Acceleration and the twin paradox

[DiracPool]

After many years of agonizing over it, I have still failed to come to terms with the twin paradox. Here's a brief review of my understanding and a few questions:

A standard story is as follows: Twin A leaves the Earth for planet Zolan 10 light years away. Twin B stays on Earth. Let's say that, over a period of a week, twin A accelerates to 0.8c (relative to Earth and Zolan, which are at rest relative to each other) and then remains cruising at that velocity, decelerates for a week upon approaching Zolan, makes a sweep around the planet (waves at everyone), begins to accelerate once more towards Earth for a week, reaches a cruising speed of 0.8c, and then finally decelerates for a week before landing safely on the Earth and having lunch with twin B.

Now, I'm not worried about the exact figures, but we can all agree that twin A will have aged more slowly than twin B and will be "younger" than twin B.

Why is this? The main problem most people seem to have is how do you get around the apparent symmetry of relative motion problem between the two twins.

In order to address this, popular consensus among physicists/cosmologists that I've seen is that this apparent symmetry becomes broken once twin A accelerates from the Earth and enters a non-inertial frame, and this is what is responsible for the slowing of twin A's aging relative to twin B. I have a few questions about this.

1) "twin A accelerates from the Earth and enters a non-inertial frame"--a non-inertial frame relative to what? To the earth/twin B? Some objective measure of spacetime displacement? To the CMB? I guess what I'm asking is that, is there a way to measure the acceleration of an object/frame without some measure of what to say it is accelerating in relationship to? I'm guessing there's a stock answer to this question, but I'm remiss to recall it at this moment.

2) Lorentz transformations (LT): The equation determining the time dilation factor in twin A's slowing of aging is, AFAIK, contained completely with the LT equations. However, I see no term or provision at all for acceleration in the time dilation equation. So how does acceleration play a role quantitatively in this paradox when it seems as if all you need is a simple velocity difference to create the dilation effect? Are there some equations I am not aware of that address this, or some kind of formalism that can be used to quantify these measures? I'll give three examples of scenarios I'm having difficulty seeing how the standard formalism can address:

2a) Say twin A were to accelerate (and decelerate) for 2 weeks instead of one week in both directions from Earth to zolan, but compensates for the lost time by cruising at a speed slightly greater that 0.8c so that the two week acceleration round trip flight time was exactly the same as the one week. Would that make any difference in the age difference between twin A and twin B upon twin A's return?

2b) Say twin A took the exact same trip as above (say the one week version), but now twin B also took a similar trip in the opposite direction to planet Xanadu 5 light years away, but accelerated at a different rate than twin A, and reached a different cruising speed that was, say, 0.5c. How would we calculate their age differences in this situation once they both returned to earth? Could we handle it with only the LT and the relativistic velocity addition formulas? If not, then how?

2c) Finally, let's say we have twin A again going on his trip to Zolan (one week acceleration edition), but now we have twin B centrifuging himself in a 2001: Space odyssey style space station, undergoing rapid centripetal (or is it centrifugal?) acceleration. And then twin A and twin B go for lunch on Earth when twin A comes back. Does that make a difference in their age? Or should twin B just as well stayed on Earth during twin A's trip?

I guess the more broader encompassing question is what it is specifically about twin A's undergoing an acceleration or transition into a non-inertial frame that sets his clock running slower than twin B's. It seems to be this act of accelerating that does it, not simply a relative difference in velocity. But this is the was the LT's lead you believe.

 

[A.T.]

After many years of agonizing over it,.

Have you ever discussed it here? Or read the other threads on it (see bottom of page)? Your questions seem like the usual ones, answered 1000 times already.

I guess what I'm asking is that, is there a way to measure the acceleration of an object/frame without some measure of what to say it is accelerating in relationship to?

Yes, with an accelerometer:
http://en.wikipedia.org/wiki/Proper_acceleration

 

[Nugatory]

You should also work through the FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html. Look especially at the Doppler and spacetime diagram sections.

Part of your problem is that

In order to address this, popular consensus among physicists/cosmologists that I've seen is that this apparent symmetry becomes broken once twin A accelerates from the Earth and enters a non-inertial frame, and this is what is responsible for the slowing of twin A's aging relative to twin B.

is not right. The acceleration is almost a red herring. It doesn't explain the differential aging; it does provide an objective basis (their accelerometers read different values) for saying that they have different experiences so it's at least possible for them to end up with different outcomes.

Another misunderstanding you need to get rid of:

twin A accelerates from the Earth and enters a non-inertial frame"--a non-inertial frame relative to what?

Velocities are frame-dependent and relative, but whether a given frame is inertial an invariant property of that frame and independent of all other frames. If I am at rest relative to the origin of a frame, and I am experiencing no acceleration as measured by an accelerometer, then that frame is inertial.

 

[m4r35n357]

In fact it is possible to observe differential aging even in inertial travel between two points in space. See the second part of this link for an example: http://mathpages.com/rr/s6-05/6-05.htm

 

[zoki85]

Ugh, beating that dead horse again

 

[Dale]

twin A accelerates from the Earth and enters a non-inertial frame

This wording is incorrect. You do not enter or leave a reference frame. A reference frame is a mathematical construct, a tool for analysis. It is not a physical object which you can enter or leave.

If being "in a reference frame" were to have any meaning then you would have to say that every object is always "in" every reference frame. They are just not at rest in most reference frames. So, the correct wording is that "twin A accelerates from the Earth so A's rest frame is non-inertial".

a non-inertial frame relative to what?

A reference frame is inertial or not inertial without reference to anything else. Similarly, a given object is inertial or non-inertial in an absolute sense, not relative to any reference frame. You do not need to specify "relative to what" because it is not relative.

 

[DiracPool]

The acceleration is almost a red herring. It doesn't explain the differential aging

Well, this is a good example of where my trouble lies because the "acceleration solution" is what I keep coming across in my perusals on the matter. Here are a couple of instances:

Fast forward to 1:29:00 to listen to Lenny Susskind's comment on the legitimacy of using the acceleration of the traveling twin to explain the twin paradox.



Also, fast forward here to 4:46 where this guy from sixty symbols starts off with, "The resolution to this paradox..."

Have you ever discussed it here? Or read the other threads on it (see bottom of page)? Your questions seem like the usual ones, answered 1000 times already.

Yes I have discussed it here, I've followed many of the related threads and have even started one or two. And I'm still not comfortable with my understanding of it or I would have not made the post. Case in point, I'm not getting from the two physicists' videos I posted that "acceleration is a red herring," I'm getting from them that it's an explanation (of sorts). And the sixty symbols guy goes on to say that you need to invoke general relativity to address the problem. But I've also heard that you don't need to invoke GR to explain the twin paradox. So...

Ugh, beating that dead horse again

From what I can see this horse is not quite dead yet. At least from the level of understanding I am able to acquire, but's why I'm asking here from those "in the know." Hopefully I can glean some insight from the links provided and/or any further discussion in this thread.

 

[PeterDonis]

I'm not getting from the two physicists' videos I posted that "acceleration is a red herring," I'm getting from them that it's an explanation (of sorts).

It is part of the explanation in this particular scenario because, as Nugatory said, it is a physical difference between the two twins: the traveling twin feels acceleration for at least some portion of his journey, while the stay-at-home twin never does--he's in free fall, feeling no acceleration, the whole time.

However, this difference is not the complete explanation: it can't be, because, as others have commented, there are other scenarios in which both "twins" are in free fall the whole time, but still have different ages when they meet up again. So just saying "acceleration explains the twin paradox" is not sufficient, because that "explanation" doesn't generalize.

Here's an explanation that does generalize: two twins that take different paths through spacetime between the same two events (where they separate, and where they meet up again) can experience different elapsed times between those two events. The role acceleration plays in the standard twin paradox is to explain how both twins can pass through the same pair of events--the traveling twin has to turn around, and he has to feel acceleration when he does, because the scenario is set in flat spacetime. In the scenarios where both twins are in free fall the whole time, spacetime is curved, so there can be multiple free-fall worldlines between the same pair of events, and they can have different elapsed times along them.

Many discussions of the twin paradox (including, apparently, the videos you linked to) don't go to this level of generality, which is a shame, because it makes it harder for people to understand the general rule. One of the reasons I like the Usenet Physics FAQ article on the twin paradox is that it explicitly talks about the general rule (in the Spacetime Diagram section).

 

[Dale]

the sixty symbols guy goes on to say that you need to invoke general relativity to address the problem

The sixty symbols guy is flat out wrong on that point if that is what he said.

 

[Nugatory]

Well, this is a good example of where my trouble lies because the "acceleration solution" is what I keep coming across in my perusals on the matter. Here are a couple of instances:

That's why I pointed at you at two specific sections of that FAQ. Read them.

 

[Dale]

DiracPool, suppose that you had a piece of paper with two points on it, one labeled "start" and one labeled "finish". Suppose further that there were two paths connecting the start and finish, one is a straight line and the other has a bend in it.

Do you understand that the bent path is longer? Could you explain why the lengths are not the same?

 

[Jimmy]

Do you understand that the bent path is longer? Could you explain why the lengths are not the same?

\rhetorical: Which part of the bent path contributes the extra length?

 

[DiracPool]

That's why I pointed at you at two specific sections of that FAQ. Read them.

I plan on it, thanks.

Do you understand that the bent path is longer? Could you explain why the lengths are not the same?

Because one is curved and one is straight, and the shortest distance between 2 points is a straight line? Is that a trick question or am I missing something? Lol

One of the reasons I like the Usenet Physics FAQ article on the twin paradox is that it explicitly talks about the general rule (in the Spacetime Diagram section).

Is that the same one Nugatory posted? Yes, I plan to read it, thanks.

 

[Dale]

Is that a trick question or am I missing something? Lol

It is not a trick question. The analogy to the twin paradox is almost exact, for reasons that hopefully will become clear to you later. If you really understand the path lengths then you will understand the twin paradox also.

Because one is curved and one is straight, and the shortest distance between 2 points is a straight line?

Why is the shortest distance between two points a straight line? If you were speaking with someone who was unfamiliar with geometry on a plane, how could you explain to them why a straight line is the shortest distance or how to determine the length of a path? In particular, what would you do to explain the asymmetry in the lengths?

 

[georgir]

Here's an explanation not involving curvature.

I will assume you know what a space-time diagram is... now think about what is "now" for a certain event on such a diagram. This is what is called a simultaneity plane in literature, and is represented as a simultaneity line on the diagram and is actually a simultaneity space in reality :p Hint, it is the horizontal line though the event on the diagram.

For the two diagrams representing the two different reference frames centered on our traveller right before he turns around and right after he turns around (assume instantaneous acceleration, for simplicity), the two simultaneity lines will be different.

You can plot each "now" on the other space time diagram to understand it better, but the specifics are not really important. It simply boils down to "relativity of simultaneity". When you accelerate, or in other words when you switch to a different reference frame, you change the definition of what you call "now". So you can say that the stay-home twin's clock goes slower before you accelerate, goes slower after you accelerate as well, and yet shows more time in the end when you return because of this jump in your "now" right when you accelerate.

 

[A.T.]

I'm not getting from the two physicists' videos I posted that "acceleration is a red herring," I'm getting from them that it's an explanation (of sorts).

Explanation of what exactly? Differential proper acceleration is the reason why there is no symmetry. And if you want to describe the whole thing from the non-inertial twin's frame, based on certain simultaneity conventions, then proper acceleration affects how space time coordinates look there.

 

[harrylin]

[..] a non-inertial frame relative to what? [..] I'm guessing there's a stock answer to this question [..]

There are several standard answers for that, all stemming from classical mechanics. There are also answers from General Relativity that are incompatible with the former. Special relativity uses (or at least, used!) the definitions of classical mechanics. IMHO it's not helpful to mix them up, but regularly answers are given in the context of "flat" GR, and that commonly adds to confusion. More likely you are searching for a clear understanding in the context of pure SR, without GR jargon.
If not, you can ignore what follows. :)

Concerning SR and classical mechanics: Newton used the concept of the "fixed stars"; and despite the fact that far away stars are not really "fixed", that serves rather well up to a very high precision. As classical mechanics and SR consider free fall in gravitational fields to be non-inertial, another way to approximate a measure of inertial motion is to consider objects that are freely moving far away from heavy masses. And of course, one can use the observed laws of effects of heavy masses to make corrections to readings of accelerometers. This should have been clearly explained when you learned classical mechanics, but many textbooks fail in this respect.

The equation determining the time dilation factor in twin A's slowing of aging is, AFAIK, contained completely with the LT equations. However, I see no term or provision at all for acceleration in the time dilation equation. So how does acceleration play a role quantitatively in this paradox when it seems as if all you need is a simple velocity difference to create the dilation effect? [..]

Your point is totally correct. SR does not account for any effect of acceleration itself on the speed of clocks; only indirectly, by means of a change of velocity, does acceleration have an effect. A change of velocity is required in order for the two to meet again, it's as simple as that! BTW, this was already pointed out by Langevin, before it became a "paradox"...

Elaboration: the point is that as long as the two are in inertial motion (SR definition), a comparison of their clocks by means of coordinate systems in which each remains in rest must be symmetrical according to the LT equations. The only way to break this symmetry (according to SR) is that at least one of the two changes velocity. Only by the rather insignificant fact that a change of velocity is called "acceleration", does acceleration play an important role in this account according to SR. No change of velocity -> no asymmetry possible - and that's all there is to the essential role of "acceleration"!

From there on it's all rather trivial, similar to, as mentioned in Susskind's video, a Pythagoras calculation. In the most simple variant you can pick an inertial coordinate system in which the one is at rest, and then you easily find how much the time of the other one will be delayed when they meet up again. And, as you noticed, quantitatively only the speed matters for that case.

Also, just as in classical mechanics, you can switch and jump frames as and when you like for your analysis: that complicates the calculation but it won't change the answer (of course).

On a sidenote, Susskind's remark about "experiencing acceleration" is a bit misleading; nothing about sensorial perception is in the SR equations or in SR theory, as you likely noticed. I doubt that he read the old literature, for in the first two examples of such calculations (by Einstein and by Langevin, neglecting the effect of gravitation on time keeping), locally no detectable acceleration is experienced by the traveling clock or by the traveler during the time under consideration.

PS while writing this, the Susskind video was still playing; I spotted a number of inaccuracies, regretfully... but most of those are of no consequence for the topic here.

 

[PAllen]

There are several standard answers for that, all stemming from classical mechanics. There are also answers from General Relativity that are incompatible with the former. Special relativity uses (or at least, used!) the definitions of classical mechanics. IMHO it's not helpful to mix them up, but regularly answers are given in the context of "flat" GR, and that commonly adds to confusion. More likely you are searching for a clear understanding in the context of pure SR, without GR jargon.
If not, you can ignore what follows. :)

Used correctly, I think relating SR and GR clarifies the issues.

Concerning SR and classical mechanics: Newton used the concept of the "fixed stars"; and despite the fact that far away stars are not really "fixed", up to a certain precision that serves rather well. As classical mechanics and SR consider free fall in gravitational fields to be non-inertial, another way to approximate a measure of inertial motion is to consider objects that are freely moving far away from heavy masses. And of course, one can use the observed laws of effects of heavy masses to make corrections to readings of accelerometers. This should have been clearly explained when you learned classical mechanics, but many textbooks fail in this respect.

Do you have a reference for Newton using fixed stars? My knowledge of history is that the distance to, and any concept of the nature of fixed stars came only with Herschel centuries after Newton's death. Einstein never referenced 'fixed stars'. Newton, I believe, discussed the idea that a spinning bucket was distinguishable in a universe with no stars at all.

A correct, IMO, way to distinguish SR inertial frame from GR inertial frame is to exploit geometric understanding from GR rather than reject it. SR is a theory of flat Minkowskian space-time in which global inertial frames are well defined, and global comparison of velocities are well defined (due precisely to absence of curvature). Given that global comparison of velocities are well defined, acceleration is globally defined relative to global inertial frames, rather than being a local observation due to accelerometers. This means that in SR (irrespective of the fact that no satisfactory theory of gravity is possible), readings of accelerometers have no fundamental significance (because they are a complex dynamical instrument, rather than in expressing - in the ideal - a feature of geometry as in GR).

Your point is totally correct. SR does not account for any effect of acceleration itself on the speed of clocks; only indirectly, by means of a change of velocity, does acceleration have an effect. A change of velocity is required in order for the two to meet again, it's as simple as that! BTW, this was already pointed out by Langevin, before it became a "paradox"...

It has also been pointed out by several posters in this thread, as well as in the referenced FAQ.

[edit: A clarification on acceleration in SR. As I mentioned, in SR an accelerometer is not a 'fundamental' instrument as is a clock or or rod. However, acceleration is a fundamental invariant in SR. Thus, irrespective of what is 'felt', a planetary flyby in SR is proper acceleration and breaks symmetry between observers. In SR, there must be acceleration in a non-geodesic path, just as in Euclidean geometry, there must be a bend in a non-geodesic path. As a final note to pedants, I am discussing only standard topology of spacetime, not e.g. cylindrical flat Minkowski spacetime.]

 

[harrylin]

[...] Do you have a reference for Newton using fixed stars? [..]

For using the concept of "fixed stars" as reference? Yes of course, nowadays the Principia is online (just text search the page for occurrences of "stars"):
- https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions

And as usual, a discussion is provided in Wikipedia:
https://en.wikipedia.org/wiki/Inertial_frame_of_reference#Newton.27s_inertial_frame_of_reference

A correct, IMO, way to distinguish SR inertial frame from GR inertial frame is to exploit geometric understanding from GR rather than reject it. [..]

No doubt so - but who wants to reject understanding of GR? Once more, there is no need for GR to clarify the standard (SR) "twin paradox" issues that the OP asked about.

SR is a theory of flat Minkowskian space-time in which global inertial frames are well defined, and global comparison of velocities are well defined (due precisely to absence of curvature). [..]

Yes indeed!

 

[PAllen]

For using the concept of "fixed stars" as reference? Yes of course, nowadays the Principia is online (just text search the page for occurrences of "stars"):
- https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions

But this discussion rejects use of fixed stars as any part of the definition of inertial frames.

 

[ghwellsjr]

After many years of agonizing over it, I have still failed to come to terms with the twin paradox. Here's a brief review of my understanding and a few questions:

I will try to answer your questions in a way that makes sense to you so that your years of agonizing can come to an end.

Before getting into your specific scenario and questions, let me give you the bottom line that you can use to easily figure out how to calculate any scenario with traveling observers. First, you want to describe the entire scenario from one Inertial Reference Frame (IRF) as you have done in your next paragraph. Then you need to determine the speed profile of each observer according to that IRF. This mean tabulating how long each observer remains at each speed. In some specifications, this is given. In others, like yours, you have to calculate it based on other factors. In your case, you had four weeks of accelerations throughout the scenario. I plan to deal with those later in another post but to simplify things to begin with, let's assume the traveler accelerates instantly. You already incorporated another simplifying factor which is to specify speeds as a fraction of the speed of light which is given the symbol beta, "β". You specified the distance that this observer traveled at a particular speed before he turned around and returned at the same speed to his starting point. Therefore the length of time it took this observer to get to his turnaround point is the distance divided by the speed and the same for the return trip.

The other observer remains at rest in your IRF so his speed is zero.

Next we have to calculate the Time Dilation factor for each different speed in the scenario. This is simply gamma:

γ = 1/√(1-β²)

For each segment of an observer's speed profile, we have to divide the IRF time interval by the Time Dilation factor to determine how much the observer ages during that segment.

So let's look at the details of your scenario (with instantaneous accelerations):

A standard story is as follows: Twin A leaves the Earth for planet Zolan 10 light years away. Twin B stays on Earth. Let's say that, over a period of a week, twin A accelerates to 0.8c (relative to Earth and Zolan, which are at rest relative to each other) and then remains cruising at that velocity, decelerates for a week upon approaching Zolan, makes a sweep around the planet (waves at everyone), begins to accelerate once more towards Earth for a week, reaches a cruising speed of 0.8c, and then finally decelerates for a week before landing safely on the Earth and having lunch with twin B.

Now, I'm not worried about the exact figures, but we can all agree that twin A will have aged more slowly than twin B and will be "younger" than twin B.

OK, but it's so easy to calculate the exact figures now that you know how so let's go ahead and do it.

Twin A is going to travel 10 light years at 0.8c. That means it will take him 10/0.8 = 12.5 years according to the IRF to get to Zolan and another 12.5 years to get back to Earth making the entire scenario last 25 years according to the IRF.

Next, we calculate gamma from beta:

γ = 1/√(1-β²) = 1/√(1-0.8²) = 1/√(1-0.64) = 1/√(0.36) = 1/(0.6) = 1.667

Now we divide Twin A's time going out by gamma, 12.5/1.667 = 7.5. We do the same thing for his trip home and we see that he has aged 15 years during his trip.

Finally we do the same thing for Twin B. Since his speed is zero the entire time, gamma equals 1, so the amount that he ages is the same as the duration of time for the scenario according to the IRF which we already determined as 25 years.

To summarize, Twin A ages 15 years and Twin B ages 25 years.

Why is this? The main problem most people seem to have is how do you get around the apparent symmetry of relative motion problem between the two twins.

What apparent symmetry? The scenario, as you specified it, is not symmetrical. If you had Twin B traveling in the opposite direction, otherwise doing the same thing as Twin A, then it would be symmetrical and they would both age the same. But I don't see the point in claiming that a problem exists when clearly there is none.

In order to address this, popular consensus among physicists/cosmologists that I've seen is that this apparent symmetry becomes broken once twin A accelerates from the Earth and enters a non-inertial frame, and this is what is responsible for the slowing of twin A's aging relative to twin B. I have a few questions about this.

Yes, as soon as Twin A accelerates from Earth, leaving Twin B at rest on Earth, symmetry is broken but Twin A is just as much in the one and only IRF that we have considered so far as Twin B is. If you want to introduce a non-inertial frame, you should understand that both twins are also in that non-inertial frame during the entire scenario. Nobody jumps from one frame to another frame. I realize that it's quite common to hear people talking about observers entering different frames but that only invites confusion and is totally unnecessary.

1) "twin A accelerates from the Earth and enters a non-inertial frame"--a non-inertial frame relative to what? To the earth/twin B? Some objective measure of spacetime displacement? To the CMB? I guess what I'm asking is that, is there a way to measure the acceleration of an object/frame without some measure of what to say it is accelerating in relationship to? I'm guessing there's a stock answer to this question, but I'm remiss to recall it at this moment.

Usually when an observer accelerates and we want to build an non-inertial frame, we would choose his rest state for the non-inertial frame. It usually starts out with him inertial and ends with him inertial but in between he accelerates. Even if this is the case, there is no standard way to define a non-inertial frame so you have to state what you have in mind. My favorite is one based on radar measurements and it does provide an objective measure of space time displacement. Yes, the non-inertial Twin A is measuring displacements to the Earth/Twin B and depicting them as the ones that are accelerating rather than himself.

2) Lorentz transformations (LT): The equation determining the time dilation factor in twin A's slowing of aging is, AFAIK, contained completely with the LT equations.

Yes, you can derive the Time Dilation factor from the LT but once you realize that it is just gamma, you don't need the LT when defining a scenario according to a single IRF as you did in your first post. I added the instructions on how to determine the aging of each twin based on their speeds in the IRF. The purpose of the LT is to see what the scenario looks like according to another IRF moving at some speed with respect to your defining IRF.

However, I see no term or provision at all for acceleration in the time dilation equation.

That's because it's based on speed, not acceleration.

So how does acceleration play a role quantitatively in this paradox when it seems as if all you need is a simple velocity difference to create the dilation effect?

Acceleration plays no role, neither does velocity, just speed according to an IRF. It's not even a velocity difference or a speed difference, just the speed(s) of each observer according to an IRF. You figure the Time Dilation of each observer independently off all other observers or objects and add up their accumulated agings as described earlier and then you compare any differences.

Are there some equations I am not aware of that address this, or some kind of formalism that can be used to quantify these measures?

Just the ones I showed you earlier, I hope you're well aware of them by now.

I'll give three examples of scenarios I'm having difficulty seeing how the standard formalism can address:

2a) Say twin A were to accelerate (and decelerate) for 2 weeks instead of one week in both directions from Earth to zolan, but compensates for the lost time by cruising at a speed slightly greater that 0.8c so that the two week acceleration round trip flight time was exactly the same as the one week. Would that make any difference in the age difference between twin A and twin B upon twin A's return?

Not enough to worry about, maybe a few days.

2b) Say twin A took the exact same trip as above (say the one week version), but now twin B also took a similar trip in the opposite direction to planet Xanadu 5 light years away, but accelerated at a different rate than twin A, and reached a different cruising speed that was, say, 0.5c. How would we calculate their age differences in this situation once they both returned to earth? Could we handle it with only the LT and the relativistic velocity addition formulas? If not, then how?

I have shown you that you don't need the LT nor the relativistic velocity addition formula. Why don't you work this out just like I showed you? Just don't worry about any week-long acceleration--make it instantaneous.

2c) Finally, let's say we have twin A again going on his trip to Zolan (one week acceleration edition), but now we have twin B centrifuging himself in a 2001: Space odyssey style space station, undergoing rapid centripetal (or is it centrifugal?) acceleration. And then twin A and twin B go for lunch on Earth when twin A comes back. Does that make a difference in their age? Or should twin B just as well stayed on Earth during twin A's trip?

Again, acceleration doesn't matter but speed does so specify the speed and you can figure it out. Remember, you can have the same speed but different accelerations, so that should tell you that acceleration doesn't matter.

I guess the more broader encompassing question is what it is specifically about twin A's undergoing an acceleration or transition into a non-inertial frame that sets his clock running slower than twin B's. It seems to be this act of accelerating that does it, not simply a relative difference in velocity. But this is the was the LT's lead you believe.

After all I have said, I hope you can see how to improve this paragraph!

 

[PhoebeLasa]

After many years of agonizing over it, I have still failed to come to terms with the twin paradox.
[...]

I think the best answer to the OP's questions are given by Brian Greene in his "The Fabric of the Cosmos" PBS NOVA series, and in his book of the same title. Brian basically says that, whenever the traveler is accelerating TOWARD the home twin, the traveler will say that the home twin's age is rapidly increasing. And whenever the traveler is accelerating AWAY FROM the home twin, the traveler will say that the home twin's age is rapidly decreasing. The rate of rapid increase or decrease depends also on how far apart the twins are. When the twins are together, there is no such effect, but the effect can be huge, even for very small accelerations, when the distance is very large (as in Brian's example).

 

[ghwellsjr]

I think the best answer to the OP's questions are given by Brian Greene in his "The Fabric of the Cosmos" PBS NOVA series, and in his book of the same title. Brian basically says that, whenever the traveler is accelerating TOWARD the home twin, the traveler will say that the home twin's age is rapidly increasing. And whenever the traveler is accelerating AWAY FROM the home twin, the traveler will say that the home twin's age is rapidly decreasing. The rate of rapid increase or decrease depends also on how far apart the twins are. When the twins are together, there is no such effect, but the effect can be huge, even for very small accelerations, when the distance is very large (as in Brian's example).

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?

 

[harrylin]

But this discussion rejects use of fixed stars as any part of the definition of inertial frames.

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.

 

[harrylin]

[..] Einstein never referenced 'fixed stars'. [..]

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

 

[PhoebeLasa]

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]

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]

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]

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.

 

[harrylin]

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!

 

[Demystifier]

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]

 

[phyti]

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.

 

[Ookke]

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.]

...do not explain why ... only ... how much ...

That is true for all of physics.

 

[Dale]

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]

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?

 

[m4r35n357]

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 into the definition of the spacetime interval.

 

[Demystifier]

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]

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.

 

[PhoebeLasa]

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]

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 into 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.

 

[jethomas3182]

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.

 

[georgir]

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).

 

[m4r35n357]

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 ;)

 

[georgir]

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.

 

[m4r35n357]

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!

 

[dmitrrr]

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]

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.

 

[dmitrrr]

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]

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.