Was Einstein lucky when not considering twin paradox as paradox?

[JesseM]

There's a widespread misconception that you need general relativity in situations involving acceleration, but it's just not true; special relativity handles acceleration just fine. You can google for "Rindler coordinates" for one example, and you'll find another example (a clock experiencing uniform circular motion due to the Earth's rotation) in Einstein's original 1905 paper to which ghwellsjr gave you a link above.

You can also forego accelerating coordinate systems, and just analyze the time elapsed on an accelerating clock using the coordinates of some inertial frame in which you know the clock's coordinate position and velocity as a function of coordinate time. The trick is to approximate a smoothly-varying path by a polygonal path made up of a bunch of short inertial segments lasting a coordinate time \Delta t, that way the time elapsed on the clock on each segment will be \sqrt{1 - v^2/c^2} \Delta t, and then you can just add up the clock times on all the segments (using the appropriate v for each segment, which can vary from one to another) to get the total time elapsed on the polygonal path. Then you let the time of each segment become infinitesimal, so the sum becomes an integral and the total time elapsed on a clock with velocity as a function of time v(t) is just \int \sqrt{1 - v(t)^2/c^2} dt.

Einstein doesn't go into detail, but he does allude to this method at the end of section 4 of the 1905 paper, when he writes:

It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be \frac{1}{2} tv^2/c^2 second slow.

(note that the factor he gives is the result of a first-order approximation to the fully accurate formula t\sqrt{1 - v^2/c^2})

 

[harrylin]

My point, as I said, is that the calculation has nothing really to do with GR [..] He's using SR in noninertial coordinates.

I see that you disagree with his argument; and for reasons different from yours, so do I. :)
In fact, I won't discuss disagreements you may have with Einstein about his theory, as you here agree with the point that I tried to make (sorry if that was not clear!): Einstein and contemporaries understood perfectly well how to handle accelerations with SR - even accelerating frames. Furthermore, he did not pretend that GR is required to handle acceleration. Nevertheless, the misconception about acceleration did come about. Thus my suggestion remains that perhaps many people who did not correctly understand SR, got the wrong impression that GR had to be used for accelerated objects frames and even accelerated objects, because Einstein argued that GR could be used like that.

 

[stevendaryl]

I see that you disagree with his argument.

No, I don't disagree with his argument. I disagree that his argument involves General Relativity. Going from SR to GR consists of two steps:

1. Allowing the metric tensor g_{\mu \nu} to be dynamic, instead of constant everywhere.
2. Describing how the metric tensor is affected by mass/energy/momentum (the field equations--these describe how spacetime curvature is affected by stress-energy, but in the most basic version of GR, the curvature tensor is a function of the metric only).

Those are the only two additions to SR to produce GR, as far as I know. (There is perhaps a little ambiguity in #1, because it's not always straight-forward how to generalize the physics of flat spacetime to curved spacetime.) Neither is relevant in the twin paradox.

 

[harrylin]

No, I don't disagree with his argument. I disagree that his argument involves General Relativity. [..].

Once more (and with this last repetition I end my discussion about this side topic): that is your main disagreement with Einstein. He admitted that " it is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K " , and the rest of that paper is his argument in defence of that position against critics.

 

[stevendaryl]

He admitted that " it is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K " , and the rest of that paper is his argument in defence of that position against critics.

I agree that it is correct from the point of view of GR, but it's ALSO correct from the point of view of SR.

 

[harrylin]

I agree that it is correct from the point of view of GR, but it's ALSO correct from the point of view of SR.

Einstein clarified that if one would make the mistake to consider the accelerating system K' as a valid "rest" system from the point of view of SR, in which only Galilean reference systems such as K are equivalent, one would create a twin paradox in SR. That has now been explained on this forum many times, even in this thread. The point from which you distracted is the fact that Einstein and his contemporaries knew very well how to handle acceleration in SR.

 

[stevendaryl]

Einstein clarified that if one would make the mistake to consider K' as a valid "rest" system from the point of view of SR, in which only Galilean reference systems such as K are equivalent, one would create a twin paradox in SR.

Yes, the equations do not have the same form in a non-inertial coordinate system. But that's a fact about SR. The derivation that Einstein gave is an SR derivation. Of course, it's valid in GR, as well, but there is nothing specifically GR about it.

His derivation is an SR derivation, because he did not make use of any of the differences between SR and GR.

 

[stevendaryl]

Yes, the equations do not have the same form in a non-inertial coordinate system. But that's a fact about SR. The derivation that Einstein gave is an SR derivation. Of course, it's valid in GR, as well, but there is nothing specifically GR about it.

His derivation is an SR derivation, because he did not make use of any of the differences between SR and GR.

The equations of SR have the same form in any inertial frame. But to figure out the form in a noninertial frame, all it takes it calculus. I consider that still SR to treat the accelerated frame as "at rest"--that's just a coordinate transformation. The fact that when using noninertial coordinates, the metric tensor becomes position-dependent just falls right of the coordinate transformation. So position-dependent time dilation in noninertial coordinates is inherent in SR. There is nothing "General Relativistic" about it. To call the position-dependence of the metric a "gravitational field" is just picturesque language. There is still no additional assumptions involved, as far as I can tell, beyond those of SR. So it's a mistake to call it a "GR" resolution, because there is nothing in it that isn't already implicit in SR.

I suppose that GR is involved when you say that the fake gravitational field that results from noninertial coordinates is no less real than the gravitational field due to planets, but since the equations don't depend on how "real" the gravitational field is, I just don't see how the derivation could be considered a GR derivation.

 

[harrylin]

Yes, the equations do not have the same form in a non-inertial coordinate system. But that's a fact about SR. The derivation that Einstein gave is an SR derivation. Of course, it's valid in GR, as well, but there is nothing specifically GR about it.

His derivation is an SR derivation, because he did not make use of any of the differences between SR and GR.

K' is invalid as "rest frame" in SR. Despite our differing disagreements with Einstein, we agreed a long time ago on the point that I made, which is that Einstein and others understood acceleration in SR. It was not in "retrospect" that General Relativity was not needed to calculate the twin problem. Einstein never suggested that GR would be needed for the calculation. However, from my discussion with you I now slightly change my hypothesis about how that misunderstanding may have come about. For it now seems plausible to me that many people may have misunderstood Einstein's arguments in his papers from 1916-1918 that GR could be used for accelerated frames and even accelerated objects, so that they misconstrued that according to Einstein GR had to be used. And that's all that I will hypothesize about that. :)

 

[stevendaryl]

K' is invalid as "rest frame" in SR.

SR as a theory of physics is not about rest frames. That's a way to talk about SR, and a way to derive the Lorentz transformations, but as a theory of physics, it makes claims about the behavior of clocks and rods and light signals and so forth. Those claims can be expressed in any coordinates you like. The fact that they were originally derived for inertial reference frames is irrelevant, except for historical interest.

So K' is not an inertial reference frame. That's certainly true. What does that fact have to do with whether using K' to describe the twin paradox involves GR or not? If I have a description of what goes on, according to frame K, and I know the coordinate transformation connect coordinates of frame K to coordinates of frame K', then I know how to describe things in frame K'. I don't need to know whether K' is a "valid rest frame". If by "valid rest frame" you mean "inertial rest frame", then It's not, and it doesn't matter.

There is nothing about using K' that requires going beyond SR.

 

[pervect]

Pedagogically, the techniques needed to handle general coordinate systems (aka non-inertial frames) are taught in GR courses, but as far as the physics go, if you have flat space time (i.e. no gravity), you can use the techniques of SR just fine. You could conceivable even do the analysis without tensors, though if you want to compare your results to textbook results, I'm not aware of any textbooks that don't use tensors (not that I've read them all).

A brief outline of one way to go about doing this:

1) Solve the relativistic rocket equation for a constant proper acceleration rocket. Check your solution against the standard ones on wikipedia and/or the relativistic rocket FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html .

I will use the following notation: let your reference inertial frame be (t,x,y,z). Let your rocket coordinates be (T,X,Y,Z). Let the momentarily co-moving inertial frame coordinates at some proper time T be represented by (t', x', y', z'). Let the acceleration of the rocket be called g.

So solving the relativistic rocket equation you want to find z(T) and t(T), the inertial coordinate frame position and time as a function of rocket time T, which is just the proper time of the accelerating rocket.2) Consider the momentarily co-moving inertial frame at proper time T on the rocket. Because of linearity, there will be a linear relationship between $\hat{z'}$,$\hat{t'}$, the components of the basis vectors in the momentarily comoving inertial frame at time T, and $\hat{z}$, $\hat{t}$ the basis vectors in the inertial frame. Basically all we really need to do is figure out the components of $\hat{z'}$ in the inertial frame, this vector will have some time components due to the relativity of simultaneity, and some space-component due to relativistic length contraction

3) Find an expression that converts rocket coordinates (T,X,Y,Z) to inertial coordinates (t,x,y,z). We will basically have
t = t(T) + Z * (t-component of $\hat{z'}$)
x = X
y = Y
z = z(T) + Z * (z-component of $\hat{z'}$)

here t(T) and z(T) are the expressions we derived in part 1, and while the relationship between changes in the Z coordinate at time T and the changes in the t and z coordinates is linear, we need to find out what the coefficients are.

At this point we've codified precisely what we mean by "rocket coordinates" by having an expression that gives the inertial coordinates (t,x,y,z) as functions of the rocket coordinates (T,X,Y,Z).

4) Optional. Using the above results from 3, find the metric in the accelerating coordinates by taking dx^2 + dy^2 + dz^2 - c^2 dt^2 in terms of dT,dX,dY, and dZ. This is just algebra (albeit lengthly without computer assistance). Compare it to the textbook results, which say that you should get dX^2 + dY^2 + dZ^2 - c^2 (1+gZ)^2 dT^2

5) Optional, but recommened. Consider whether the mapping derived in part 3 is a 1:1 mapping (hint: we need to insist that 1+gZ is positive if we want to have a 1:1 mapping).

 

[harrylin]

SR as a theory of physics is not about rest frames. That's a way to talk about SR, and a way to derive the Lorentz transformations, but as a theory of physics, it makes claims about the behavior of clocks and rods and light signals and so forth. Those claims can be expressed in any coordinates you like. The fact that they were originally derived for inertial reference frames is irrelevant [..] There is nothing about using K' that requires going beyond SR.

Einstein certainly agreed with that. I promised to leave our disagreement about the issue that you next brought up, as it is irrelevant for my clarification that obviously this was not "retrospectively" understood - instead it was understood right from the start.

 

[stevendaryl]

Einstein certainly agreed with that. I promised to leave our disagreement about the issue that you next brought up, as it is irrelevant for my clarification that obviously this was not "retrospectively" understood - instead it was understood right from the start.

I know, I'm not arguing about that. I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

 

[stevendaryl]

I know, I'm not arguing about that. I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

On the other hand, I can't figure out what it could possibly mean to say that "the traveling twin is at rest in a valid rest frame", other than "there is a coordinate system in which the traveling twin is at rest".

 

[harrylin]

I know, I'm not arguing about that.

OK. :)
I though that you were still trying to avoid that conclusion.

I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

It is misleading to pretend that there is no difference at all between 1916 GR and modern GR... In a nutshell:

1. The frame of the traveling twin is invalid for SR's laws of nature (even the second postulate doesn't work with it!).
That simple fact has also been elaborated many times on this forum.

2. The frame of the traveling twin is valid for GR's laws of nature according to 1916 GR.
The trick: "we can "create" a gravitational field by a simple variation of the co-ordinate system." -E. 1916
I briefly discussed this in the beginning of a post some time ago in a different thread: https://www.physicsforums.com/threa...solution-compatible-with-einsteins-gr.656240/

 

[stevendaryl]

1. The frame of the traveling twin is invalid for SR's laws of nature (even the second postulate doesn't work with it!).
That simple fact has also been elaborated many times on this forum.

What does "invalid" mean? The traveling twin's frame is not an inertial frame. It's not an inertial frame in GR, either. The second postulate states that the speed of light has speed "c" in any inertial frame. That doesn't mean that a noninertial frame is "invalid", it just means that the speed of light doesn't necessarily have speed c in that frame. You don't need a separate law to deal with a noninertial frame, you just need calculus. Calculus plus SR is still SR.

There is no such thing as a valid or invalid coordinate system. There is only valid or invalid reasoning. If you reason about a noninertial coordinate system as if it were inertial, then you've engaged in invalid reasoning.

So it is true that SR, when expressed as laws about inertial reference frames, tempts people into invalid reasoning if they try to apply the laws, as written, in a noninertial reference frame. That means that you need to understand what the laws say in a way that it is independent of coordinate systems. That was not completely understood at the time Einstein wrote SR. He did not know how to formulate laws that worked in any coordinate system. But that's a limitation of his mathematics, not his physics.

2. The frame of the traveling twin is valid for GR's laws of nature according to 1916 GR.

It's neither more nor less valid according to SR than GR. There is NO difference between SR and GR when it comes to noninertial frames. In the limiting case of flat spacetime, they are the SAME physical theory. You are perpetuating a misconception. That's my original point about Einstein's "GR solution to the twin paradox". It introduced a misconception that apparently persists to this very day.

 

[harrylin]

I think there was a specific reason Einstein used GR to resolve the twin paradox. He wanted to construct an analogous scenario (via the equivalence principle) in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time that the twins were separated. When he fired his rocket engine, he was doing it strictly to counteract the spatially-uniform gravitational field that is somehow momentarily switched on, so that the rocket-twin would remain stationary and unaccelerated. That momentarily switched-on gravitational field causes the "home twin" (the twin who has no rocket) to accelerate, reverse course, and move toward the "traveler". The resulting conclusion using this GR scenario is that the rocket-twin will say that the "home-twin" suddenly gets much older while that gravitational field is switched on.

The exact same result (regarding the rocket-twin's conclusion about the home-twin suddenly getting much older during the turnaround) is obtained without recourse to GR (and without any gravitational fields), purely from SR, using a non-inertial reference frame for the rocket-twin which is formed by piecing together multiple inertial frames that are each momentarily co-moving with the rocket-twin at different instants of his life. The rocket-twin is always at the spatial origin of his non-inertial reference frame, but he never contends that he doesn't accelerate. He knows that he accelerates, and reverses course, when he turns on his rocket. And he knows that it is the home-twin who is unaccelerated for the whole trip.

There is a difference between being "always absolutely at rest" (Einstein's GR scenario for the rocket-twin) versus "being always at the spatial origin of your own personal reference frame, but accelerating at will using your rocket engine" (the SR scenario). But what the rocket-twin says about the home-twin suddenly getting much older during the turnaround is exactly the same for both scenarios (even though it's a different twin doing the turnaround in the two cases).

I had missed that post of yours. Good summary of Einstein's paper! :)

PS. Note that I don't agree with that paper; but before criticizing an opinion, one has to correctly present it first.

 

[harrylin]

What does "invalid" mean? [..]

I gave the example of invalid application of SR's second postulate on accelerating frames, and the same happens with invalid application of the Lorentz transformations. But of course, you know that:

So it is true that SR, when expressed as laws about inertial reference frames, tempts people into invalid reasoning if they try to apply the laws, as written, in a noninertial reference frame. [..]

Exactly - that was the point Einstein made; and this was at that time rather well understood (but not anymore, see my next post!).

It's neither more nor less valid according to SR than GR. There is NO difference between SR and GR when it comes to noninertial frames. In the limiting case of flat spacetime, they are the SAME physical theory. You are perpetuating a misconception. That's my original point about Einstein's "GR solution to the twin paradox". It introduced a misconception that apparently persists to this very day.

Sorry, the link I gave in my last post clarifies that the misconception here is yours, as you project your understanding of modern GR on Einstein's GR v.1.0 of 1916. And maybe you overlooked, like I did, the clear summary by PhoebeLasa? A few minutes ago I now also commented on that post.

 

[harrylin]

[..] Perhaps many people who did not correctly understand SR, got the wrong impression that GR had to be used for accelerated objects frames and even accelerated objects, because Einstein argued that GR could be used like that?

[..] it now seems plausible to me that many people may have misunderstood Einstein's arguments in his papers from 1916-1918 that GR could be used for accelerated frames and even accelerated objects, so that they misconstrued that according to Einstein GR had to be used. And that's all that I will hypothesize about that. :)

By pure coincidence (for my brother in law I'm checking a book), I stumbled yesterday evening on a case that supports my hypothesis about that misconception in a strong way:

"[SR] accounted for straight-line uniform motion, but it did not account for non-uniform or accelerated motion. And as everybody knows, accelerated motion is common on Earth and throughout the universe.
[..]
The [twin] paradox, as it existed before general relativity, was that you couldn't be sure which of the twins would be the youngest when they got back together after the flight [...] since all motion is relative according to special relativity. But when Einstein formulated his general theory of relativity he showed that there was, indeed, a distinct difference between the twins." -Einstein's Brainchild, Barry Parker.

And note that the author was physics professor from 1967 to 1997! :(

 

[stevendaryl]

I gave the example of invalid application of SR's second postulate on accelerating frames, and the same happens with invalid application of the Lorentz transformations. But of course, you know that:

There is invalid reasoning, but what does an invalid "frame" mean?

Sorry, the link I gave in my last post clarifies that the misconception here is yours

It's a misconception that GR is equivalent to SR in the limit of vanishing spacetime curvature? It's provably the case.

And note that the author was physics professor from 1967 to 1997

Well, the misconception is perpetuated by many people who should know better.

 

[stevendaryl]

Look, tell me what aspect of Einstein's derivation involves GR. If you agree that it is a GR derivation, then point to a step in the derivation that is not valid in SR. The key remark by Einstein is this:

‘according to the general theory of relativity, a clock works faster the higher the gravitational potential at the place where it is situated’

That result is derived from SR. The way you get that result is by:

1. Calculating time dilation in SR using noninertial coordinates.
2. Noting that in these coordinates, there is a "pseudo-gravitational field", and that clocks higher up in this "field" run faster. (This is pure calculus--it follows from SR plus the calculus of coordinate changes).
3. Invoke the equivalence principle, to get the analogous fact about "real" gravitational fields.

Gravitational time dilation was a derivation of SR + the equivalence principle. It was a step toward GR, but this step required almost nothing beyond SR and the insight of the equivalence principle.

Now, when Einstein (or whoever) invokes GR to explain the twin paradox, what is he doing? He's saying:

1. The accelerating twin feels a pseudo-gravitational field.
2. By the equivalence principle, this is like real gravitational fields.
3. By GR, there must be time dilation for clocks that are higher up in this "field".

That is what is going on in the GR explanation for the twin paradox. Use the equivalence principle to transform an accelerated reference frame to one in a gravitational field, then use gravitational time dilation. But when you consider that gravitational time dilation is derived from SR plus the equivalence principle, this explanation is seen to be completely circular! You start with a pure SR problem--acceleration in flat spacetime--then transform to a GR problem, which is then solved by transforming BACK to SR.

It makes sense to view the twin paradox as a way to explain GR in terms of SR concepts. But the other way around is nonsense.

 

[harrylin]

There is invalid reasoning, but what does an invalid "frame" mean?

It means that its use as reference for those laws does not give correct results.

It's a misconception that GR is equivalent to SR in the limit of vanishing spacetime curvature? It's provably the case.

As explained, Einstein's 1916 GR claimed to be more than that.

Well, the misconception is perpetuated by many people who should know better.

That's all you have to say about it? I'll elaborate. I don't think that he is a lone exception, and neither do you.
Thus, I just found a probable cause of the modern "twin paradox" miscomprehension. Apparently, physics professors taught their students for decades that according to SR's first postulate, all motion is relative.

 

[stevendaryl]

It means that its use as reference for those laws does not give correct results.

You have a law saying "The speed of light is c in every inertial reference frame". It is DERIVABLE from this law that in a noninertial reference frame, the speed of light is not c. If a fact is a derivation from SR, how can it be contrary to SR?

As explained, Einstein's 1916 GR claimed to be more than that.

But it's not. That's a misconception. If Einstein believed it, then he was mistaken. In the absence of spacetime curvature, GR is equivalent to SR. From the point of view of the twin paradox in flat spacetime, there is no difference between GR and SR. None.

 

[harrylin]

Look, tell me what aspect of Einstein's derivation involves GR. If you agree that it is a GR derivation, then point to a step in the derivation that is not valid in SR. The key remark by Einstein is this:

‘according to the general theory of relativity, a clock works faster the higher the gravitational potential at the place where it is situated’

That result is derived from SR. The way you get that result is by:

1. Calculating time dilation in SR using noninertial coordinates.
2. Noting that in these coordinates, there is a "pseudo-gravitational field", and that clocks higher up in this "field" run faster. (This is pure calculus--it follows from SR plus the calculus of coordinate changes).
3. Invoke the equivalence principle, to get the analogous fact about "real" gravitational fields.

Gravitational time dilation was a derivation of SR + the equivalence principle. It was a step toward GR, but this step required almost nothing beyond SR and the insight of the equivalence principle.

That part of your argument looks totally correct to me.

Now, when Einstein (or whoever) invokes GR to explain the twin paradox, what is he doing? [..]

Once more: Einstein did not "invoke GR" to "explain the twin paradox" - quite the contrary! Critics of 1916 GR invoked GR's "general principle of relativity" to create the twin paradox. So, please take my advice and don't quickly reply. Instead verify that Phoebelisa's summary of the 1918 paper is correct, next ponder over it, and then re-read the last part of our discussion. Also, recall my earlier clarification:

[..] I don't agree with that paper; but before criticizing an opinion, one has to correctly present it first.

[stevendaryl:] when you consider that gravitational time dilation is derived from SR plus the equivalence principle, this explanation is seen to be completely circular! You start with a pure SR problem--acceleration in flat spacetime--then transform to a GR problem, which is then solved by transforming BACK to SR. [..]

That is absolutely correct; the same was argued in a paper on the clock paradox by Builder, in the fifties.

You have a law saying "The speed of light is c in every inertial reference frame". It is DERIVABLE from this law that in a noninertial reference frame, the speed of light is not c. If a fact is a derivation from SR, how can it be contrary to SR?

Exactly: it is contrary to SR to use that law with non-inertial frames.

But it's not. That's a misconception. If Einstein believed it, then he was mistaken. [..].

It's inherent in his postulate of GR. I don't think that you actually pretend that he was mistaken about his belief of what he postulated. ;)

 

[stevendaryl]

Once more: Einstein did not "invoke GR" to "explain the twin paradox" - quite the contrary! Critics of 1916 GR invoked GR's "general principle of relativity" to create the twin paradox.

Well that was a big mess that is at least partly Einstein's fault. You don't need a theory of physics to permit you to use arbitrary coordinates. And there is no content to the claim "all motion is relative" above and beyond the claim that you can use arbitrary coordinates. The paradox-mongers went from

Coordinates in which the traveling twin is at rest are just as valid as coordinates in which the stay-at-home twin is at rest​

which is true, to

Therefore, there is no way to say that one twin should be older than the other when they reunite.​

which is false. The whole argument has nothing to do with GR, really. It has to do with the use of arbitrary coordinates. The question is: how to describe the source of the asymmetry between the twins in general (noninertial) coordinates.

The modern resolution is that computations of numeric quantities such as elapsed times for spacetime paths involve tensor quantities (the metric tensor in this case), and the components of a tensor have different values for different coordinate systems. So it's a MATHEMATICAL mistake to transform from inertial coordinates to noninertial coordinates without making the corresponding change to the metric tensor.

The terminology of "gravitational fields due to acceleration" is simply a nonmathematical way to talk about the metric tensor in noninertial coordinates (technically, the pseudo-gravitational field corresponds to the connection coefficients, which are computed from the metric tensor).

So both invoking GR to create the twin paradox and invoking GR to resolve it are misconceptions. Neither involves GR. The whole discussion is about general coordinate systems, so it's not GR at all.

 

[JesseM]

So both invoking GR to create the twin paradox and invoking GR to resolve it are misconceptions. Neither involves GR. The whole discussion is about general coordinate systems, so it's not GR at all.

It's a misconception under the modern definition of the distinction between GR and SR, which revolves around whether spacetime has inherent curvature or not. But it seems to be a historical fact that Einstein and other physicists didn't always define the distinction that way. And while defining it differently may be less elegant, it's really only a "misconception" about terminology (and only relative to the modern usage of the terms, so it can't be labeled a 'misconception' on the part of Einstein), and not a misconception about physics--it's just about what theory you say you're using when you do a particular analysis, not about any differences in any actual physical predictions (although obviously it would be a genuine misconception about physics if someone claimed that the SR time dilation formula should still work in a non-inertial coordinate system).

This section of the twin paradox FAQ entry on the site of physicist John Baez has a good discussion of this history:

Einstein worked on incorporating gravitation into relativity theory from 1907 to 1915; by 1915, General Relativity had assumed pretty much its modern form. (Mathematicians found some spots to apply polish and gold plating, but the conceptual foundations remain the same.) If you asked him to list the crucial features of General Relativity in 1907, and again in 1915, you'd probably get very different lists. Certainly modern physicists have a different list from Einstein's 1907 list.

Here's one version of Einstein's 1907 list (without worrying too much about the fine points):

General Principle of Relativity

All motion is relative, not just uniform motion. You will have to include so-called pseudo forces, however (like centrifugal force or Coriolis force).

Principle of Equivalence

Gravity is not essentially different from any pseudo-force.

The General Principle of Relativity plays a key role in the Equivalence Principle analysis of the twin paradox. And this principle gave General Relativity its name. Even in 1916, Einstein continued to single out the General Principle of Relativity as a central feature of the new theory. (See for example the first three sections of his 1916 paper, "The Foundation of the General Theory of Relativity", or his popular exposition Relativity.)

Here's the modern physicist's list (again, not sweating the fine points):

Spacetime Structure

Spacetime is a 4-dimensional riemannian manifold. If you want to study it with coordinates, you may use any smooth set of local coordinate systems (also called "charts"). (This free choice is what has become of the General Principle of Relativity.)

Principle of Equivalence

The metric of spacetime induces a Minkowski metric on the tangent spaces. In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime. Freely falling bodies follow geodesics.

Gravitation = Curvature

A gravitational field due to matter exhibits itself as curvature in spacetime. In other words, once we subtract off the first-order effects by using a freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field.

The third feature finds its precise mathematical expression in the Einstein field equations. This feature looms so large in the final formulation of GR that most physicists reserve the term "gravitational field" for the fields produced by matter. The phrases "flat portion of spacetime", and "spacetime without gravitational fields" are synonymous in modern parlance. "SR" and "flat spacetime" are also synonymous, or nearly so; one can quibble over whether flat spacetime with a non-trivial topology (for example, cylindrical spacetime) counts as SR. Incidentally, the modern usage appeared quite early. Eddington's book The Mathematical Theory of Relativity (1922) defines Special Relativity as the theory of flat spacetime.

 

[stevendaryl]

It's a misconception under the modern definition of the distinction between GR and SR, which revolves around whether spacetime has inherent curvature or not. But it seems to be a historical fact that Einstein and other physicists didn't always define the distinction that way.

I suppose that there is an ambiguity about what constitutes a "physical theory" and when one physical theory is the same or different from another. My inclination is to think that developing more sophisticated mathematical way of working with a theory is still the same theory. So Lagrangian mechanics is still Newtonian mechanics, and GR in flat spacetime is still SR.

 

[PhoebeLasa]

The Dolby & Gull SR answer to the question "How does the home twin's age vary during the traveler's turnaround, according to the traveler" seems to be very popular on this forum, but I've never seen a GR solution (via the equivalence principle) that gets the same answer that the D&G SR method gets. The only GR solution that I've ever seen agrees with the co-moving inertial frames SR solution, which is very different from the D&G SR solution. Has a GR solution that agrees with the D&G SR solution been given anywhere?

 

[Dale]

What is a GR solution in your view? Can you provide a reference for this "GR solution" that agrees with the SR comoving frames as an example?

I am just not sure what you are asking since to me all twin paradox solutions are inherently SR solutions.

 

[Nugatory]

but I've never seen a GR solution (via the equivalence principle) that gets the same answer that the D&G SR method gets

There's no reason that you expect that you would. The equivalence principle says that gravity can be modeled locally as acceleration; it does not say that all acceleration can be modeled as gravity, especially not globally.

Whether we attack the twin paradox in flat spacetime using the methods of special or general relativity, we're going to end up computing the same coordinate-independent quantities, namely the proper times along the worldlines of the two twins. The GR machinery just gives us a bit more latitude in choosing coordinates on our way to the solution.