Twin Paradox Problem: Do Twins Age Differently?

[TrickyDicky]

Sorry for misunderstanding you. When you said "the real situation where the Earth twin is not inertial", in what sense were you saying that the Earth twin is not inertial?

I'm just separating the "formal plane" of the paradox solution in SR versus the real universe where unlike in flat Minkowski space, there are no pure inertial frames but they are nevertheless used in idealized situations where its use is irrelevant to the problem at hand.
When there is no actual inertial reference, I at least wouldn't know how to compute the time dilation that accumulates between the twins.

 

[stevendaryl]

The problem with Relativity's explanation for the Twin paradox, is that, once back on earth, for the traveling twin's clock to have a lesser time than the stay at home twin's clock, it can be deduced that the rate of time on the traveling twin's clock must have slowed down at some point during the journey.

No, that's not true. Not in any objective sense.

Here's an analogy. Suppose you have a system of roads, and every road has markers on it every 100 meters. You have two roads that meet at a point, diverge, and then come back together at a second point. You can compare the number of markers along the two different roads between the first meeting and the second meeting. You might find that one road has a greater number of markers between the two points than the other road. Does that mean that the road with the greater number of markers must have its markers closer together, or that the road with the small number of markers must have its markers farther apart? No, it's just that the distance between two points depends on the path taken. The number of markers is an accurate measure of these two distances.

In SR, the analogy of "number of markers along a road" is "number of ticks of a clock along a spacetime path". Two spacetime paths meet at some point, diverge, and then come back together later. The number of clock ticks between the two points is different for the two different paths. Does this mean that one clock's ticks come closer together, or that the other clock's ticks come farther apart? No, it's just that the proper time between two points depends on the spacetime path taken.

 

[TrickyDicky]

See the citations above. Once more: in post #62 I did not assume that the Earth is an inertial frame. Thus there is no need to think that it is important that the Earth twin ("the other observer") was inertial, contrary to what you continued to say. I picked the solar system rest frame, which is practically inertial over the course of two hundred years; and in that protocol neither twin was at any time "inertial".

Because you picked the solar system as inertial rest frame to refer and compute the accumulated time dilations of both twins, it makes no difference, I was stressing you need at least one inertial reference.

I'm not a mathematician. Here it is:
https://en.wikisource.org/wiki/On_the_Dynamics_of_the_Electron_%28June%29

You don't have to be to know that, none of us(surely I'm not) may be physicists and we are talking about SR right? ;-)

That must be the Lorentz group, but since you mentioned Poincare I thought you might be talking about the Poincare group which in this case is not needed because the origin is fixed in this problem.

 

[stevendaryl]

I'm just separating the "formal plane" of the paradox solution in SR versus the real universe where unlike in flat Minkowski space, there are no pure inertial frames but they are nevertheless used in idealized situations where its use is irrelevant to the problem at hand.
When there is no actual inertial reference, I at least wouldn't know how to compute the time dilation that accumulates between the twins.

I still don't understand what you're talking about. The reason that there are no inertial frames in the real universe is because of gravity. In the absence of gravity, Earth would be at rest in an inertial frame. But you seem to want to say that Earth is not inertial, and ALSO you want to ignore gravity.

There is no way to accurately compute elapsed times on clocks near the Earth without taking gravity into account. The elapsed time on a clock as it travels between events A and B is given by:

$\tau = \int_A^B \sqrt{g_{ij} dx^i dx^j}$

where $g_{ij}$ is the metric tensor coefficients.

For the Earth, a good approximation to $\tau$ for slow velocities is given by:

$\tau = \int_A^B (1 - \dfrac{G M}{c^2 r} - \dfrac{1}{2} \dfrac{v^2}{c^2}) dt$

 

[TrickyDicky]

I still don't understand what you're talking about. The reason that there are no inertial frames in the real universe is because of gravity. In the absence of gravity, Earth would be at rest in an inertial frame. But you seem to want to say that Earth is not inertial, and ALSO you want to ignore gravity.

There is no way to accurately compute elapsed times on clocks near the Earth without taking gravity into account. The elapsed time on a clock as it travels between events A and B is given by:

$\tau = \int_A^B \sqrt{g_{ij} dx^i dx^j}$

where $g_{ij}$ is the metric tensor coefficients.

For the Earth, a good approximation to $\tau$ for slow velocities is given by:

$\tau = \int_A^B (1 - \dfrac{G M}{c^2 r} - \dfrac{1}{2} \dfrac{v^2}{c^2}) dt$

I want to ignore gravity when talking about the SR solution, not in general.

 

[harrylin]

No, that's not true. Not in any objective sense. [..]

For me, "objective" in the context of SR is similar to "absolute": if all inertial frames agree that a statement is true. Evidently you mean something else with "objective sense", but what?
Your example didn't clarify that, as you merely explained a certain sense of interpreting the statement that you claim to be "not true" (and probably not corresponding to the way it was meant).

 

[ghwellsjr]

You mean the real situation where the traveling twin instantly accelerates to 90%c?

No. I mean what I said, the real situation where the Earth twin is not inertial. Besides, harrylin already gave the pertinent quote from Einstein himself stating wrt the results of the time dilation it doesn't matter whether the traveling twin moves in a curve or in a polygon line (which we all know is an unphysical way of accelerating), the important thing was that the traveling one was noninertial and the other observer was inertial.

Of course it's a purely imaginary exercise intended like all exercises to ignore all irrelevant factors.

Would you say the inertiality of the Earth twin is an irrelevant factor in the "twin paradox"?

Yes, it is, if you are referring to your comment, "the important thing was that the traveling one was noninertial and the other observer was inertial". It is neither important, relevant, significant, nor is it a requirement for the Twin Paradox.

In the simplest presentation of the Twin Paradox, we talk about the Earth twin as if the Earth had no gravity and no acceleration, which are of course not true and so the Earth and the Earth twin are considered to be inertial for the purpose of discussing the scenario. In this simplest presentation the other twin is non-inertial and so without knowing anything else, we can always say that the traveling twin is the one who ages less because he experienced acceleration whereas the Earth twin did not. This leads many people to falsely jump to the conclusion that it is the acceleration that causes the difference in aging between the twins and they look for explanations (a non-inertial frame or jumping between two frames) that support that incorrect idea. These people tend to reject the simple explanation that I offered in post #42 and quoted in post #52.

But we can complicate the Twin Paradox by having both twins follow exactly the same accelerations. They can both take off in identical spaceships and achieve 90%c. Then one of them immediately turns around and lands back on Earth for the rest of the time while the other twin continues on for a long time before repeating what the first twin did. So now both twins are non-inertial in exactly the same way so we can't jump to the false conclusion that the acceleration is what caused the difference in aging. But we can still analyze this more complicated Twin Paradox using the same inertial Earth reference frame as before. It's the simplest one to use, mainly because it is the one that is used to describe their motions.

In fact, it doesn't matter how the two twins accelerate or what speeds they achieve or what directions they travel in (polygons or circles or some of each) or where they end up together. We can always analyze their individual aging during the entire process from the standpoint of the inertial frame that we use to describe their activities.

Furthermore, with a little more work, we can determine what each twin sees of the other ones clock during the entire scenario, not just at the beginning and the end, and any analysis we do (whatever frame(s) we use) will all provide the same answers. We can make this as complicated as we want. But the more complicated analyses will not provide any more insight or information into what is happening.

 

[harrylin]

[..]I was stressing you need at least one inertial reference. [..]

Sorry that I understood what you said, and not what you meant.

So yes, SR refers to inertial frames, just like classical physics. And you now understand my explanation of how to calculate it accounting for the non-inertial Earth observer. Good.

[EDIT:] However, why then do you state in-between:

When there is no actual inertial reference, I at least wouldn't know how to compute the time dilation that accumulates between the twins.

That was what I thought to have explained you in post #62, and it appeared that you understood it...

But perhaps you mean that if we lack some necessary data (such as specifications that can be converted to an inertial frame), then we can't compute the outcome? That's very true!

 

[TrickyDicky]

ghwell, You completely missed my point, see my previous posts.
I was referring to the fact that there must be some inertial frame in the problem, not that it must be one of the twins.

 

[TrickyDicky]

Sorry that I understood what you said, and not what you meant.

So yes, SR refers to inertial frames, just like classical physics. And you now understand that the non-inertiality of the Earth twin is not a problem. Good.

Great, but I understood that before.

 

[stevendaryl]

For me, "objective" in the context of SR is similar to "absolute": if all inertial frames agree that a statement is true.

I don't consider that good enough for a statement to be objective. I think that it must also be the case that the terms mentioned in that statement have a meaning that is independent of observers. A statement involving "clock rate" can't be an objective statement, because there is no such thing as a clock rate. There is only a clock rate relative to a coordinate system.

It is objectively true that an inertial path connecting two spacetime points has a longer proper time than an accelerated path connecting the same two points. It is not objectively true that a clock following the accelerated path has a lower clock rate.

 

[stevendaryl]

I want to ignore gravity when talking about the SR solution, not in general.

I really don't understand what you are saying, then. For the twin paradox, you can either include gravity or not. Whether you do or not, you can calculate the elapsed times for the two twins.

 

[TrickyDicky]

I don't consider that good enough for a statement to be objective. I think that it must also be the case that the terms mentioned in that statement have a meaning that is independent of observers. A statement involving "clock rate" can't be an objective statement, because there is no such thing as a clock rate. There is only a clock rate relative to a coordinate system.

It is objectively true that an inertial path connecting two spacetime points has a longer proper time than an accelerated path connecting the same two points. It is not objectively true that a clock following the accelerated path has a lower clock rate.

I agree with this and in a way summarizes what I was saying in my previous posts.

 

[TrickyDicky]

I really don't understand what you are saying, then. For the twin paradox, you can either include gravity or not. Whether you do or not, you can calculate the elapsed times for the two twins.

Let's drop it, I don't have the slightest idea what you are talking about, sorry.

 

[harrylin]

[..] I was referring to the fact that there must be some inertial frame in the problem, not that it must be one of the twins.

Yes, I think that I understand you now. Your remark is true in general for physics. Without any idea of the state of motion of the participating objects, little can be predicted with certainty about the physical effects. In that sense is motion not just "relative", and Newton illustrated that famously with the bucket experiment. Einstein tried to solve that with GR, but if I correctly understand his later clarifications, he did not manage to make it truly Machian.

 

[stevendaryl]

Let's drop it, I don't have the slightest idea what you are talking about, sorry.

What I'm talking about is that I don't have the slightest idea what you are talking about.

 

[Nugatory]

When there is no actual inertial reference, I at least wouldn't know how to compute the time dilation that accumulates between the twins.

I want to ignore gravity when talking about the SR solution, not in general.

When we're talking about the SR solution, either we're conducting a thought experiment in an idealized flat space or we're conducting a real experiment in an environment where the gravitational effects are negligible. Either way, we use the Minkowski metric. (That's pretty much the definition of special relativity as a special case of the general theory).

We calculate the time elapsed for each twin as $\tau = \int_A^B \sqrt{g_{ij} dx^i dx^j}$ where the g_ij are the Minkowski metric components... and that calculation requires no actual inertial reference.

[Thanks to stevndaryl for the plagiarized latex]

 

[stevendaryl]

Because you picked the solar system as inertial rest frame to refer and compute the accumulated time dilations of both twins, it makes no difference, I was stressing you need at least one inertial reference.

What you need to compute proper times is a coordinate system with a known metric tensor.

 

[TrickyDicky]

Yes, I think that I understand you now. Your remark is true in general for physics. Without any idea of the state of motion of the participating objects, little can be predicted with certainty about the physical effects. In that sense is motion not just "relative", and Newton illustrated that famously with the bucket experiment. Einstein tried to solve that with GR, but if I correctly understand his later clarifications, he did not manage to make it truly Machian.

Right.

 

[TrickyDicky]

What I'm talking about is that I don't have the slightest idea what you are talking about.

It's nice when people reaches this level of agreement.

 

[TrickyDicky]

When we're talking about the SR solution, either we're conducting a thought experiment in an idealized flat space or we're conducting a real experiment in an environment where the gravitational effects are negligible. Either way, we use the Minkowski metric. (That's pretty much the definition of special relativity as a special case of the general theory).

We calculate the time elapsed for each twin as $\tau = \int_A^B \sqrt{g_{ij} dx^i dx^j}$ where the g_ij are the Minkowski metric components... and that calculation requires no actual inertial reference.

[Thanks to stevndaryl for the plagiarized latex]

Of course you can substitute the inertial reference in SR with Minkowski metric tensor, that is known since 1907.

 

[harrylin]

For me, "objective" in the context of SR is similar to "absolute": if all inertial frames agree that a statement is true. [..]

I don't consider that good enough for a statement to be objective. I think that it must also be the case that the terms mentioned in that statement have a meaning that is independent of observers. [..]

Perhaps for you an "objective statement" may only relate to invariants? I'm not that demanding... Check,http://dictionary.reference.com/browse/objective?s=t :
5. not influenced by personal feelings, interpretations, or prejudice; based on facts; unbiased

Based on the dictionary, I conclude that definitely also the sense in which robinpike seems to have meant the statement of post #68, is "objective". It's merely a different kind of "being objective" than yours.

Even more, with your definition, SR-type "time dilation" isn't even possibly part of an "objective statement", or am I mistaken?

 

[ghwellsjr]

ghwell, You completely missed my point, see my previous posts.
I was referring to the fact that there must be some inertial frame in the problem, not that it must be one of the twins.

I can only go by what you say, not by what you are thinking. But even the statement that there must be some inertial frame in the problem is not correct. There doesn't have to be any frame. Consider this:

Two observers are traveling toward each other at a relative speed of 90%c. They each observe the other ones clock running at 4.359 times their own. When they pass, they reset their clocks to zero. Now they each observe the other ones clock running at 0.2294 times their own. After a while, one of them turns around such that they are now approaching each other at 90%c and, like initially, the one that turned around immediatly sees the other ones clock running at 4.359 times his own. When they pass again, they compare the accumulated times on their respective clocks. The one that turned around sees the time on the other ones clock as 2.2942 times his own. You will note that this is exactly a restatement of Rishavutkarsh's problem in which he did not specify an inertial frame and you will note that I did not use any frame in my analysis in post #5, inertial or non-inertial.

If you want you can use any frame to analyze the problem that Rishavutkarsh stated but it will not provide any more insight or information into what is happening. For example, you could use a frame in which the inertial observer remains at rest. Or you can use a frame in which the non-inertial observer is at rest during the first part of the scenario. Or you can use a frame in which the non-inertial observer is at rest during the last part of the scenario. Or you can use a frame in which during the first part of the problem both observers are traveling in opposite directions at the same speed or in which this is true for the last part of the problem. And note that I always say "a" frame for each of these case because there are an infinite number of frames for each one that you can choose from. Frames are arbitrary and they don't change or influence what the observers see, measure, or observe. No frame is preferred over any other, even the rest frame(s) of the observers.

 

[ghwellsjr]

What you need to compute proper times is a coordinate system with a known metric tensor.

No you don't, I just did it in post #5 and again in post #93.

 

[stevendaryl]

Perhaps for you an "objective statement" may only relate to invariants?

I consider other things besides invariants to be objective, namely vectors and tensors. They have a meaning that is independent of observer (although their components are relative to a coordinate system).

I'm not that demanding... Check,http://dictionary.reference.com/browse/objective?s=t :
5. not influenced by personal feelings, interpretations, or prejudice; based on facts; unbiased

Well, it seems to me that the term "clock rate" is a subjective notion, and so IS based on interpretations.

Based on the dictionary, I conclude that definitely also the sense in which robinpike seems to have meant the statement of post #68, is "objective". It's merely a different kind of "being objective" than yours.

Well, it seems to me that you can always paraphrase a statement that is objective in your sense into a statement that is objective in my sense. You can say "For any coordinate system, there is a time at which the clock rate of the traveling twin is less than the clock rate of the stay-at-home twin, according to that coordinate system." That's objectively true.

Even more, with your definition, SR-type "time dilation" isn't even possibly part of an "objective statement", or am I mistaken?

There is a corresponding objective statement, which is that an inertial path connecting two spacetime events has a greater proper time than an accelerated path connecting the same two points.

 

[stevendaryl]

I wrote:

What you need to compute proper times is a coordinate system with a known metric tensor.

No you don't, I just did it in post #5 and again in post #93.

No, you didn't. You wrote:

Two observers are traveling toward each other at a relative speed of 90%c. They each observe the other ones clock running at 4.359 times their own. When they pass, they reset their clocks to zero. Now they each observe the other ones clock running at 0.2294 times their own.

How did you compute those two numbers, 4.359 and 0.2294, if not by using a metric?

Maybe you are just saying that you can observe the effects of time dilation without using any metric. I wouldn't say that you were computing it.

Maybe I should rephrase what I was saying. If you want to PREDICT how much elapsed time will occur on a moving clock, then you need to know a coordinate system for describing the clock's motion, and you need to know a metric for that coordinate system.

 

[TrickyDicky]

...even the statement that there must be some inertial frame in the problem is not correct. There doesn't have to be any frame. you will note that I did not use any frame in my analysis in post #5, inertial or non-inertial.
...

No frame is preferred over any other, even the rest frame(s) of the observers.

These statements don't make any sense within SR.
As soon as you are talking about percentages of c you are using inertial frames.
The last statement is true as long as there are only inertial frames, all equally valid of course.

 

[D H]

How did you compute those two numbers, 4.359 and 0.2294, if not by using a metric?

No metric is needed. For example, suppose the two ships regularly send time tagged messages to one another. Each message is time tagged with a time of transmission by the sending ship per that ship's clock. A time of receipt is added to the message by the receiving ship per that ship's clock. Computing those rates is simple; it's just a matter of looking over the message log files and comparing transmission times versus reception times. When the ships are approaching one another, each ship will say the other ship's clock is running fast; when moving away from one another, each ship will say the other ship's clock is running slow.

 

[ghwellsjr]

These statements don't make any sense within SR.
As soon as you are talking about percentages of c you are using inertial frames.
The last statement is true as long as there are only inertial frames, all equally valid of course.

Yes, within SR, you need to use frames. But you don't need to use SR to present or solve every relativistic problem which was the case in this thread. Of course, you can use SR if you want.

I disagree that talking about percentages of c means that I am using either SR or frames. Are you saying that prior to Einstein, nobody could talk about percentages of c? Never heard of such a thing. And if what you say is true, then what frame was I talking about in my response to you in post #93?

I wish your point that SR permits only inertial frames so that we could assume an inertial frame whenever in the context of SR we are talking about frames but unfortunately, I have learned that non-inertial frames are also just as valid within the context of SR.

 

[stevendaryl]

No metric is needed. For example, suppose the two ships regularly send time tagged messages to one another. Each message is time tagged with a time of transmission by the sending ship per that ship's clock. A time of receipt is added to the message by the receiving ship per that ship's clock. Computing those rates is simple; it's just a matter of looking over the message log files and comparing transmission times versus reception times. When the ships are approaching one another, each ship will say the other ship's clock is running fast; when moving away from one another, each ship will say the other ship's clock is running slow.

Yes, if you have a detailed record of transmission and reception times on the two ships, you can use that to demonstrate time dilation. What I mean is that there is no way to compute those transmission and reception times without using a coordinate system and a metric for that coordinate system. If I just tell you that I have two ships that start together, move apart, and then come back together, can you tell me what the elapsed times on the two ships will be? No, not without more information.

 

[TrickyDicky]

Yes, within SR, you need to use frames. But you don't need to use SR to present or solve every relativistic problem which was the case in this thread. Of course, you can use SR if you want.

I disagree that talking about percentages of c means that I am using either SR or frames. Are you saying that prior to Einstein, nobody could talk about percentages of c? Never heard of such a thing. And if what you say is true, then what frame was I talking about in my response to you in post #93?

I wish your point that SR permits only inertial frames so that we could assume an inertial frame whenever in the context of SR we are talking about frames but unfortunately, I have learned that non-inertial frames are also just as valid within the context of SR.

Classical Newtonian mechanics assumed infinite light-speed, it wasn't until Maxwell eq. that a finite constant c came to dispute that to Newton, and precisely that discrepancy triggered Einstein SR. Empirically of course it was known before Maxwell and Einstein that light speed was not infinite, but had not been introduced yet in the scientific equations of mechanics.

I never said that SR only admits inertial frames but that when other frames are used, inertial ones are the preferred ones.

 

[TrickyDicky]

Yes, if you have a detailed record of transmission and reception times on the two ships, you can use that to demonstrate time dilation. What I mean is that there is no way to compute those transmission and reception times without using a coordinate system and a metric for that coordinate system. If I just tell you that I have two ships that start together, move apart, and then come back together, can you tell me what the elapsed times on the two ships will be? No, not without more information.

Exactly, and this extra information is surely needed to solve the twin paradox, that is, to break the symmetry.

 

[harrylin]

[..] Well, it seems to me that you can always paraphrase a statement that is objective in your sense into a statement that is objective in my sense. You can say "For any coordinate system, there is a time at which the clock rate of the traveling twin is less than the clock rate of the stay-at-home twin, according to that coordinate system." That's objectively true.

Two people, one idea - I had in mind a similar reformulation of his original statement with which I think everyone would agree.

Reformulated with greater economy of words:

"it can be deduced that according to any inertial coordinate system the clock rate of the traveling twin's clock must have slowed down at some point during the journey."

As a matter of fact, that was the way in which it made "objective sense" to me; I took "any inertial coordinate system" as implied.

 

[harrylin]

Classical Newtonian mechanics assumed infinite light-speed [..] [that speed] had not been introduced yet in the scientific equations of mechanics.

That is definitely wrong: it was included since Huygens and Newton.

Right. [Without any idea of the state of motion of the participating objects, little can be predicted with certainty about the physical effects. In that sense is motion not just "relative"]

As a matter of fact, it was the very purpose of Langevin's "twin" example to demonstrate the "absoluteness" of non-inertial motion; and apparently it was that issue that led to the "paradox". As his paper is a bit long-winded I'll give a synopsis that is focussed on the topic here:

The Foucault pendulum and the gyroscope demonstrate that although uniform translation has no absolute sense, rotation does have such an absolute sense.

For systems in uniform translational motion it is as if they are stationary relative to the aether: uniform motion in the aether has no experimental sense. However, one should not conclude that the aether concept must be abandoned, that the aether is non-existent and inaccessible to experiment. Uniform velocity relative to it cannot be detected, but any change of velocity, any acceleration has an absolute sense. We have therefore hold on the ether by means of accelerations; acceleration has an absolute sense as it causes the production of electromagnetic waves by matter that undergoes a change in velocity, and the ether manifests its reality as a vehicle, as support for the energy that is carried by these waves. We will see this absolute character of acceleration manifest itself in another form.

Whoever of us who is willing could explore the future Earth by making a leap forward in time which for the Earth will last two centuries but for him will last two years, however without any hope of coming back to inform us of the outcome of the voyage.
He should agree to be locked up in a projectile that will be launched from Earth with a velocity close to that of light, arranging an encounter with a star after one year of the traveller's life and which sends him back to Earth with the same velocity. Back on Earth, he will find our world aged by two hundred years if his velocity was only one twenty-thousandth less than the velocity of light. It is fun to describe how our explorer and the Earth would see each other live if they could [etc.,]
- http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

This demonstration of the "absoluteness" of acceleration became paradoxical ("Twin Paradox") due to Einstein's contrary claim that "all motion is relative".

Einstein defended that contrary view in 1918 with the following paper:

http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

The historical Twin paradox is essentially about Newton vs. Mach (or "absolutism" against "relativism"), as presented by Langevin vs. Einstein.

 

[TrickyDicky]

That is definitely wrong: it was included since Huygens and Newton.

Nope, you got that wrong about Newton mechanics and I didn't mention optics or Huygens.