction when it got close to A so it was at rest with respect to A at some finite distance apart. Meanwhile C could accelerate later, turn back towards A, and then when C got close to A again B could accelerate a third time, so that both B and C met A at the same time. In this case, since B spent a lot of the time at rest relative to A, B would have aged less than A but more than C, but in this scenario it seems to me that B "accelerated more", especially since we could imagine that the magnitude and period of B's acceleration during all three acceleration phases was the same as the magnitude and period of C's single acceleration.

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Passionflower said:
The differential aging depends on the total diversion from the geodesic path.

If three people go from event E1 to E2 and one goes inertially, one accelerates a little bit and the third one accelerates the most than it is always true that the third one ages less.

I completely agree. This whole discussion goes back back to a some earlier posts (not by you) that claimed there was some meaningful way to identify which parts of the paths from E1 to E2 were responsible for the aging differences. Specifically, the period of acceleration or the coasting after acceleration were suggested. I claim this is meaningless. The total deviation from geodesic from E1 to E2 is the only thing that is meaningful. There is no meaning to saying it is because of the accelerated part or the 'second half' of the path rather than the first part of the path.

ghwellsjr said:
It is not the acceleration that causes the difference in the ages of the twins, it's the difference in the length of time that one twin spends coasting than the other twin that causes a difference in their ages.

Consider this variant (which I have read on this forum) to show that this is true:

Both identical twins start out at rest with respect to each other at the same age in the same (approximate) location. They both accelerate identically for some period of time. They then coast. During this interval of time, they will again be at rest with respect to each other and at the same age as each other and in the same (approximate) location as each other. Then one of the twins accelerates in the opposite direction as before until traveling back toward the starting point. Now both twins are coasting but in opposite directions. Then, at the appropriate time, the first twin decelerates to come to rest at the starting point. The second twin is still coasting away from the first twin and their common starting point. Later on, the second twin undergoes exactly the same acceleration in the opposite direction that the first twin experienced to bring him back home. He coasts until the approprate time comes for him to decelerate in exactly the same manner as the first twin to bring him to rest at the starting point. Now both twins will be at rest in their initial starting points but the second twin that spent more time coasting will be younger.

Note that in this variant, both twins experience exactly the same accelerations and deceleration so this cannot explain the difference in age. Rather, it's the difference in time that each twin spends coasting between the accelerations and deceleration that produces the different ages. The accelerations cause different aging rates in the twins but until they spend time at those different aging rates, their ages will remain the same.

PAllen, it looks to me like you didn't follow my variant of the Twin Paradox because all of your comments seem to me to be addressing a different variant that has nothing to do with what I said. Please study it carefully again and then if you think I said something that is not true, point it out specifically. I admit that sometimes my posts are ambiguous and open to different interpretations, but if that is the case here, I'd rather clarify it and fix it rather than have it dismissed out of hand with no idea what the problem is.

ghwellsjr said:
PAllen, it looks to me like you didn't follow my variant of the Twin Paradox because all of your comments seem to me to be addressing a different variant that has nothing to do with what I said. Please study it carefully again and then if you think I said something that is not true, point it out specifically. I admit that sometimes my posts are ambiguous and open to different interpretations, but if that is the case here, I'd rather clarify it and fix it rather than have it dismissed out of hand with no idea what the problem is.

What I said applies to this scenario and any scenario. Imagine you draw two lines on a plane between the two points. One line is longer than the other. It happens that there are some curves and segments between them that are similar. It is not meaningful in any way to say that the dissimilar parts are responsible for the extra length. Or, I have two containers of water, one bigger. The bigger one has a section the same shape as the smaller one. Is it meaningful to say the the water in the other part is the extra water, and is more responsible for the extra volume than the other water?

In your case, the only statement that has meaning is that one path has a greater total deviation from the geodesic path. All parts of the path contribute to the aging, and it is truly meaningless to single out particular sections as responsible for the smaller aging. Every part of the path contributes additively to the aging.

PAllen said:
What I said applies to this scenario and any scenario. Imagine you draw two lines on a plane between the two points. One line is longer than the other. It happens that there are some curves and segments between them that are similar. It is not meaningful in any way to say that the dissimilar parts are responsible for the extra length.
I see now what you meant, yes I fully agree with that.

PAllen said:
What I said applies to this scenario and any scenario. Imagine you draw two lines on a plane between the two points. One line is longer than the other. It happens that there are some curves and segments between them that are similar. It is not meaningful in any way to say that the dissimilar parts are responsible for the extra length. Or, I have two containers of water, one bigger. The bigger one has a section the same shape as the smaller one. Is it meaningful to say the the water in the other part is the extra water, and is more responsible for the extra volume than the other water?

In your case, the only statement that has meaning is that one path has a greater total deviation from the geodesic path. All parts of the path contribute to the aging, and it is truly meaningless to single out particular sections as responsible for the smaller aging. Every part of the path contributes additively to the aging.

In my variant it happens that all the acceleration intervals are identical between the twins, not just similar, and the two coasting parts are different between the twins and that is why I say that it is exclusively during the coasting intervals that the aging difference between the twins accumulates. Of course, all intervals contribute to their total ages, including the ones before the scenario began (from the birth of the twins up to the beginning of the scenario and the ones after the scenario ended up to the present moment). Why don't you say they all need to be included, too, to explain the age difference?

ghwellsjr said:
In my variant it happens that all the acceleration intervals are identical between the twins, not just similar, and the two coasting parts are different between the twins and that is why I say that it is exclusively during the coasting intervals that the aging difference between the twins accumulates. Of course, all intervals contribute to their total ages, including the ones before the scenario began (from the birth of the twins up to the beginning of the scenario and the ones after the scenario ended up to the present moment). Why don't you say they all need to be included, too, to explain the age difference?
Here's a nice spacetime diagram DrGreg once posted--if we flip it over vertically so the worldlines of A and B coincide when they initially depart rather when they finally return to meet C, would it match the variant you're talking about? You can see in this diagram that both A and B have identical accelerations, just at different times:

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