Acceleration and the twin paradox

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Can you please go through the steps to show us how you would get the answer to the OP's first question in his section 2b?
When each of the twins is accelerating, it's harder to determine the perspective of each twin, but it is possible. When I joined this forum, I spent a lot of time doing searches on the twin paradox, and I remember that several old posts did talk about how to get the perspective of each twin when they are both accelerating. So far, I haven't been able to find any of those posts again. Maybe some other forum members remember them, and maybe they will have better luck finding them than I've had. I don't think Brian Greene talks about that kind of situation in his book or in the NOVA series, though.
 

ghwellsjr

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When each of the twins is accelerating, it's harder to determine the perspective of each twin, but it is possible. When I joined this forum, I spent a lot of time doing searches on the twin paradox, and I remember that several old posts did talk about how to get the perspective of each twin when they are both accelerating. So far, I haven't been able to find any of those posts again. Maybe some other forum members remember them, and maybe they will have better luck finding them than I've had. I don't think Brian Greene talks about that kind of situation in his book or in the NOVA series, though.
Then can you do it for any of the other scenarios listed in this thread where only one twin accelerates?
 

PAllen

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Not at all. The "fixed stars" are supposedly fixed in a reference frame that Newton defines to be in rest. He mentions several ways to detect acceleration relative to it. And he next provides laws (effectively equations) that are supposed to hold relatively to the so defined rest. The concept of what in the 20th century was called "inertial frames" is introduced in a corollary as follows:
"Corollary V:
The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion."
He thereby reused Galileo's illustration: "A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line."

For completeness, apart of the fact that SR does "not require an “absolutely stationary space” provided with special properties", these are the coordinate systems of the Lorentz transformations that the OP asked about:
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. [..] two systems of co-ordinates in uniform translatory motion" - Einstein 1905.
To me it seems he says we have no way of knowing if even the fixed stars are at rest, and that absolute rest can never be determined:

"But we may distinguish Rest and Motion, absolute and relative, one from the other by their Properties, Causes and Effects. It is a property of Rest, that bodies really at rest do rest in respect of one another. And therefore as it is possible, that in the remote regions of the fixed Stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body; it follows that absolute rest; cannot be determined from the position of bodies in our regions."
 

PAllen

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As a matter of fact, Einstein explicitly referenced "fixed stars" in 1919, even in the way that I answered the OP's question:

"Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer to another body. The motion of a railway train is described with reference to the ground, of a planet with reference to the total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termed systems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system of coordinates. The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system." - https://en.wikisource.org/wiki/Time,_Space,_and_Gravitation
But this formulation gives no privilege to fixed stars as a reference point at all.
 
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To me it seems he says we have no way of knowing if even the fixed stars are at rest, and that absolute rest can never be determined
To me it seems that Newton assumed the "fixed stars" to be fixed at rest; and despite the fact that there are no truly fixed stars, it works rather well in practice as also Einstein explained.
But this formulation gives no privilege to fixed stars as a reference point at all.
Question 1 was not about a reference point but about how to determine SR's inertial frames, which serve as reference for acceleration. Einstein there answers Diracpool's question in roughly the same way as I did earlier in this thread, only I had added some other ways to detect acceleration; I hope that he/she finds it useful!
 
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Motion is a phenomenon requiring multiple objects, with one serving as a reference.
The “fixed” stars are labeled as such only as an approximation, since their motion is imperceptible from local observations. They serve the purpose of a fixed inertial frame very well, if the experiments don’t take too long!
The universe as an integrated entity (all things) is then by definition the only fixed object.
 
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After many years of agonizing over it, I have still failed to come to terms with the twin paradox.
I think I can understand the agony of OP. Or maybe we have different reasons, but I'm not comfortable with twin paradox either. Lorentz transformations, spacetime diagrams etc. do not explain why there is a difference in aging. They only describe what kind of difference or how much there is difference.

The closest thing of explanation that I have found so far about "why" part is this: The universe insists that light speed must be the same in every inertial frame and in order to achieve this, it's willing to give up pretty much everything else, provided only that the system must remain consistent. This is "why" there is the strange difference in aging which doesn't seem to make any sense.

If this sounds bitter, it's probably because it is. I really hope that time dilation and other aspects of relativity are understood some day much better than now.
 

A.T.

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The closest thing of explanation that I have found so far about "why" part is this: The universe insists that light speed must be the same in every inertial frame
Yes, the explanation "why" in the traditional formulation of SR is always the two postulates.

I really hope that time dilation and other aspects of relativity are understood some day much better than now.
I think that they are understood as completely as anything in science is ever understood. SR is actually very easy to understand. It is just geometry, but instead of using the Euclidean metric you use the Minkowski metric. All of SR is contained in a single equation (##ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2##) which is reasonably easy to understand.
 

ghwellsjr

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Instead of answering the same questions about twin paradox again and again, I will just put a link to my already written (and published) answer:
http://lanl.arxiv.org/abs/physics/0004024 [Found.Phys.Lett. 13 (2000) 595-601]
Can you please go through the steps to show us how you would get the answer to the OP's first question in his section 2b?
 
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If this sounds bitter, it's probably because it is. I really hope that time dilation and other aspects of relativity are understood some day much better than now.
It is pretty well understood by now I think ;) You just need to understand the concept of spacetime interval, which is to spacetime what pythagoras is to space: $$d \tau ^2 = dt^2 - dx^2$$ and computes the elapsed proper time between two events. That one equation is all there is to it, you just need to imagine two different paths through spacetime between the same two events. Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations. You don't even need to calculate or measure anything, just keep saying the previous sentence as many times as it takes.

Forget talk of acceleration, seriously; that is just overcomplicating things. You don't need to agonize over the Lorentz transform either, as it's built in to the definition of the spacetime interval.
 

Demystifier

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Can you please go through the steps to show us how you would get the answer to the OP's first question in his section 2b?
I can only sketch the procedure and leave the details as an exercise.

1) Take first the traveler A. Evaluate (6) for t'=t'_A and insert the result in (5) to obtain t=f_A(t'_A) with some explicit function f_A.
2) Now take the traveler B and repeat the same procedure to get t=f_B(t'_B) with another explicit function f_B.
3) To compare 1) and 2) use f_A(t'_A)=f_B(t'_B), which gives you an implicit relation between t'_A and t'_B. Try to rewrite this relation in an explicit form as t'_A=F(t'_B).
 

WannabeNewton

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Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations.
It's seriously as simple and as elegant as that. If one can understand this then I don't see any reason to find the twin paradox conceptually non-trivial.
 
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Then can you do it for any of the other scenarios listed in this thread where only one twin accelerates?
I can't remember how the calculations are done. I do remember seeing some old posts on this forum where it was explained how to do it, though, and it wasn't very hard. And I remember that the result was that the while the traveling twin is accelerating toward the home twin, the traveler will say that the home twin is rapidly getting older, and that that explains how the traveler can find the home twin older at the end of the trip, even though the traveler says that during most of the trip (when he isn't accelerating), the home twin is aging more slowly than the traveler.
 

Demystifier

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It is pretty well understood by now I think ;) You just need to understand the concept of spacetime interval, which is to spacetime what pythagoras is to space: $$d \tau ^2 = dt^2 - dx^2$$ and computes the elapsed proper time between two events. That one equation is all there is to it, you just need to imagine two different paths through spacetime between the same two events. Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations. You don't even need to calculate or measure anything, just keep saying the previous sentence as many times as it takes.

Forget talk of acceleration, seriously; that is just overcomplicating things. You don't need to agonize over the Lorentz transform either, as it's built in to the definition of the spacetime interval.
This, indeed, is the simplest way to understand the twin paradox, provided that you accept the geometrical Minkowski view of special relativity. But the original Einstein formulation of special relativity did not have such a geometrical form, moreover Einstein at first didn't like the Minkowski geometrical formulation (he later changed his mind when he applied such a view to construct general relativity), and most importantly, special relativity in introductory textbooks is usually not taught in such a geometrical form. That's why many people still seek an explanation in terms of Lorentz transformations or something alike. My post #31 above offers such a coordinate-transformation explanation, for those who want it.
 
The Twin Paradox is an application of the simple math of lorentz transforms, and unfortunately it was designed to mystify.

The lorentz transform says that when you assign velocity to another observer, his distances contract and his clock slows. At the same time, when he assigns the velocity to you, he calculates that your distances contract and your clock slows. The situations are strictly symmetric, but that does not mean that you have the same distances and clock speeds. To him, yours are shorter and to you, his are shorter. They are shorter than each other. In reality, of course, we can't tell who has the velocity, we can only assign it arbitrarily. What we can measure is relative velocity.

We don't need to consider acceleration, we can do the trick with magic. He starts out beside you, with no velocity. By magic, create a relative velocity that's close to lightspeed. You calculate that his clock runs slower.

Then after a suitable long time, by magic reverse his velocity so he approaches you again. You still calculate that his clock runs slower. the

t' = gamma (t-vx)

formula gives for him t+vx in place of t-vx on the return trip, which might appear to reverse his clock slowing because of your random choice of x axis, but the gamma part stays the same and makes the sum of leaving and coming back smaller.

So when he arrives at your location, still speeding by at close to lightspeed, you will calculate his time has passed slower while he calculates that your time has passed slower. Then by magic you cancel his velocity, leaving him with a slower clock than you. It follows straight from the math, and the magic.

We can do some of this without the magic. Like in the classic homework problem of the mesons arriving from space at nearly lightspeed. They weren't accelerated to nearly lightspeed, they were born that way. We calculate that their clocks run slow so they decay slow. Somebody traveling with them would decide that our clock runs slow, and the world has become paper-thin so they just punch right through before they decay. Which version is right? They both fit the facts, of course. From our point of view they "really" have their clocks run slow so they decay slow. We leave out the step where we magically slow their velocity to zero while resetting their own clocks to fit our calculation of them. ;-)

If you complicate it with acceleration, you can get reasonable results without magic. But that's a complication on the original idea.
 
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That's why many people still seek an explanation in terms of Lorentz transformations or something alike. My post #31 above offers such a coordinate-transformation explanation, for those who want it.
More specifically, I think the key to it is understanding what coordinate transformations do to a "now" (simultaneity plane), and from that how you have two completely different definitions of "now" right before and right after switching reference frames (acceleration).
 
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As a learner, I find myself disagreeing more and more strongly about this Lorentz Transform approach to the twin "paradox". IMO SR needs to be taught in an operationally practical way, not as a "like wow, weird" sort of way. We are some way towards ditching the concept of relativistic mass, but we're still stuck with the Mr Tompkins view of moving clocks going slower and length contraction which we surely by now all know is NOT what you would really see. Relying on this more complicated Lorentz Transform analysis puts me in mind of Lorentz Ether theory. There's nothing to stop you doing it, but it's neither necessary or helpful to a novice, it comes across to me as just a historical throwback to the older, harder way of doing things (which of course is interesting in its own right to a more advanced student).

Relativity is confusing enough, let's keep it as simple as we can when talking to beginners. Let's not put artificial difficulties in their path. The key to my understanding of this was plotting events on a spacetime diagram and putting in light lines to observe the doppler effect and see the delay and rate changes that an observer would really see on various clocks in the system. These are real things, seen with the eyes or a telescope, rather than unobservable coordinate artifacts. I am unrepentant ;)
 
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Relying on this more complicated Lorentz Transform analysis puts me in mind of Lorentz Ether theory.
But that's part of the beauty of it - you can start to think of it as LET centered on some preferred frame, and then realize that any other inertial frame will do the job equally well because of the symmetry of the transforms, getting to the "relativity" part of it in a natural way.
 
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I have no issue whatsoever with that approach for those who are interested, just don't expect poor newbies (like the ones in this thread) to follow that kind of discussion!
 
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My post #31 above offers such a coordinate-transformation explanation, for those who want it.
About arXiv: physics/0004024 The paper was published in reputable journal [Found.Phys.Lett. 13 (2000) 595-601]. It has no evident flaws. It has following characteristics: 1) the effect of memory in form of integration. 2) The Special Relativity is based on two postulates, isn't it? How many postulates has this paper? It can be important theoretical construction for wide range of the natural processes.
 

Nugatory

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About arXiv: physics/0004024 The paper was published in reputable journal [Found.Phys.Lett. 13 (2000) 595-601]. It has no evident flaws. It has following characteristics: 1) the effect of memory in form of integration. 2) The Special Relativity is based on two postulates, isn't it? How many postulates has this paper? It can be important theoretical construction for wide range of the natural processes.
I don't understand what you mean by "the effect of memory in the form of integration". as for you rsecond question, Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.
 
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I don't understand what you mean by "the effect of memory in the form of integration".
Thank You and I am sorry for lack of clarity. Under the effect of memory is meant: the coming behavior of a body depends on the previous history of this body. In common worldview it is enough to know position and velocity of a body to know its future evolution. Here it is not the case, am I right? I understand, that it is not easy question for the rapid answering. Author of this "memory" remark is Dmitri Martila.
 

Demystifier

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I don't understand what you mean by "the effect of memory in the form of integration". as for you rsecond question, Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.
I agree.
 

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