Using General Relativity to analyze the twin paradox

[PeterDonis]

If Einstein means that a uniform gravitational field permeates the whole of space

This is one of the subtleties that only really comes out when you try to do the actual math. It turns out that the expression "uniform gravitational field" does not have a straightforward translation into math. The obvious translation is "g constant everywhere", which translates into "Christoffel symbols constant everywhere", but AFAIK there is no valid coordinate chart on Minkowski spacetime for which this is true (except for the trivial case of an inertial frame in which they are all zero). So there appears to be an unavoidable ambiguity in which coordinate chart corresponds to Einstein's ordinary language description.

Does the requirement that the Doppler shifts make sense in this GR view impose a restriction on the g-field?

I don't know. As I said in my previous post, I don't know that Einstein or any of those who criticized his article considered this, and I have not tried to work through an analysis myself.

 

[harrylin]

[...]
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.

He certainly didn't mention it; he only mentioned the aspects that did seem to work. Builder pointed out that those aspects automatically work by design, thanks to the (weak) equivalence principle.

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.

I expanded on Builder's analysis as now repeated here; it looks clear to me that the "gravitational field" that Einstein invoked cannot explain all the phenomena - as it should in order to be on the same footing as kinetic energy and other physical fields. My analysis was however only qualitative; it did not seem necessary to go beyond that. I wrote in my post to which you link in the OP: "As far as I know, none of the involved authors (Einstein, Tolman, Moller, ...) addressed that self-contradiction."

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

Any clock is at rest in its co-moving frame, by mere definition! The claim of Einstein that raised the criticisms, is that, in contradiction to SR, observers of K' are justified to hold that it is system K that does all the acceleration: from that perspective it is clock U1 that does all the traveling and K' is all the time in rest, it does not accelerate. But then all the laws of nature must work correctly to predict the same as SR in calculations from that perspective.

 

[vanhees71]

Well, the proper time of each twin gives his/her aging (at least according to standard interpretation of "aging") is simply given by
$$\Delta \tau=\int_{\lambda_0} ^{\lambda_0+\Delta \lambda} \mathrm{d} \lambda \sqrt{g_{ \mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \lambda} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \lambda}},$$
when simplifying the description of the twin to a world line of a point particle (take his/her center of momentum as this point). That's it. There's not more to the "twin paradox" than that that the aging depends on the world line of each twin in pseudo-Riemannian space, no matter what Einstein might have said about it. Particularly one should be aware of the fact that everything said about relativistic gravity/general relativity before 1916 has to be taken with some care, because before that the full theory was not developed yet!

Already in his famous 1905 paper is a wrong statement, when seen from the point of view of fully developed GR 10 years later. So what? Even a genius as Einstein can be wrong in some aspects of an issue on the forefront of research. It's hard to imagine somebody to find out something new without making mistakes sometimes :-).

 

[harrylin]

[..] Particularly one should be aware of the fact that everything said about relativistic gravity/general relativity before 1916 has to be taken with some care, because before that the full theory was not developed yet!

The issue concerns 1916 GR (see my post #3). IMHO modern GR is somewhat different, and in part for a good reason!

Already in his famous 1905 paper is a wrong statement, when seen from the point of view of fully developed GR 10 years later. So what? Even a genius as Einstein can be wrong in some aspects of an issue on the forefront of research. It's hard to imagine somebody to find out something new without making mistakes sometimes :-).

The important difference is that later papers and textbooks do not seem to agree on this one (e.g. Builder <-> Moller), and as a result it is still a discussion topic in the literature.

PS. one of the recent papers is by Pesic, http://iopscience.iop.org/0143-0807/24/6/004
Regretfully it appears that Pesic doesn't understand Einstein's 1918 paper (nor Langevin's 1911 paper), and so he remarks "It is perplexing to think that [Einstein] might have forgotten the full force of his own earlier argument". (No he surely had not forgotten that!)

 

[maline]

If Christoffel-symbol fields are allowed to simply appear in regions of space, with no cause, then why don't we see objects randomly accelerating? Does the arbitrary field have to extend throughout space? Wouldn't that require simultaneity?
Also, what do the "distant stars" have to do with anything? If they accelerated in frame K', wasn't that a result of the same field?

 

[stevendaryl]

If Christoffel-symbol fields are allowed to simply appear in regions of space, with no cause, then why don't we see objects randomly accelerating? Does the arbitrary field have to extend throughout space? Wouldn't that require simultaneity?
Also, what do the "distant stars" have to do with anything? If they accelerated in frame K', wasn't that a result of the same field?

Christoffel symbols are not physical fields. Think about this: you're walking directly away from a tree, and then you decide to turn around and start walking toward the tree. From your point of view, the tree is behind you, moving away from you, then stops and moves in a big circle until it is in front of you. What force could possibly cause a huge tree to behave that way? The answer is that there is nothing happening to the tree, it's only the location of the tree relative to a you-centered coordinate system that is changing.

That's all that Christoffel symbols do, is describe the aspects of the motion of objects that are artifacts of your coordinate system.

 

[stevendaryl]

Also, what do the "distant stars" have to do with anything? If they accelerated in frame K', wasn't that a result of the same field?

Unlike velocity, which is relative to a coordinate system, there is an absolute notion of acceleration. You can measure it using an accelerometer. Einstein used the phrase: acceleration relative to the "distant stars" to describe this absolute aspect of acceleration. It's not really relative to the stars; it's relative to the local spacetime.

 

[Dale]

This is one of the subtleties that only really comes out when you try to do the actual math. It turns out that the expression "uniform gravitational field" does not have a straightforward translation into math. The obvious translation is "g constant everywhere", which translates into "Christoffel symbols constant everywhere", but AFAIK there is no valid coordinate chart on Minkowski spacetime for which this is true (except for the trivial case of an inertial frame in which they are all zero). So there appears to be an unavoidable ambiguity in which coordinate chart corresponds to Einstein's ordinary language description.

This is precisely the reason I have always taken the position that the outline given in this pop-sci paper does not provide enough information to infer any specific coordinate chart or simultaneity convention.

 

[vanhees71]

Usually Rindler coordinates define uniform acceleration in SRT:

https://en.wikipedia.org/wiki/Rindler_coordinates

 

[Dale]

it looks clear to me that the "gravitational field" that Einstein invoked cannot explain all the phenomena - as it should in order to be on the same footing as kinetic energy and other physical fields. My analysis was however only qualitative; it did not seem necessary to go beyond that. I wrote in my post to which you link in the OP: "As far as I know, none of the involved authors (Einstein, Tolman, Moller, ...) addressed that self-contradiction."

If you are claiming a self contradiction then it certainly is necessary to show it mathematically. Frankly, I don't see even a possibility that it is correct. Once you have the Christoffel symbols you have covariant derivatives, and can calculate null geodesics and therefore Doppler shifts.

The problem with the pop-sci paper is that it is too vague to translate uniquely into math. Not that the underlying math is wrong. Also, his terminology and interpretation is outdated, but "self-contradictory" is an unjustified claim.

 

[harrylin]

Unlike velocity, which is relative to a coordinate system, there is an absolute notion of acceleration. You can measure it using an accelerometer. Einstein used the phrase: acceleration relative to the "distant stars" to describe this absolute aspect of acceleration. It's not really relative to the stars; it's relative to the local spacetime.

Although I agree with you, Einstein fully disagreed with you at the time that he wrote his 1918 paper. If this is not clear, I'll gladly contrast his statements with yours.

 

[maline]

If Christoffel-symbol fields are allowed to simply appear in regions of space, with no cause, then why don't we see objects randomly accelerating? Does the arbitrary field have to extend throughout space? Wouldn't that require simultaneity?
Also, what do the "distant stars" have to do with anything? If they accelerated in frame K', wasn't that a result of the same field?

Christoffel symbols are not physical fields. Think about this: you're walking directly away from a tree, and then you decide to turn around and start walking toward the tree. From your point of view, the tree is behind you, moving away from you, then stops and moves in a big circle until it is in front of you. What force could possibly cause a huge tree to behave that way? The answer is that there is nothing happening to the tree, it's only the location of the tree relative to a you-centered coordinate system that is changing.

That's all that Christoffel symbols do, is describe the aspects of the motion of objects that are artifacts of your coordinate system.

I am trying to understand Einstein's perspective, in which there are real albeit frame-dependent "gravitational forces", as you explained here:

Whether you put the terms FgravF_{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 FgravF_{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, FgravF_{grav} is real, in the sense that it is measurable, to the same extent that coordinate acceleration is.

So again, if we're allowed to describe a "physical field" with no source that exerts forces, what are the limitations on that?

And of course the metric in your vicinity is indeed determined by the propagation (to the extent things even have to propagate--see below) of spacetime curvature from those sources in your past light cone, to your current spacetime location.

So are Einstein's "fields" simply arbitrary or are they caused by past motion of the stars in this frame? That idea would seem helpful for explaining the doppler shift that the traveller sees, as well. But why would the stars have moved if not as a result of the same type of "gravitational field"? Is there an infinite regress here?
(Actually, that might not be so bad. Einstein believed in a static, eternal universe, so if you can push the problem infinitely back in time, you've solved it!)

 

[PeterDonis]

Any clock is at rest in its co-moving frame, by mere definition!

We're not talking about momentarily comoving inertial frames, we're talking about non-inertial frames. You appeared to be saying that K' could not be considered to be "at rest" in a non-inertial frame, which is not correct. There is no unique way to define such a non-inertial frame, but it is certainly possible to do so.

 

[PeterDonis]

Einstein used the phrase: acceleration relative to the "distant stars" to describe this absolute aspect of acceleration.

Actually, I'm not sure if Einstein meant to equate "acceleration relative to the distant stars" with proper acceleration measured by an accelerometer, at least not by definition. I think he considered it a genuine physical question why the two should be the same, i.e., why, when we measure how we are accelerating, in a coordinate sense, relative to a frame in which the distant stars are at rest, we get the same answer as when we measure the proper acceleration we actually feel, with an accelerometer. Those two measurements are obviously not the same measurement, and I think the first is what he meant by "acceleration relative to the distant stars".

(The GR answer to the question of why they are the same, of course, is that the distant stars, or more precisely the distant stars in the past light cone, determine the local spacetime geometry, which in turn determines the proper acceleration you feel, so the two measurements will give the same answer because the frame in which the distant stars are at rest is measured using the light from those stars, which arrives along with the information that determines the local spacetime geometry.)

 

[PeterDonis]

are Einstein's "fields" simply arbitrary or are they caused by past motion of the stars in this frame?

They're arbitrary in the sense that you can change them by changing coordinates, without changing any physics. But they're "caused by past motion of the stars" in the sense that any valid coordinate chart you choose will be describing the same spacetime geometry, which determines the relationship between the "fields" in one chart and the "fields" in another.

 

[stevendaryl]

So again, if we're allowed to describe a "physical field" with no source that exerts forces, what are the limitations on that?

If the physical field is coordinate-dependent, then the limitation is that any coordinate-independent measurement must (tautologically, I guess), give the same value regardless of the choice of coordinate systems. So I can magically make a gravitational field appear that can accelerate stars light-years away from here. That sounds pretty powerful, but anything of consequence has to be independent of that choice. So, my choice definitely cannot force two planets to smash into each other (since the fact that two planets collided is coordinate-independent).

 

[harrylin]

We're not talking about momentarily comoving inertial frames, we're talking about non-inertial frames. You appeared to be saying that K' could not be considered to be "at rest" in a non-inertial frame, which is not correct. [...]

I certainly did not suggest such a thing! First of all, K' is itself a frame. You appeared to be saying that K' according to Einstein considers itself to be an accelerating frame, which is not correct. Instead, K' considers K to be an accelerating frame (note that the acceleration in Einstein's description is coordinate acceleration, but not merely so: K' considers K to be falling while it stays itself in rest).

 

[harrylin]

If you are claiming a self contradiction then it certainly is necessary to show it mathematically.

I'll gladly do so with approximate equations that already more than suffice (later).

The problem with the pop-sci paper is that it is too vague to translate uniquely into math. [..]

Builder refers to several follow-ups in the literature that corroborate Einstein's calculation. As its scope is limited to the equivalence between stationary fields and acceleration, exactly that aspect can probably not go wrong (for the same reason it cannot add anything useful to the SR analysis, as Builder explained).
- I already mentioned Moller's "The theory of relativity", referenced by Builder (I don't think that Builder had seen Einstein's paper), and which is easily found on archive.org (is that a legal copy? to make sure I don't give the link), from p.258
- I also found that Einstein's calculation was recently corroborated by Unnikrishnan, in eq.1-8: http://www.iisc.ernet.in/~currsci/dec252005/2009.pdf

 

[maline]

If the physical field is coordinate-dependent, then the limitation is that any coordinate-independent measurement must (tautologically, I guess), give the same value regardless of the choice of coordinate systems. So I can magically make a gravitational field appear that can accelerate stars light-years away from here. That sounds pretty powerful, but anything of consequence has to be independent of that choice. So, my choice definitely cannot force two planets to smash into each other (since the fact that two planets collided is coordinate-independent).

Sure, that's obviously the bottom line. But within one frame, what are the rules that make sure of this? Where can or cannot such fields appear?

they're "caused by past motion of the stars" in the sense that any valid coordinate chart you choose will be describing the same spacetime geometry, which determines the relationship between the "fields" in one chart and the "fields" in another.

I am working in Einstein's frame K', with some valid chart, say the "MCIF solution". Can the "fields" here be explained as having a "physical" cause? The traveler's rockets & choice of frame are not candidates; we are already discussing this frame. Is there a narrative here in which the stars accelerated in the past, all along the traveler's light-cone, and these motions caused the "field"? If so, would this be a sort of gravitational wave?

 

[harrylin]

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. [..].

Here's my reconstruction of how the "twin" astronaut example became a "paradox" (= an apparent contradiction). I figured it out from studying later commentaries as well as several original papers of that time (If anyone knows of a paper of the period 1907-1916 that seems to be in conflict with my reconstruction, I'll be all ears!).

1. Einstein-1905 and Langevin-1911 describe with SR how an asymmetry arises when one clock or astronaut is in inertial motion and the other not. There is no paradox (at least, not for most physicists; some people probably had problems with mutual time dilation which is also portrayed as a paradox).
2. The standard explanation that is given for the asymmetry is that the laws of nature of SR relate to inertial frames; it is faulty to apply the Lorentz transformations from an accelerating frame. According to SR only the "stay-at-home" may pretend to be "truly in rest"; the "traveler" may not claim that it is the "stay-at-home" who accelerates instead.
3. But then Einstein claims (already before finishing the theory) that thanks to his theory of GR (the general relativity of motion), the "traveler" may just as well pretend to be "truly in rest", and that it is the "stay-at-home" who accelerates instead! With that the standard SR explanation doesn't work anymore and from the confusion that it caused, a "paradox" is born.
4. Apparently Einstein received a lot of criticism, and so he wrote in 1918 a paper to answer those criticisms. It's the one (English translation) that is linked to in the OP.

 

[vanhees71]

2. The standard explanation that is given for the asymmetry is that the laws of nature of SR relate to inertial frames; it is faulty to apply the Lorentz transformations from an accelerating frame. According to SR only the "stay-at-home" may pretend to be "truly in rest"; the "traveler" may not claim that it is the "stay-at-home" who accelerates instead.

Why shouldn't you be allowed to use non-inertial frames within SR? It's also allowed in Newtonian mechanics which also obeys the special relativity principle as does SR. With your argument you'd even be forbidden to describe non-uniform (free-particle) motion at all. This is not right for sure, since SRT works well for accelerated particles. Otherwise LHC and other accelerators wouldn't work.

It doesn't matter, who is "truly at rest" or not. This doesn't make sense already in Newtonian physics. The usual hypothesis is that aging is given by the proper time of the object under consideration. It has been proven for unstable particles to very high accuracy ("age" = "mean lifetime"). Whether it has ever been checked for living organisms, I don't know, and I guess, it's hard to invent an experiment.

 

[harrylin]

Why shouldn't you be allowed to use non-inertial frames within SR? It's also allowed in Newtonian mechanics which also obeys the special relativity principle as does SR. With your argument you'd even be forbidden to describe non-uniform (free-particle) motion at all. [..]

Evidently you misunderstand "my" (Einstein's) argument. The papers that I referred to in post #90 show that SR is perfectly capable of describing non-uniform motion. There is also no issue with mapping to non-inertial frames in SR, nor can there have been such an issue; I gave the example of how common and accepted that was in Newtonian mechanics in post #34.

The "twin paradox" is a common textbook example of misapplying non-inertial frames so that some students wrongly conclude that clocks in rest in the inertial frame S will age less. Another example of misapplying accelerating frames was given in a parallel thread by me and Nugatory here :
#49 https://www.physicsforums.com/threa...-spaceship-paradox.804582/page-3#post-5054605
#54 https://www.physicsforums.com/threa...-spaceship-paradox.804582/page-3#post-5054628

 

[stevendaryl]

Although I agree with you, Einstein fully disagreed with you at the time that he wrote his 1918 paper. If this is not clear, I'll gladly contrast his statements with yours.

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

 

[vanhees71]

I haven't followed all these arguments. But, again, proper time and proper distance are independent of the chosen coordinates, it's a Minkowski invariant (scalar). So there's no difference in the result, if I choose non-inertial coordinates. Of course, the same holds even true in general relativity, which is covariant wrt. to general diffeomorphisms. It's so by construction!

 

[harrylin]

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

That empty space still has a "notion" of acceleration is just what Einstein denied - and it's that denial that led to the criticisms that he tried to counter with his 1918 paper.

 

[maline]

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

How can this make sense? If we mean coordinate acceleration, then it is only relative, as Einstein explained in our article. If we mean proper acceleration, how could anyone have thought it relative? Any one of us standing on Earth is undergoing proper acceleration that isn't relative to any object!

 

[harrylin]

I haven't followed all these arguments. But, again, proper time and proper distance are independent of the chosen coordinates, it's a Minkowski invariant (scalar). So there's no difference in the result, if I choose non-inertial coordinates. Of course, the same holds even true in general relativity, which is covariant wrt. to general diffeomorphisms. It's so by construction!

Here we are dealing with something entirely different! What would you think if I claimed that one may equally well hold that your "non-inertial" coordinate frame K' is in fact not "non-inertial" but in rest? And that as a consequence, the "inertial" coordinate frame K can be considered to be an accelerating frame (in other words, the clock in rest in K has non-inertial coordinates)? That's what Einstein did.

 

[stevendaryl]

How can this make sense? If we mean coordinate acceleration, then it is only relative, as Einstein explained in our article. If we mean proper acceleration, how could anyone have thought it relative? Any one of us standing on Earth is undergoing proper acceleration that isn't relative to any object!

Well, Mach thought that acceleration should be relative. He didn't actually have a theory that worked that way, though. Mach's reasoning was that there should be no observable difference between:

1. Hopping into rocket ship and accelerating in a straight line in the x-direction.
   
2. Somehow contriving to get all the masses in the universe except for the rocket to accelerate in the negative x-direction.

As I said, General and Special Relativity are not Machian in this sense, because a rocket that is alone in the universe can still feel acceleration. (Actually, that's a bad example, since you can't accelerate without throwing mass behind you, in which case, there is some other mass that you can be accelerating relative to. A better example is rotation. If you are on a space station that is rotating, you can feel the rotation, even in the case where the space station is the only object in the universe, and so there is nothing that it is rotating relative to.)

 

[stevendaryl]

Here we are dealing with something entirely different! What would you think if I claimed that one may equally well hold that your "non-inertial" coordinate frame K' is in fact not "non-inertial" but in rest? And that as a consequence, the "inertial" coordinate frame K can be considered to be an accelerating frame (in other words, the clock in rest in K has non-inertial coordinates)? That's what Einstein did.

You are using "non-inertial" and "at rest" as if they were mutually exclusive. But in General Relativity, they are not. If you are at rest on the surface of a planet, the natural coordinate system is non-inertial.

 

[stevendaryl]

Well, Mach thought that acceleration should be relative. He didn't actually have a theory that worked that way, though. Mach's reasoning was that there should be no observable difference between:

1. Hopping into rocket ship and accelerating in a straight line in the x-direction.
   
2. Somehow contriving to get all the masses in the universe except for the rocket to accelerate in the negative x-direction.

As I said, General and Special Relativity are not Machian in this sense, because a rocket that is alone in the universe can still feel acceleration. (Actually, that's a bad example, since you can't accelerate without throwing mass behind you, in which case, there is some other mass that you can be accelerating relative to. A better example is rotation. If you are on a space station that is rotating, you can feel the rotation, even in the case where the space station is the only object in the universe, and so there is nothing that it is rotating relative to.)

It seems to me that for a theory to be Machian, space can't be a Riemannian manifold, but must be Euclidean. A machian theory would have to be formulated in terms of relative position vectors: the vector separation between objects. But the separation between two objects is only an unambiguous vector if parallel transport is path-independent. I'm not sure if that uniquely would make space Euclidean, but it surely constrains the geometry considerably. Also, if what's relevant is the separation at a particular time, it would seem to require that simultaneity be absolute, and not relative.

So based on a few moments thought, it seems to me that a machian theory would have to be formulated in something like Galilean spacetime, rather than a general pseudo-Riemannian spacetime.

 

[harrylin]

You are using "non-inertial" and "at rest" as if they were mutually exclusive. But in General Relativity, they are not. If you are at rest on the surface of a planet, the natural coordinate system is non-inertial.

The language in Einstein's paper is consistent with classical mechanics and SR; and there is negligible nearby mass in the discussion. However, Einstein does not use the word "inertial" there, so it would perhaps have been clearer if I had replaced the cited "inertial motion" by "uniform motion" or "Galilean motion", as "inertial" has become ambiguous.

In fact, if one considers "in rest" to mean the same as "accelerating" in this context then one mixes up the two different points of view that Einstein distinguishes. Then it's quite impossible to even understand what the discussion was about.

 

[stevendaryl]

Perhaps you use modern jargon that is incompatible with Einstein's 1918 paper and that could hinder a correct understanding.

I don't think so. In the dialog written by Einstein, he says the following: (from the point of view of an accelerated reference frame, $K'$):https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

1. 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. An external force acts upon clock U2, preventing it from being set in motion by the gravitational field. When the clock U1 has reached velocity v the gravitational field disappears.

2. U2 moves with constant velocity v up to point B of the positive x-axis. U1 remains at rest.

He's saying, that from the point of view of $K'$, it is U2 that is at rest. But U2 is certainly NOT inertial. So it's a big mistake to conflate "being at rest" with "moving inertially".

I also disagree with you that the two mean the same thing in Newtonian physics, either.

You can write the Newtonian equations of motion in an arbitrary coordinate system as follows:

$m \frac{d^2 x^j}{dt^2} = F^j + F_{fict}^j$

where $F^j$ is the same force that would be present in an inertial coordinate system, and $F_{fict}^j$ is the extra terms due to curvilinear, noninertial coordinates. Being intertial means that $F_{fict}^j = 0$, while being at rest means that $\frac{dx^j}{dt} = 0$. Those aren't the same, at all.

 

[PeterDonis]

you appeared to be saying that K' according to Einstein considers itself to be an accelerating frame,

I said K' was a non-inertial frame. I did not say that means it is "accelerating". The only one who is equating "non-inertial" with "accelerating" is you.

 

[PeterDonis]

the separation between two objects is only an unambiguous vector if parallel transport is path-independent.

This is true for Minkowski spacetime, so a "Euclidean" manifold is not required. But a flat manifold (zero Riemann tensor) certainly is.

 

[stevendaryl]

This is true for Minkowski spacetime, so a "Euclidean" manifold is not required. But a flat manifold (zero Riemann tensor) certainly is.

But in Minkowsky spacetime, the separation vector between two objects is ambiguous if the objects are moving relative to one another. The vector will be frame-dependent. (I don't just mean that the components are frame-dependent---that's always the case.) The separation between EVENTS is unambiguous any Minkowsky spacetime, but not the separation between OBJECTS. That's the reason that in SR, forces can't really be direct interactions between objects; they have to be mediated by fields, which propagate.