Using General Relativity to analyze the twin paradox

[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...

 

[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.