Using General Relativity to analyze the twin paradox

[stevendaryl]

In that case Einstein would have been debating about nothing - apart of complexity, nobody has or had a problem with non-inertial coordinates! It's even commonly used in classical mechanics. Mapping to the geoid by means of Newton's mechanics is right at the start of many textbooks.

Well, I don't see any content to the "induced gravitational field due to acceleration" above and beyond what was already known in Newtonian physics in noninertial coordinates. I really do think that Einstein's GR resolution to the twin paradox had no content above and beyond SR in noninertial coordinates. Now, I think that the discussion was useful, in that it shows how the same situation can be viewed as velocity-dependent time dilation in one set of coordinates, and "gravitational" time dilation in another set of coordinates. But that doesn't actually provide any new insight about the twin paradox. Instead, it provides insight about GRAVITY -- real gravity due to masses. To me, the usefulness of equating "fictitious forces" with "gravitational field" is not that it provides any new insight about SR, but that it provides insight about the nature of gravitational fields. Using GR to solve an SR problem is ridiculous, in my opinion. But using SR to solve (approximately) a problem involving clocks at different altitudes on Earth is a big deal. The problem can't be solved without the equivalence principle, unless you go all the way to full GR.

 

[Dale]

A real gravitational field must obey GR. In GR, cause and effect is assumed and gravitational fields propagate at local speed c.

You are starting with a valid premise but making an incorrect conclusion here.

In GR whatever you mean by the term "gravitational field" clearly must obey GR, however what that term refers to has changed over time. Einstein used the term "gravitational field" to refer to the Christoffel symbols. According to GR the Christoffel symbols are not required to "propagate at local speed c". Thus the statement that the "gravitational fields propagate at local speed c" is false using Einstein's terminology.

More modern usage would be to either not use the term "gravitational field" at all or to use it to refer to the Riemann curvature tensor. The statement that the "gravitational fields propagate at local speed c" would be correct using that terminology. But that is not the argument that Einstein is making nor the terminology that he was using.

 

[PeterDonis]

Others including me and you clarified how a gravitational field cannot explain the observed phenomena.

Huh? Where did I say that?

one should not confound SR's use of non-inertial frames with Einstein's claims about a physical explanation by means of induced gravitational fields.

Since by "gravitational field" Einstein meant "Christoffel symbols", and since those are only nonzero in flat spacetime in a non-inertial frame, I don't see how this is "confounding" at all; it's just matching up the math with Einstein's ordinary language explanation.

In contrast, Einstein: or when you fire your rocket and feel the induced gravitational field!

No, Einstein did not say you "feel" a gravitational field. He said you feel the acceleration required to hold yourself at rest in the gravitational field. The gravitational field is introduced in the non-inertial frame in which you are at rest, in order to explain how you can be at rest while at the same time feeling acceleration. But the field itself is not "felt", any more than the field of the Earth is "felt"--the field of the Earth is introduced to explain how you can be sitting at rest on the Earth while at the same time feeling acceleration. The fact that a gravitational field, by itself, is not felt, is the whole point of Einstein's "happiest thought", that a person falling freely will not feel his own weight.

A real gravitational field must obey GR.

Minkowski spacetime does obey GR: the stress-energy tensor is zero and the Einstein tensor is zero. This is true whether you use inertial or non-inertial coordinates, since tensor equations are covariant.

In GR, cause and effect is assumed and gravitational fields propagate at local speed c.

No, changes in spacetime curvature propagate at local speed c. But in flat spacetime, the curvature is always zero, so there are no changes to propagate.

light from distant stars also propagates at local speed c.

Yes, it does. So what?

 

[stevendaryl]

I agree of course. In contrast, Einstein: or when you fire your rocket and feel the induced gravitational field!

This is perhaps a subtle distinction, but you don't "feel" any induced gravitational field on board an accelerating rocket, and you don't "feel" a gravitational field when standing on the Earth. What you feel is the force of the floor pushing up against the bottom of your feet. Gravity doesn't explain this force, it explains why, in spite of the force, you stay in the same location (relative to Earth-fixed coordinates).

 

[Jimster41]

The SR equations of motion for a test mass are simplest when you use inertial cartesian coordinates and parametrize using proper time. Then it's just:

$m \frac{d U^\mu}{d\tau} = F^\mu$

where $F$ is the 4-force. So it looks just like Newton's F=ma.

If you switch to using curvilinear, noninertial coordinates and use some other parameter $s$ besides proper time, you have, instead:

$m (\frac{d \tilde{U}^\mu}{ds} + \Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu) = \tilde{F}^\mu$

where $\tilde{U}$ and $\tilde{F}$ is rescaled versions of $U$ and $F$, and where $f = \frac{ds}{d\tau}$

The "induced gravitational field" due to acceleration just amounts to moving terms from the left-hand side to the right-hand side, and writing:

$m \frac{d \tilde{U}^\mu}{d\tau} = F_{eff}^\mu = \tilde{F}^\mu + F_{grav}^\mu$

where

$F_{grav}^\mu = - m(\Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu)$

Whether you put the terms $F_{grav}$ on the left side, and call them connection coefficients, or put them on the right side, and call them gravitational forces, is just a matter of taste, but it doesn't change the physics.

Is $F_{grav}$ a real force, or not? Well, it's not real, in that it's not due to any source. People talk about it being "induced by acceleration", but that's not true, really. They are induced by the choice of the noninertial coordinate system. That choice isn't forced on you by the fact that you're in an accelerating rocket. A person inside a rocket can use inertial coordinates just as well as someone floating inertially. Anybody can use any coordinates they like; you don't have to use coordinates in which you, personally, are at rest.

On the other hand, $F_{grav}$ is real, in the sense that it is measurable, to the same extent that coordinate acceleration is.

This was helpful. I really appreciate this kind of relatively gentle use of formal symbolism in context, to illuminate the ambiguity of perception being discussed. I'd love to have that more often, though I can see why it's a bit painful.

When you say gravity is not real because it's not due to any "source", There is a temperamental rub or me, (if I am following) in terms of what metaphors or adjectives adhere...for me the bizarre elasticity of inertial perspective in space time, counts as a "source" of a pretty interesting sort. It's not like when you change coordinate frames the physics of acceleration changes. They can be described differently, but when you accelerate, (or are near mass) spacetime elasticity and its distortion is causing things that are recognizably "real" to happen (maybe even "most real"). It seemed for a moment you were suggesting space-time distortion was "not real".

Just watched Interstellar last night. Lots of chuckles and cringes at the physics references, but some of the visualizations were lovely. One sequence about GR, did a good job of giving me the willies.

 

[PhoebeLasa]

I really do think that Einstein's GR resolution to the twin paradox had no content above and beyond SR in noninertial coordinates.

I think Einstein's use of GR to explain the twin "paradox" has an additional importance that has not yet been mentioned here: Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull. And Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

 

[Jimster41]

I think Einstein's use of GR to explain the twin "paradox" has an additional importance that has not yet been mentioned here: Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull. And Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

Yeah, now I'm confused again. If acceleration is only a figment due to non-inertial choice of reference frame why does the "travelling" twin age more slowly. If gravitational force due to acceleration is "fictitious" and only due to choice of reference frame, Just because he had a rocket firing at his brother doesn't mean he was the one accelerating.

I had been picturing that he was, then you convinced me that one could just as easily imagine that his brother could feel like the one zooming away, if the right frame was chosen, but now I'm confused as to how the physical effects of acceleration were assigned to one twin and not the other (regardless of rockets). I realize now I had actually been imagining a physical substance, a sort of geometry-ether made of little tets of space-time rest-frames, both twins are composed of these real objects with identity, relation to each other, and some sense of sequential "history". The twin with the rockets was distorting his using energy - applied to those objects, so his clock slowed down.

 

[PeterDonis]

Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR.

Yes, but obtaining that result required choosing a particular non-inertial coordinate chart. See below.

Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

Not explicitly, but he did implicitly when he specified the "gravitational field" that appears when the rocket twin chooses to use (non-inertial) coordinates in which he is always at rest (or, equivalently, when he specified the home twin's aging as a function of the rocket twin's time). When he did that, he was implicitly assuming a particular choice of non-inertial coordinates, which in turn implies a particular choice of simultaneity convention. Different choices of non-inertial coordinates (such as Dolby & Gull's), with different simultaneity conventions, would also give different "gravitational fields" (i.e., different connection coefficients), and different behavior of the home twin's aging as a function of the rocket twin's time. The fact that Einstein didn't spell all those implicit assumptions out does not mean he wasn't making them.

 

[PeterDonis]

If acceleration was only a perspective change, why does the "travelling" twin age more slowly.

Acceleration in the sense of proper acceleration--acceleration that you feel--is not a "perspective change". It's a direct observable.

The "perspective change" is the choice of coordinate chart; but coordinates in themselves have no physical meaning. All the physics is in the direct observables. And "relative rate of aging" is not a direct observable. There is no direct observable the traveling twin can use to tell him "how fast the home twin is aging at this moment". The home twin's "rate of aging" depends on which coordinates the traveling twin chooses to use.

The direct observable related to "aging" is the fact that, when the two twins meet up again, the traveling twin's clock shows less elapsed time. But that is a property of the two twins' respective paths through spacetime as a whole; it is not a property of any particular point on the twins' paths.

It's just as if you and I both started out from New York City with cars whose odometers read zero, and met up again in Los Angeles to find that your odometer read more miles than mine, because you took a longer route than I did. The difference in odometers is a direct observable, but it is meaningless to ask at what point on our respective paths the difference in odometers "happened". Nor is it meaningful to ask, during either of our journeys, what the other's odometer reading is "at the same point" in his journey. There is no unique mapping between points on the two paths; there is only the comparison of the total path lengths.

Similarly, the two twins follow different paths through spacetime, and those paths have different lengths. But there is no unique mapping between points on the two paths, so there's no way to tell "which twin is older" in any invariant sense at any particular point. You can only compare the total path lengths.

 

[Jimster41]

Ah that helps.

So this is the problem with intrinsic curvature... It can't be measured from some exterior perspective. It can only be evaluated by comparing end results of closed paths. Is it incorrect to say that it is a property of the points on their path (what else could it be caused by)... but that property is not observable - because for us it is intrinsic?

 

[PeterDonis]

So this is the problem with intrinsic curvature

With path curvature, not spacetime curvature. We are considering a scenario in flat spacetime; spacetime curvature is zero. But the traveling twin's path is curved, whereas the home twin's path is straight. (In the idealized case where the traveling twin's turnaround is instantaneous, his path is composed of two straight legs plus a "corner", and all the path curvature is at the corner. In a more realistic case, the corner would be "rounded off" to be smooth because the twin's acceleration is limited to some finite amount--a "corner" would mean infinite acceleration.)

 

[PeterDonis]

Is it incorrect to say that it is a property of the points on their path (what else could it be caused by)

Path curvature requires the path to be a curve in a higher-dimensional manifold; there is no such thing as completely "intrinsic" path curvature. (This is in contrast to spacetime curvature; for a manifold of two or more dimensions, intrinsic curvature is meaningful, and when we talk about spacetime curvature, we are talking about intrinsic curvature.)

but that property is not observable

Sure it is. Path curvature is just proper acceleration, which is directly observable.

 

[Dale]

Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull.

Do you have a reference for any of this?

As far as I am aware none of it is correct and you are again simply pushing an odd personal agenda that you have been repeatedly told is wrong.

 

[PeterDonis]

As far as I am aware none of it is correct

I was interpreting this as meaning simply that the Dolby & Gull simultaneity convention is different from the "comoving inertial frame" simultaneity convention, which is true, and that the simultaneity convention assumed by "Einstein's GR solution" is the same as the latter one, which is true as far as I know (at least to the extent that Einstein specified a coordinate chart at all).

This does not imply that I agree with the poster's interpretation of what all that means, of course.

 

[Dale]

the simultaneity convention assumed by "Einstein's GR solution" is the same as the latter one

I don't even think that that much is clear. In Einstein's "pop-sci" work there is not enough math provided to identify what simultaneity convention he is using, and in his non "pop-sci" work it is clear that he allows for all coordinate choices.

 

[PeterDonis]

I don't even think that that much is clear.

Hm, you're right; looking at the Einstein article I linked to in the OP, he doesn't commit himself to any particular simultaneity convention. In the Usenet Physics FAQ article, the "comoving inertial frame" convention is assumed, but of course Einstein didn't write that.

 

[PhoebeLasa]

[...] looking at the Einstein article I linked to in the OP, he doesn't commit himself to any particular simultaneity convention.

Einstein (in the posted reference) says that ALL of the extra aging of the home twin (according to rocket twin) happens while the rocket is firing. That's the co-moving-inertial-frames solution, NOT the Dolby & Gull solution.

 

[stevendaryl]

Einstein (in the posted reference) says that ALL of the extra aging of the home twin (according to rocket twin) happens while the rocket is firing. That's the co-moving-inertial-frames solution, NOT the Dolby & Gull solution.

Okay. Yes, you're right--if you have different simultaneity conventions, then the accounting for when the extra aging takes place is different.

 

[Dale]

Einstein (in the posted reference) says that ALL of the extra aging of the home twin (according to rocket twin) happens while the rocket is firing. That's the co-moving-inertial-frames solution, NOT the Dolby & Gull solution.

Not enough information is given to show that it is the co-moving-inertial-frames solution. You are simply assuming that based on your own desire for it to be so. Do you have some reference that proves that all of the extra aging being during the rocket firing implies the co-moving-inertial-frames and excludes all other possibilities?

More problematic, however, is your continued assertion that it is the only valid convention, which is certainly false and certainly not supported by Einstein's writings, either this "pop-sci" reference or his technical papers.

 

[PeterDonis]

Einstein (in the posted reference) says that ALL of the extra aging of the home twin (according to rocket twin) happens while the rocket is firing.

Are you sure you're talking about the Einstein article and not the Usenet Physics FAQ article? The latter is the one that explicitly talks about "where the aging occurs". Einstein's article does not, as far as I can see.

 

[PhoebeLasa]

Are you sure you're talking about the Einstein article and not the Usenet Physics FAQ article? The latter is the one that explicitly talks about "where the aging occurs". Einstein's article does not, as far as I can see.

You're reading too fast. Slow down and smell the roses.

 

[Ibix]

PhoebeLasa's answer is unhelpful, but I guess he's referring to:

During the partial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the resting clock U2. However, this is more than compensated by a faster pace of U1 during partial process 3.

The "partial processes 2 and 4" are the unaccelerated out-and-back phases; process 3 is the turnaround phase.

Einstein says that during partial process 3 a homogeneous gravity field appears, which precisely balances the thrust of the traveling twin's rocket and accelerates the stay-at-home twin. What he does not do, as far as I can see, is specify a simultaneity convention. So while he says that the aging happens while the gravity field is present, he doesn't specify when the gravity field is present for the stay-at-home twin in terms of the stay-at-home twin's clocks.

Is there a simultaneity convention implicit in the homogeneity of the gravity field? Or will any convention do?

 

[PeterDonis]

What he does not do, as far as I can see, is specify a simultaneity convention...he doesn't specify when the gravity field is present for the stay-at-home twin in terms of the stay-at-home twin's clocks.

Exactly.

 

[harrylin]

[..] Einstein used the term "gravitational field" to refer to the Christoffel symbols. [..].

Please provide a reference, thanks!

 

[stevendaryl]

Please provide a reference, thanks!

In the dialog that is linked to in the very first post, Einstein doesn't explicitly use the word "Christoffel symbol", but he does say, from the point of view of the "traveling" twin:

A gravitational field appears, that is directed towards the negative x-axis. Clock U1 is accelerated in free fall, until it has reached velocity v.

That is clearly using "gravitational field" to mean "acceleration due to gravity". The equations of motion for a test mass in SR in general, non-inertial, curvilinear coordinates attributes the (coordinate) acceleration due to gravity to the Christoffel symbols:

$\frac{d^2 x^j}{dt^2} = - \Gamma^j_{kl} \frac{dx^k}{dt} \frac{dx^l}{dt} - \frac{d log(\gamma)}{dt} \frac{dx^j}{dt}$

(The second term is due to using the non-affine parameter $t$ rather than proper time $\tau$; $\gamma$ is the conversion factor: $\frac{dt}{d\tau} = \gamma$)

 

[Dale]

Please provide a reference, thanks!

A. Einstein. "The Foundation of the General Theory of Relativity". Annalen der Physik 354 (7), 769-822. (Translated version at: http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity ). Section 13, last sentence "If $\Gamma_{\mu\nu}^{\tau}$ vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field."

As stevendaryl mentioned. In his pop sci paper he never uses the technical term. But he describes the properties of the gravitational field sufficiently clearly to make it understood.

 

[vanhees71]

Without having read the thread, I just would like to know, what's the big issue with the socalled twin paradox? I don't understand, why it should be a paradox at all! The physical statement is that the age of a system (as in this case a living organism) is identical with its proper time. I'm not aware, whether ever an experiment was made in the context of the age of living organisms. I guess it's hard to define what "age" means. For sure, there are very accurate measurements made with clocks, showing that, if properly constructed, show their proper time and compare well with the difference to clocks in other frames of reference according to both the time-dilation and gravitational effects. Also the mean lifetime of unstable particles have been considered, and always it was found that their proper lifetime is an invariant and that it follows the usual time dilation according to the Lorentz factor (particularly high accuracy is reached in storage rings of particle accelerators). Last but not least the proper function and accuracy of the Global Positioning System (GPS) proves that both the relativistic time-dileation and gravitational effects are very accurately describing the behavior of clocks.

So there's no paradox! On the theory side there cannot be one, because the mathematics of relativistic space-time is a very well defined mathematical model, and the correctness in sense of a physical theory is well justified by many high-accuracy measurements.

 

[Ben Niehoff]

Is this really an entire thread arguing over whether Einstein used a particular argument in some popsci article from 100 years ago, rather than arguing over the actual physics? Wow.

Plenty of smart people use wrong arguments to arrive at right conclusions (including Einstein), and plenty of smart people arrive at wrong conclusions as well (including Einstein, but I don't know about this specific reference). I guess historians are welcome to take an interest in how Einstein made his way, eventually, to the right physics, but as physicists learning GR, I don't see the value in this.

The best GR answer to the twin paradox, in my opinion, is this: Two travellers take two paths through spacetime, and those paths are different lengths. The twin on the longer path ages more, because "length" in this case is proper time. This, frankly, is a general answer that applies to any situation whatsoever; in curved space, in flat space, with arbitrary acceleration. Just integrate the proper time along the path taken. That's it.

There is no unambiguous notion of "when" the path difference occurs. This fact is not in any way mysterious. You can visualize this in normal, Euclidean space, because the phenomenon is exactly the same: You have two curves of different lengths, that start at one endpoint and meet at the other. As you sweep along the points of one curve, there is no natural way to correspond these points to the points of the other curve. So you can't say "where" the path difference happened; it is a global property of the curve.

 

[harrylin]

Huh? Where did I say that? [..]

By now I thought to have made that pretty clear, but here once more: You disagreed with Einstein's "physical entity" (similar to any such entities like energy, magnetic fields etc. which have to propagate as that's what such physical entities do, even if they are "relative"!) by starting out with arguing to the contrary that "The "field" that appears in the analysis does not have to propagate, because it is a coordinate effect, not a physical effect." Right from the start you denied Einstein what he claimed. In fact, what you wrote basically agrees with Builder, Baez etc. against Einstein, but you didn't realize it because you could not believe that he could have meant what he wrote.

Now, as you really do think that Einstein's GR resolution to the twin paradox had no content above and beyond SR in noninertial coordinates, I must conclude (since you wrote that in answer to my earlier statement which I rewrite here as follows), that according to you Einstein's debate with other physicists was a debate about nothing despite the fact that nobody has or had a debatable problem with non-inertial coordinates! I'll try one last time to clarify the difference between a trivial mapping to accelerating frames (which can't be the cause of dispute) and the dispute that Einstein obviously discussed.

No, Einstein did not say you "feel" a gravitational field. He said you feel the acceleration required to hold yourself at rest in the gravitational field. [..]

Oops yes indeed he did not say what I said - but neither what you said! (for Einstein, acceleration during a certain time is not "in rest"). Thus he wrote: "An external force acts upon clock U2, preventing it from being set in motion by the gravitational field."

Minkowski spacetime does obey GR

I referred to his criticism on the preferred status of the class of inertial frames in SR. As I pointed out it is that what he claimed to have gotten rid of, and it is that claim that caused the criticism, just as he elaborated.

No, changes in spacetime curvature propagate at local speed c. [..] Yes, [light from distant stars also propagates at local speed c.]. So what?

Can you explain how with Einstein's illustration (he didn't give the numbers but that's not the sticking point, and Moller gave them later in his textbook), a gravitational field that was induced by firing the rocket of U2 can provide a valid explanation of the instant Doppler effect of light from distant stars according to K'? That the speed of induction isn't even important is what I last discussed in the other thread. And please don't forget that K' doesn't accelerate in that explanation: K' remains in rest, and instead K accelerates with U1.
Thanks!

 

[harrylin]

A. Einstein. "The Foundation of the General Theory of Relativity". Annalen der Physik 354 (7), 769-822. (Translated version at: http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity ). Section 13, last sentence "If $\Gamma_{\mu\nu}^{\tau}$ vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field."

As stevendaryl mentioned. In his pop sci paper he never uses the technical term. But he describes the properties of the gravitational field sufficiently clearly to make it understood.

Thanks! Somehow I regard such symbols to be mere mathematical representations of the physical entities that they describe.

 

[Dale]

Thanks! Somehow I regard such symbols to be mere mathematical representations of the physical entities that they describe.

Yes. That's true of all of the mathematical quantities used in any theory.

 

[PeterDonis]

You disagreed with Einstein's "physical entity"

No, I disagree that Einstein said a "physical entity" must propagate. You are saying that, but I don't see where Einstein said that. What I see is that your definition of "physical entity" disagrees with the definition implicit in Einstein's terminology; by his terminology, since he uses "physical entity" to include things like Christoffel symbols which do not propagate, he clearly does not intend the implication that anything that is a "physical entity" must propagate.

a gravitational field that was induced by firing the rocket of U2 can provide a valid explanation of the instant Doppler effect of light from distant stars according to K'?

I don't know that Einstein's analysis even covers this question. Einstein was using the "gravitational field" to account for the differential aging of the two twins, using the non-inertial frame in which the traveling twin is at rest. But he does not, as far as I can see, even consider how, or whether, the "gravitational field" can account for the change in Doppler shift that the traveling twin sees when he turns around.

I'm not clear about whether any of the criticisms of Einstein that you mentioned raise this issue either. I can see that it's a valid question, but if Einstein doesn't address it, and none of the criticisms address it, the only option we would have to analyze it would be to try to construct an analysis on our own.

 

[PeterDonis]

what's the big issue with the socalled twin paradox?

There isn't one. This thread isn't about whether the paradox is actually a paradox; we know it isn't. It's about a particular interpretation that Einstein used to explain how the events could be viewed by the traveling twin, if that twin wanted to use a (non-inertial) frame in which he was always at rest.

arguing over whether Einstein used a particular argument in some popsci article from 100 years ago

It's not about whether Einstein used the argument; he did. It's about whether a particular criticism of it, published much later, was valid (or perhaps about for what, if any, range of values of "valid" the criticism was valid).

 

[PeterDonis]

for Einstein, acceleration during a certain time is not "in rest"

I'm not sure I understand this; doesn't Einstein say the traveling twin is "at rest" in the system K'?

 

[Ibix]

Einstein was using the "gravitational field" to account for the differential aging of the two twins, using the non-inertial frame in which the traveling twin is at rest. But he does not, as far as I can see, even consider how, or whether, the "gravitational field" can account for the change in Doppler shift that the traveling twin sees when he turns around.

This is related, I think, to my question in #57. If Einstein means that a uniform gravitational field permeates the whole of space, he can calculate the path of the stay-at-home and can tell how long (by the stay-at-home's clocks) the field must have been present to distort the path of the stay-at-home such that the final meet-up happens as it does. That seems to me to imply a simultaneity convention implicit in Einstein's model. On the other hand, a weaker gravitational field present for longer will certainly turn the stay-at-home twin around just as well, and one could make the case that the only thing that the traveller can tell from his accelerometers is that the g-field was constant at his location.

Does the requirement that the Doppler shifts make sense in this GR view impose a restriction on the g-field? I'm aware I'm slightly out of my depth here - I hope this isn't terribly naive...