Twin Paradox with "home time"

[PAllen]

Try that explanation on your grandmother. ;)
While it's true that he does not present the detailed calculations, a careful reading makes it obvious what he's trying to do.

I'm surprised that you don't see the circularity. Einstein argues that he can view the accelerating traveler as stationary in a uniform gravitational field (by applying the equivalence principle). He then applies the physical laws of a gravitational field (that presumably come from GR) to the problem .

The circularity is that these physical laws of a gravitation field were not independently derived. They were derived directly from the physics of uniform acceleration in SR. The physics of uniform acceleration were transported to apply to a gravitational field (by invoking the equivalence principle) in the first place. That is, the gravitational law is founded on uniform acceleration in SR.

So by invoking a gravitation field, Einstein has added nothing to the analysis already done using acceleration alone in SR. In other words, the impression that he is applying something new that comes from GR is an illusion. Perhaps he had a sly smile on his face when answering his critics in this way?

You are missing a subtlety of the history of GR. In the modern view, GR is the theory of curved spacetime, SR the theory of flat spacetime that (only) is locally true in GR. However, Einstein had a different way of looking at it, which (to the best of my knowledge) he never abandoned. In his view, the derivation of the properties of non-inertial frames in SR was part of GR, which also extended this to cover significant mass with curved spacetime. Thus, he is using the features of accelerated coordinates in SR displaying a position dependent potential (which you can see in the Rindler metric - to which I believe I referred you earlier), all derived for flat spacetime (no Einstein field equations of GR involved). Einstein just viewed this physics as special case of GR rather than part of SR.

Thus, with Einstein's packaging, there is no circularity.

It seems you have still not fully grappled with my posts #5 and #11. Especially the point that in the non-inertial coordinates in which the metric shows a potential, the traveling twin is not accelerating. It is the home twin that is accelerating in these coordinate, and that acceleration plays no role in the clock rate of the home twin.

 

[Dale]

Try that explanation on your grandmother. ;)

My grandmother is dead, but were she alive I am sure that I would be able to explain to her how to add 1 and divide by 2 to go from the metric to the potential.

While it's true that he does not present the detailed calculations, a careful reading makes it obvious what he's trying to do.

Then please present the detailed calculations that are so obvious to you. They are far from obvious to me.

I'm surprised that you don't see the circularity. Einstein argues that he can view the accelerating traveler as stationary in a uniform gravitational field (by applying the equivalence principle). He then applies the physical laws of a gravitational field (that presumably come from GR) to the problem .

The circularity is that these physical laws of a gravitation field were not independently derived. They were derived directly from the physics of uniform acceleration in SR. The physics of uniform acceleration were transported to apply to a gravitational field (by invoking the equivalence principle) in the first place. That is, the gravitational law is founded on uniform acceleration in SR.

That is a matter of historical happenstance, not logical circularity. There is no reason that you cannot logically start with GR and use it to derive results that apply to the special case of flat spacetime. In fact, logically it is more sound to start with a general principle and derive special cases than to start with a special case and then generalize (despite the fact that historically it rarely happens that way).

So by invoking a gravitation field, Einstein has added nothing to the analysis already done using acceleration alone in SR.

I agree completely on this.

 

[CKH]

Sure. SR is these two statements: (1) spacetime is globally flat; (2) the laws of physics are Lorentz invariant. Neither of those statements requires inertial frames to be defined (though of course it's much easier and more intuitive to model them mathematically with inertial frames).

1) Spacetime is globally flat.

That's a geometric statement about a mathematical abstraction called spacetime (see "block spacetime" for a debate about its physical reality). I'm guessing it would be described mathematically as a four dimensional manifold with 0 curvature throughout. How does this statement acquire a meaning in the physical world? Physics in SR is about the behavior of objects (while ignoring their gravitational properties).

Allow me to attempt to relate these mathematical abstractions to the physical world and replace that abstract description with a physical one.

Under the conditions that no forces act on objects, objects at mutual rest remain at mutual rest. Objects in motion move at uniform speed in straight lines. These statements appear to be equivalent to "each object remains at rest in an inertial frame". This is the physical equivalent to the abstract mathematical statement "spacetime is flat".

2) The laws of physics are Lorentz invariant.

Let's translate that statement into a more detailed one as suggested in the Lorentz Covariance wiki article which states:

An equation [a physical law in this case] is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term "invariant" here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity, i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.

(I believe that this second statement is what distinguishes SR from Newton's theory.)

This is like a game of hide and seek. We can attempt to hide the inertial frames in some abstract mathematical statement. But when we dig into the physics behind the abstraction, we find that the inertial frames are still there.

I had the same difficulty in the thread Einstein says objects do not fall to the Earth? In that thread, a 4-vector equation for force, mass and acceleration was presented which includes a term which compensates for the fictitious forces that appear in any specific non-inertial or non-cartesian coordinate system. This is done in order to make the law "coordinate independent". I claim that the equation is dependent on the concept of inertial motion. No one wanted to admit that this "fictitious force" term is constructed relative to inertial motion, even though they stated that the term vanishes in all inertial frames. It was claimed that inertial frames or inertial motion or whatever you want to call the "absence of proper acceleration" was nowhere to be found in that equation. I do not understand that claim.

No, it isn't, not if by "appearances" you mean the actual direct observations the traveler makes. The traveler cannot directly observe what time it is on the home twin's clock "right now". He can only directly observe light signals arriving at his worldline that came from the home twin. All of those direct observations can be described without using an inertial frame, or indeed without using any frame at all. The only reason the traveler would need to define a frame is if he insists on asking questions that have no uniquely determined answers, like "what time is it on the home twin's clock right now?" But there is no need to ask any such question in order to describe or predict direct observations.

I can't help it if you don't like such questions. You seem to feel that such questions should be prohibited because the answers are coordinate dependent or because they only work in flat spacetime?

I agree with you that observations made at a single point in spacetime are very limited. In that case you can only speak of about signals or objects that intersect your worldline. You can use "radar" as means of measurement. It is a constrained method of observation but by no means the only way to make observations. So, must we dispense with a concept like an inertial frame in which time is defined at more than one position? When I look at my watch, would you argue that it says nothing about what time it is a block away? I think that's taking it a bit too far.

I understand that questions like "what time is it on the home twin's clock right now?" are intuitively appealing; but that doesn't mean they have uniquely determined answers. Part of understanding relativity is understanding that some of your Newtonian intuitions about what concepts are meaningful or what questions have uniquely defined answers need to be discarded.

If you really think I don't understand that, then I suspect you only skimmed the OP and just jumped to the conclusion that I was deluded and that I claimed uniqueness. All measurements are relative to something, right? There are no absolutes. There are invariants, but they too have conditions of measurement that are relative to something.

No, but that in itself is not sufficient to combine MCIFs. MCIFs do not just cover points on the curve; they also cover points off the curve.

So do tangents; do you have something against them? They answer questions like "which way am I going now"? "What points in space are normal to my current direction and how far away are they?" "If I had some clocks available, synchronized with my own in my momentary state of motion, what would they read someplace else?"

I suspect you don't like these questions because you believe that they mislead novices and you want to avoid that (which is of course a noble cause). Well, perhaps that is true, but the questions are not without meaning when properly stated. If you claim meaninglessness (for example with the reply "mu") to such carefully worded statements then you may also create confusion in the minds of novices who think they now require a zen master to convey to them some ineffable truth.

Pervect's post in response to this gave a good explanation of the limitations of MCIFs and Fermi Normal Coordinates.

I will get to it. Responding to all these objections is time consuming. However, I also must admit that I was a bit put off by pervect's repeated references to "high school" and "Newtonian physics". Along the way in these discussion, I am learning from experts like yourself, but it is not helpful to ask me to respond to specious complaints about carefully worded statements that apparently have not been carefully read.

The goal itself is not well-defined, because you are assuming that "the perspective of an observer" has a unique definition for events not on the observer's worldline. It doesn't. The intuitive feeling that it does is one of those intuitions that you need to discard.

"Simultaneous" does not have a unique physical meaning. The intuitive feeling that it does is another of those intuitions that you need to discard.

See above. This is not news to me. I did not speak of intuitions in my OP. I spoke of conditions and results in the context of those conditions. If you and other posters continue to deny that then it's very hard to communicate usefully.

In fairness to you, you wish to exclude my statements because you believe they may be misinterpreted (oddly even by myself). It may well be that other skimmers might draw incorrect conclusions, but there is nothing unusual about that problem.

You may be misunderstanding Einstein's reason for introducing his simultaneity convention (which basically amounts to the "radar" convention, as you have correctly noted). He was not doing it to propose a unique physical meaning for simultaneity, not even a relative one. He was doing it to show that the obvious pre-relativistic meaning of simultaneity, as applied to light signals received by an observer from two events equidistant from him, when combined with the observed fact that light propagates at the same speed in all inertial frames, requires relativity of simultaneity--i.e., it requires that the Newtonian concept of an absolute physical meaning for simultaneity must be abandoned. That doesn't mean that "simultaneity is still real, but it's relative". It means simultaneity is not "real"; it's just a convention.

Perhaps, but I think you are conjecturing what Einstein intended in the context of mathematical approaches developed long after 1906 (that is an anachronism). My own interpretation is he established a non-local definition of time (as well as distance, which he does not explicitly define) for an inertial frame. You believe he did so only to appease those stuck on Newtonian notions? Did he not use these very constructs (inertial frames with their coordinates of time and space) to derive the Lorentz transformations from the PoR and the constancy of the speed of light?

Peter, before you lecture me about these fact of non-uniqueness could you do me the courtesy of actually reading my OP? I think you will find that every time I mentioned "home time" it was qualified with the frame that defines it.

They are the rules that are necessary to have a mathematically valid coordinate chart. An inertial frame, as it is defined in SR, is just a particular kind of valid coordinate chart, which satisfies some additional constraints.

Perhaps you mean "mathematically tractable"? The fact that different inertial frames assign different coordinates to the same events is not "un-mathematical".

SR does not "define" simultaneity in a particular way, because SR does not require you to use inertial frames. All you are really saying is that, if a convention of simultaneity is different from the one for an inertial frame, then a frame using that other convention can't be an inertial frame. You're right; it can't. That's just a fact about non-inertial frames.

But you appear to insist that we cannot find instantaneous inertial frames in arbitrary motion and work with them mathematically. I have no idea why you want to insist on that. In SR, how do you determine that rate of a clock moving in a circle relative to a stationary clock at the center? How do you add up those differences around the circle?

You are confusing "SR" with "SR as applied using inertial frames". The two are not the same. The fact that SR was initially introduced using inertial frames, and that it is still widely taught in introductory courses using inertial frames, does not mean that inertial frames are required for SR. It just doesn't.

Here again, I feel that you attempt to hide the centrality of inertial frames to SR. You claim that this concept is somehow transcended in a more lofty mathematical formulation, but you haven't convinced me, yet. You need only look at the foundation of the mathematics to see that the concept is still there (see above).

Your resistance to acknowledging carefully worded conclusions doesn't help my understanding and it may confuse others as well.

I am indeed appreciative of the time and work it requires of you and others to respond. But please don't put statements in my mouth that I did not make and then declare me wrong or misguided.

 

[CKH]

I agree completely on this.

Well, then you understand why it is circular. GR does not derive by some special magic what happens in a uniform gravitational field. It comes directly from SR and requires no knowledge about gravity itself aside from the belief in the equivalence principle.

 

[Dale]

Well, then you understand why it is circular.

Not at all. It is completely unnecessary to bring in GR and I agree that it "adds nothing" to the SR analysis to talk about, but it is not circular.

GR does not derive by some special magic what happens in a uniform gravitational field. It comes directly from SR

I think that you have the logic here exactly backwards. From a logical standpoint it is actually impossible to derive GR from SR. GR is not a derivation of SR, it is a generalization of SR. Do you understand the difference? You can derive a special-case from a general-case (GR logically implies SR), but you simply cannot logically derive a general-case from a special-case (SR does not logically imply GR).

It is clear that historically SR came first. However, it seems that you are being distracted by the historical order of the development and mistaking that for a logical dependency. A good special-case (e.g. SR) and some reasonable intuition (e.g. the equivalence principle) can allow a good guess at a useful generalization (e.g. GR), and this is historically what happened. But intuition is not derivation and historical order is not logical inference.

GR is not logically derivable from SR, so it is not circular logic for Einstein to analyze a scenario in flat spacetime using GR and his gravitational fields. It is just unnecessary (it "adds nothing" as you said and to which I agreed).

 

[PeterDonis]

How does this statement acquire a meaning in the physical world?

Spacetime curvature is the same thing as tidal gravity. If spacetime is flat, with zero curvature, that means no tidal gravity is present anywhere. Of course the real universe we live in does not have that property; SR, as a theory, does not describe our actual world in terms of global properties.

Physics in SR is about the behavior of objects (while ignoring their gravitational properties).

And, as I just noted, SR does not actually apply to the actual universe, because there is tidal gravity present in our actual universe, so the behavior of objects does not exactly match the predictions of SR. (Note that we can detect the presence of tidal gravity by making measurements on very small objects whose gravity is negligible; attributing the presence of tidal gravity to the presence of gravitating masses is separate, conceptually speaking, from detecting the presence of tidal gravity itself.)

Under the conditions that no forces act on objects, objects at mutual rest remain at mutual rest.

This is ok except for the term "forces"; a better way of stating it would be "under the conditions that objects have zero proper acceleration". With that stipulation, yes, this is equivalent to stating that there is zero tidal gravity, and therefore zero spacetime curvature.

We can attempt to hide the inertial frames in some abstract mathematical statement. But when we dig into the physics behind the abstraction, we find that the inertial frames are still there.

But "inertial frame" in the quote you gave has a subtly different meaning than the one we've been using up to now. Note that the quote specifies measurements made at a single spacetime event. So the two different "inertial frames" being used to describe measurements at that spacetime event don't have to cover all of spacetime; they only have to cover an infinitesimal region of spacetime around the chosen event (enough to define derivatives of quantities at that event).

In other words, the "inertial frame" here is really what is called in GR a "local inertial frame"--by definition it only covers a small patch of spacetime. So the claim that the laws of physics look the same in all inertial frames is a much weaker statement here than it would be if "inertial frame" had its usual meaning (which we have been using up to now) of a global inertial frame, covering all of spacetime. And a better way of saying what I have been saying is that SR does not require global inertial frames. It does require local inertial frames, or something equivalent, in order to make sense of the concept of Lorentz invariance. But, as I just noted, that is a much weaker requirement.

No one wanted to admit that this "fictitious force" term is constructed relative to inertial motion

That's because it isn't. It arises out of perfectly general terms in tensor equations that are written without making any assumptions about the state of motion. The fact that those terms happen to vanish for a coordinate chart constructed in a particular way in a particular spacetime--the kind of chart that defines a global inertial frame in flat spacetime--does not mean those terms require the concept of inertial motion for their definition.

I suggest taking some time to get familiar with differential geometry. I learned it from the section in MTW on differential geometry, which may not be the best reference; Carroll's online lecture notes also cover it.

You seem to feel that such questions should be prohibited because the answers are coordinate dependent or because they only work in flat spacetime?

I have never said such questions should be prohibited. I have only said you should not expect the answers to mean something they don't mean. If you're okay with that, ask away. But when you talk as though there is some preferred definition of simultaneity, for example, based on inertial frames, you are attributing a meaning to the answers to those questions that is simply not there. There is no preferred definition of simultaneity; there just isn't. If asking those questions and getting answers to them makes you think there is, then you need to either stop asking the questions, or stop attributing a meaning to the answers that they don't have.

must we dispense with a concept like an inertial frame in which time is defined at more than one position?

I have never said we must dispense with the concept of an inertial frame. I have only said you should not attribute a meaning to it that it doesn't have. See above.

So do tangents

No, they don't. Tangent vectors only "cover" a single point. This is another area where you need to learn some differential geometry; learning it will show you why the concept of "vector" you may be used to, where a vector is an arrow going from one point in space (or spacetime) to another, doesn't work, and needs to be replaced with the concept of "tangent vector", which is only "attached" to a single point in spacetime. (More precisely, at each point in spacetime, there is something called the "tangent space", and tangent vectors--and all other vectors, tensors, and geometric objects used in the math of differential geometry--are defined in the tangent space.)

They answer questions like "which way am I going now"? "What points in space are normal to my current direction and how far away are they?" "If I had some clocks available, synchronized with my own in my momentary state of motion, what would they read someplace else?"

The first question is answered by the tangent vector to your worldline, yes.

The second is not answered by a tangent vector by itself. There are a couple of different ways to answer it using differential geometry, but a tangent vector alone is not enough.

The third is also not answered by a tangent vector by itself. You need a synchronization convention. The Einstein convention is one possible one, but not the only one.

Once again, I strongly recommend taking some time to learn differential geometry.

I did not speak of intuitions in my OP. I spoke of conditions and results in the context of those conditions.

But you keep on talking as if you think those conditions are somehow privileged or preferred. You keep on talking as if the Einstein synchronization convention, and the other machinery that defines an inertial frame, are somehow privileged or preferred. They aren't.

before you lecture me about these fact of non-uniqueness could you do me the courtesy of actually reading my OP? I think you will find that every time I mentioned "home time" it was qualified with the frame that defines it.

That's not the kind of non-uniqueness I'm talking about. I know that you know that different inertial frames have different simultaneity conventions, based on the Einstein simultaneity definition. What I'm not sure you understand is that there is nothing that requires an observer, even if he is moving inertially, to use the Einstein simultaneity definition. Using that definition is a choice--a convention. There is nothing in physics that requires it. Two observers both moving inertially, and both at rest relative to each other, and both in flat spacetime, could perfectly well choose different simultaneity definitions; and as long as they both constructed valid coordinate charts based on their respective definitions, they could both make correct physical predictions.

Perhaps you mean "mathematically tractable"?

No, I meant exactly what I said.

The fact that different inertial frames assign different coordinates to the same events is not "un-mathematical".

Different inertial frames are different coordinate charts. The rules I gave apply to a single coordinate chart. Different coordinate charts give different descriptions of spacetime and what happens in it; the rules I gave are what is required for a single description to be valid.

you appear to insist that we cannot find instantaneous inertial frames in arbitrary motion and work with them mathematically.

I have never said that. I have only said that you can't combine multiple "instantaneous inertial frames" along a non-inertial worldline into a single consistent "frame".

In SR, how do you determine that rate of a clock moving in a circle relative to a stationary clock at the center?

First you need to decide what "rate" means. If it means "rate in the inertial frame in which the clock at the center is at rest", then it's easy. If it means something else, you need to decide what. For example, the two clocks could exchange light signals and use the round-trip travel times to determine their rates.

How do you add up those differences around the circle?

You integrate the rate of the clock moving in a circle (determined based on how you define "rate", as above) along its worldline.

You need only look at the foundation of the mathematics to see that the concept is still there (see above).

See my response above.

please don't put statements in my mouth that I did not make and then declare me wrong or misguided.

I'm not sure what statements you think I've misattributed to you. Are you saying that inertial frames are central to SR (which you just said in your post), but are not "required" for SR (which is how I worded your claim in the quote you gave)? If that's your position, it seems odd.

 

[CKH]

However, it seems that you are being distracted by the historical order of the development and mistaking that for a logical dependency. A good special-case (e.g. SR) and some reasonable intuition (e.g. the equivalence principle) can allow a good guess at a useful generalization (e.g. GR), and this is historically what happened. But intuition is not derivation and historical order is not logical inference.

If indeed GR is an independent theory (from SR), then you should not require the equivalence principle to arrive at GR. Is it true the the equivalence principle is unnecessary for GR? I'm asking because I haven't actually followed any derivation of GR (the math is still difficult for me). So, I was under the perhaps false impression that the equivalence principle is the central justification (the bridge that allows us beginning with SR to discover how gravity behaves in general).

It's possible however that the equivalence principle is actually just "window dressing" while GR has an origin in which that principle is entirely absent and unnecessary, but some other principles are used instead.

What do you think? Is GR justified (derived) is some completely independent way (how?) but after the fact, we notice that the equivalence principle happens to be consistent with this independently derived theory?

 

[PAllen]

If indeed GR is an independent theory (from SR), then you should not require the equivalence principle to arrive at GR. Is it true the the equivalence principle is unnecessary for GR? I'm asking because I haven't actually followed any derivation of GR (the math is still difficult for me). So, I was under the perhaps false impression that the equivalence principle is the central justification (the bridge that allows us beginning with SR to discover how gravity behaves in general).

It's possible however that the equivalence principle is actually just "window dressing" while GR has an origin in which that principle is entirely absent and unnecessary, but some other principles are used instead.

What do you think? Is GR justified (derived) is some completely independent way (how?) but after the fact, we notice that the equivalence principle happens to be consistent with this independently derived theory?

One arrives at a theory via heuristics, guesses, etc. (hopefully good ones!). Once you have a theory, you derive consequences, and the initial heuristics become irrelevant - unless they are also formally derived consequences.

The mathematical statement of GR nowhere has the principle of equivalence. However, it is a derivable approximate, local, consequence. Given the caveats, some highly esteemed GR experts (e.g. J. L. Synge), argued it should be abandoned, because as formal, mathematical statement it is simply false everywhere in GR (due the the approximate, local nature). Most physicists find the principle of equivalence remains a highly useful guide to intuition in GR, but one that always must be used with care, and no derivation based on it can be relied on without some additional formal validation.

 

[PeterDonis]

Is it true the the equivalence principle is unnecessary for GR?

Yes. There are actually multiple ways of arriving at GR from a simple set of starting assumptions; MTW lists six of them. The first two, which are the methods that Einstein and Hilbert, respectively, used in 1915 to obtain the Einstein Field Equation, are:

(1) Use automatic conservation of the "source" of gravity as the key requirement. This involves finding a tensor to describe gravity that is constructed from the metric and its derivatives and whose covariant divergence is identically zero; this tensor turns out to be the Einstein tensor, $G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$. Then the field equation for gravity is obtained by simply setting this tensor equal to the tensor describing the source (the stress-energy tensor), times a proportionality constant whose value can be fixed by looking at the weak-field, slow-motion Newtonian limit.

(2) Use the principle of least action. This involves constructing a Lagrangian for gravity, and then using standard variational methods to find the field equation corresponding to that Lagrangian, which turns out to be the Einstein Field Equation.

Notice that the EP does not appear anywhere in the above. Notice also that neither of the above methods look anything like "start with SR and add gravity". Neither do three of the other four methods MTW lists.

The fifth method MTW lists is the only one that involves anything like the EP or starting with SR and adding gravity; it is to start with the field theory of a massless, spin-2 field in flat spacetime, and notice that such a theory makes correct predictions for gravity in the weak-field, slow-motion Newtonian limit. However, this theory is not self-consistent, and there turns out to be no way to make it self-consistent without making the "background" flat spacetime that we started with completely unobservable--all objects move as though they were in the curved spacetime obtained from the full metric tensor $g_{\mu \nu}$ that includes all the contributions from the spin-2 field (which is nonlinear because it interacts with itself, so there are an infinite series of contributions, and it took some very smart physicists about a decade to figure out how to extract a self-contained answer from all this). The fact that all objects move in this curved spacetime can be viewed as a manifestation of the EP--all objects see the same (curved) spacetime geometry, regardless of their mass or composition.

The problem with viewing the above as showing that the EP is "required" for GR, or that GR can be derived by taking SR and adding gravity, is twofold. First, as noted, there are multiple ways of deriving the Einstein Field Equation, and the others do not require the EP or taking SR and adding gravity. Second, if you interpret the above spin-2 field method as "SR plus gravity", then you are implicitly limiting yourself to curved spacetimes that have the same topology as flat Minkowski spacetime. Unfortunately, that excludes most of the key solutions to the EFE that are extensively used in GR, including all the black hole solutions and the FRW solutions used in cosmology. So the best that the spin-2 field approach can give us is a way of understanding, at least heuristically, how gravity might arise locally from some underlying spin-2 field; it can't give us the full range of predictive power that GR can.

 

[CKH]

Spacetime curvature is the same thing as tidal gravity. If spacetime is flat, with zero curvature, that means no tidal gravity is present anywhere. Of course the real universe we live in does not have that property; SR, as a theory, does not describe our actual world in terms of global properties.

Which is of course the same as "objects at rest remain at rest" etc. in ordinary language (of physics). It is nothing more. The statement "spacetime is flat" is couched in a more sophisticated geometric language which has been found useful as a mathematical tool in GR. Manifolds, curvature, tensors etc. are concepts are not needed in SR. No one needs to speak of gravity in SR.

Agreed, SR does not describe "our actual world in terms of global properties". We think GR does, but not with absolute certainty. However, SR is a pretty good approximation for some purposes.

I conjecture that your emphasis on GR in a discussion about SR arises from your concern that novices may misapply it to GR. That's OK, but why keep repeating it and why apparently contradict statements in the OP on that basis? Why not just accept them for what they are by acknowledging correctness under the conditions repeatedly specified? Or , show these conclusions to be actually wrong under the conditions specified.

In the OP, we have something more than an abstract mathematical argument. We have one in which physical clocks have been placed and synchronized in a well-defined manner. We can then make conclusions about the coincident (same event) readings on both physical clocks. If you are saying that those conclusion are wrong, then it is almost certainly myself who is wrong because you are an expert. In that case I need your help to fix my error.

And, as I just noted, SR does not actually apply to the actual universe, because there is tidal gravity present in our actual universe, so the behavior of objects does not exactly match the predictions of SR.

Right, objects at rest do not remain at rest. You don't even need to know what gravity is or a tide is.

(Note that we can detect the presence of tidal gravity by making measurements on very small objects whose gravity is negligible; attributing the presence of tidal gravity to the presence of gravitating masses is separate, conceptually speaking, from detecting the presence of tidal gravity itself.)

Well yes, but if we are interested in a physical cause then we need to explain this through the presence of gravitating masses.

This is ok except for the term "forces"; a better way of stating it would be "under the conditions that objects have zero proper acceleration". With that stipulation, yes, this is equivalent to stating that there is zero tidal gravity, and therefore zero spacetime curvature.

I'm OK with that, but I hope you also see that "objects at rest stay at rest..." is equivalent to the geometric expression "zero spacetime curvature". The former is a physical statement, the latter is an abstract mathematical statement that needs some "translation" to become physically meaningful.

But "inertial frame" in the quote you gave has a subtly different meaning than the one we've been using up to now. Note that the quote specifies measurements made at a single spacetime event. So the two different "inertial frames" being used to describe measurements at that spacetime event don't have to cover all of spacetime; they only have to cover an infinitesimal region of spacetime around the chosen event (enough to define derivatives of quantities at that event).

In other words, the "inertial frame" here is really what is called in GR a "local inertial frame"--by definition it only covers a small patch of spacetime. So the claim that the laws of physics look the same in all inertial frames is a much weaker statement here than it would be if "inertial frame" had its usual meaning (which we have been using up to now) of a global inertial frame, covering all of spacetime. And a better way of saying what I have been saying is that SR does not require global inertial frames. It does require local inertial frames, or something equivalent, in order to make sense of the concept of Lorentz invariance. But, as I just noted, that is a much weaker requirement.

Agreed. In the terminology used in physics, an "inertial frame" has global extent in time and space. The concept (if there is one so-called) of "inertial motion" is not global. If "local inertial frame" is the correct terminology for "a local cartesian coordinate system in which proper acceleration is zero", then that term is usually preferable for the sake of generality and applicability in GR as well as SR. However, in SR we can always speak of the global concept "inertial frame" which is otherwise a term of very limited applicability.

That's because it isn't. It arises out of perfectly general terms in tensor equations that are written without making any assumptions about the state of motion. The fact that those terms happen to vanish for a coordinate chart constructed in a particular way in a particular spacetime--the kind of chart that defines a global inertial frame in flat spacetime--does not mean those terms require the concept of inertial motion for their definition.

You still disagree that the "fictitious force" term has anything to do with "inertial motion"? How do you calculate or measure the coefficients in that term without reference to some local cartesian coordinates with "0 proper acceleration" (i.e. local inertial motion)? If you allow the coefficients to be computed or measured in a local coordinate system which is non-inertial or non-cartesian, the law will be wrong.

So if you want to insist, that no concept of inertial motion exists in that equation, then tell me how you physically measure the coefficients with no such reference.

I suggest taking some time to get familiar with differential geometry. I learned it from the section in MTW on differential geometry, which may not be the best reference; Carroll's online lecture notes also cover it.

I will, just that it takes time and perhaps I waste far too much time arguing against misinterpretations of my own statements.

I have never said such questions should be prohibited. I have only said you should not expect the answers to mean something they don't mean. If you're okay with that, ask away. But when you talk as though there is some preferred definition of simultaneity, for example, based on inertial frames, you are attributing a meaning to the answers to those questions that is simply not there. There is no preferred definition of simultaneity; there just isn't. If asking those questions and getting answers to them makes you think there is, then you need to either stop asking the questions, or stop attributing a meaning to the answers that they don't have.

This is not so mysterious as you make it sound. You make it sound as if "simultaneity" (more generally an interval of time) is some especially slippery undefinable measurement with the implication that the concept is best abandoned.

So let me make an analogy. Suppose there are two trees in the distance and I measure the angle between them. Would you so vigorously complain that that the angle is meaningless because if you measure it from another point it is different? There is no prefered definition of the angle between the trees, but that angle is perfectly meaningful given it's conditions of measurement. We could not survey land without the belief that these angles have meaning. Of course, in a (small) survey we take for granted the "flatness of space".

In the OP, the conditions have not been left to implication. The condition of SR is explicitly stated and the use of inertial frames to make measurements is explicitly stated.

I have never said we must dispense with the concept of an inertial frame. I have only said you should not attribute a meaning to it that it doesn't have. See above.

Do you mean by this that "inertial frame" implies "global inertial frame" which is different from "local inertial frame"? What meaning have I attributed to "inertial frame" that it doesn't have, globalness? If so, perhaps I misused the word. and should have said "local inertial frame". Does that completely throw you off the rails of comprehension?

No, they don't. Tangent vectors only "cover" a single point. This is another area where you need to learn some differential geometry; learning it will show you why the concept of "vector" you may be used to, where a vector is an arrow going from one point in space (or spacetime) to another, doesn't work, and needs to be replaced with the concept of "tangent vector", which is only "attached" to a single point in spacetime. (More precisely, at each point in spacetime, there is something called the "tangent space", and tangent vectors--and all other vectors, tensors, and geometric objects used in the math of differential geometry--are defined in the tangent space.)

OK. No real problem here.

We know that a tangent can only intersect a curved line at a single point, fine. But as soon as you introduce the notion of "tangent space" you are not sticking with your rule that "tangent vectors only 'cover' a single point". You are using them as a basis in this tangent space which extends from that point to points off of the curve.

The first question is answered by the tangent vector to your worldline, yes.

The second is not answered by a tangent vector by itself. There are a couple of different ways to answer it using differential geometry, but a tangent vector alone is not enough.

The third is also not answered by a tangent vector by itself. You need a synchronization convention. The Einstein convention is one possible one, but not the only one.

Once again, I strongly recommend taking some time to learn differential geometry.

You mean "not enough" because we need orientation, rotation etc.? What an MCIF is, is more than a vector. Is that what you mean?

Consider the consequences is using a different convention than Einstein's in a local inertial frame. Let's imagine two mile markers along the x-axis within the boundaries within this local inertial frame. We will have one clock at each marker, but they will be de-synchronized wrt to Einstein's convention by 1 hour to satisfy some other "convention of simultaneity". We perform two experiments in which a car (starting from rest in the frame) undergoes a certain proper acceleration for a certain amount of proper time and then stops accelerating (all within the local inertial frame). After accelerating, in each experiment the car cruises past the two milestones (but in the opposite direction).

The measurement system (the milestones/clocks) have no proper acceleration. We are forced to conclude (prior to experiment) by symmetry that the transits times are equal (whether in proper time or coordinate time in that local frame). But using these clocks they are not. So as a "convention for simultaneity for these clocks" one might ask "what good is that one"? It's "meaningless nonsense" or if you prefer "arbitrary and inconsistent with the physical symmetry". In what sense can you claim that they are synchronized?

But you keep on talking as if you think those conditions are somehow privileged or preferred. You keep on talking as if the Einstein synchronization convention, and the other machinery that defines an inertial frame, are somehow privileged or preferred. They aren't.

I just explained why I think the Einstein synchronization convention is "preferred" in a local inertial frame, so you can address the weakness in that explanation. Actually I'd go so far as to say it is the required convention for consistency of physical measurement. Without consistency of measurement, what physics can you apply?

I also explained that the angle between two trees is not a privileged measurement. What is it that you think I'm missing? You project onto my mind by implication that I "think those conditions are somehow privileged". You must be talking to someone else for I certainly never claimed any such thing.

What I'm not sure you understand is that there is nothing that requires an observer, even if he is moving inertially, to use the Einstein simultaneity definition. Using that definition is a choice--a convention.

Then justify your other convention in the above example of a moving car.

Nobody can force you to do anything (in a free world), but if your results are nonsense it places your choice in doubt.

There is nothing in physics that requires it.

Not even consistency? Not symmetry? Not anything? That makes "simultaneity" a word devoid of all meaning, does it not?

Take my tree analogy. If you say there is nothing forcing me to measure an angle between trees in a certain way from a certain point in space, then there is no conclusion that can be reached about angles and the concept of angle is useless physically. It cannot provide any consistent result.

Two observers both moving inertially, and both at rest relative to each other, and both in flat spacetime, could perfectly well choose different simultaneity definitions; and as long as they both constructed valid coordinate charts based on their respective definitions, they could both make correct physical predictions.

Would you call it a "correct physical prediction" that the car in the above example moves at different speeds depending upon the direction it is going? You could, I suppose but I see no purpose in so degrading the meaning of "physical prediction" as to make the prediction arbitrary and claim that all such predictions are equally correct. When you do that you say bye bye to physics, for you can no longer measure anything in a consistent manner.

Different inertial frames are different coordinate charts. The rules I gave apply to a single coordinate chart. Different coordinate charts give different descriptions of spacetime and what happens in it; the rules I gave are what is required for a single description to be valid.

The rules you impose are not required for nature to be valid, they are imposed by your difficulties in mathematics with multi-value relations.

Sure you can define any coordinate chart you want, but there is a convenient family of charts defined by nature herself, without need of reference to some other chart, a chart in which all proper acceleration vanishes. Any chart is related to this chart by it's non-zero proper accelerations. Proper acceleration is an absolute because it is measured in a specified manner that does not allow for different conditions of measurement. Zero proper acceleration is particularly easy to measure. We need only free an object at a point and see if it moves from that point (to eliminate rotations we actually need more than one point). We care not how by much it moves or in what direction, nor how fast.

In the "coordinate independent" equation of inertial motion (you know, f=ma in modern form), there is an implicit local chart which is any of the family of charts with 0 proper acceleration and cartesian coordinates. The coefficients in the "fictitious force" term are calculated with reference to anyone of those special charts. Or to put it another way those coefficients are calculated wrt to 0 proper acceleration and cartesian coordinates. All measurements are relative to something. This is precisely why the coefficients are 0 in any local inertial frame. It's no coincidence and that fact the they are 0 is not meaningless. That fact allows me to claim that the equation is defined wrt to 0 proper acceleration (i.e. any local inertial frame). Why do you deny that?

I have never said that. I have only said that you can't combine multiple "instantaneous inertial frames" along a non-inertial worldline into a single consistent "frame".

By your definition of "consistent", no you cannot. You don't "like" the non-uniqueness of coordinates, but it nevertheless exists. There is no mathematically flaw that I see, simply a mathematical inconvenience.

First you need to decide what "rate" means. If it means "rate in the inertial frame in which the clock at the center is at rest", then it's easy. If it means something else, you need to decide what. For example, the two clocks could exchange light signals and use the round-trip travel times to determine their rates.

If we get desperate we can do that experimentally. How would you calculate the result without doing that experiment?

You integrate the rate of the clock moving in a circle (determined based on how you define "rate", as above) along its worldline.

My point is this:
1) How do you compute the rate of the clock moving in a circle (relative to the "stationary" clock at the center)? (I suspect you apply a MCIF and the Lorentz transformation)
2) How can you integrate those rates? (I suspect you integrate over these tangent MCIFs (or, in case you complain, "tangent local MCIFs"), and yet you claim that is not valid because they cannot be combined)

These are just suspicions, not claims of course.

I'm not sure what statements you think I've misattributed to you.

Here's one:

But you keep on talking as if you think those conditions are somehow privileged or preferred. ​

You keep implying that I believe there is something unique about simultaneity when I never said any such thing. And, if you read my OP you would see that I qualified the meaning of clock readings every time I referred to them, specifically to avoid such a misinterpretation of my words. Put yourself in my shoes and see if you find it annoying.

What you are attributing to me in that statement exist in your head. I don't know why, but I'll take a guess: 1) your failure to carefully read my words, 2) you projection upon me of the confusion of others with whom you have discussed this in the past.

I'd appreciate it if you would not attribute the mistakes of others in other threads to me. It's not helpful to respond with complaints that result from you own failure to read carefully (if that is actually the case). And it's especially annoying to attribute thoughts to me that are manifestly not mine.

Are you saying that inertial frames are central to SR (which you just said in your post), but are not "required" for SR (which is how I worded your claim in the quote you gave)? If that's your position, it seems odd.

It seems (from my perspective of course) that it is your business to find something odd about about things I say, almost as a matter policy.

From my perspective, if I make a mistake in terminology, it is rather like talking to a computer that says "that does not compute". If and when I make a mistake in terminology, perhaps you could use your intellect to see through the mistake and correct the terminology without denying all meaning to my statement with "I find that odd". I understand that can be difficult. Nevertheless, I am truly puzzled by the difficulties you express (in spite of your apparent intellectual abilities) in understanding me. Why do you keep misinterpreting my OP and ascribing misperceptions to me?

If I did contradict myself, could you show me where so I can try to straighten out what I may have misstated?

 

[CKH]

Yes. There are actually multiple ways of arriving at GR from a simple set of starting assumptions; MTW lists six of them. The first two, which are the methods that Einstein and Hilbert, respectively, used in 1915 to obtain the Einstein Field Equation, are:

I can read your words but they are mostly over my head. So I will take your word for it that EP is unnecessary without being able to understand what the other principles of derivation actually are. My problem of course is that when you raise the discussion to levels of abstraction not familiar to myself, I cannot tell what they conceal. ;)

But seriously, I do stand corrected. It was my belief that EP is what opened the door to understanding gravity. Since there are yet other ways, that is interesting. It seems as if Einstein built GR on a foundation of EP. But I doubt we can claim that it has since been discovered that no foundation is required. Rather, I would conjecture this, these alternative foundations contain assumptions about gravity they can be shown to be equivalent to EP.

Concerning the historical view of the foundation of Einstein's GR, do you believe that Einstein did not develop his theory around EP and therefore his use of pseudo-gravitational fields was indeed non-circular in the context of his own derivation?

As an aside, I read (actually sort of read because the math was over my head) a paper not too long ago, in which the author presented a different theory of relativity (actually slightly different in physical prediction) and remarked that in his theory, the EP is a consequence, not an assumption. Given what you have said, I suppose that is also true for GR when derived in some other way. In each of these derivations there of course must be some assumptions about gravity, for we cannot go on to describe gravity without assumptions (however well-founded in experiment).

I have a question about EP that is bothering me. EEP speaks of "uniform acceleration". Objects with rigidity (total resistance to mechanical stress) such as a rocket ship cannot accelerate uniformly. So, in a cartoon version of EEP we have an observer in a spaceship who cannot distinguish his accelerated motion from a uniform gravitational field. What coordinate system is it in which there is no distinction? The coordinates of the rigid rocket or exactly uniformly accelerated coordinates?

 

[PeterDonis]

If you are saying that those conclusion are wrong

Remember that I came into this thread because you asked about simultaneity conventions other than the Einstein one. I responded by pointing out that the very scenario you described in the OP, where the traveling twin uses clocks synchronized with the home twin's clock using the Einstein simultaneity definition, to see how much time his clock has "lost" compared to the home twin's clock, amounted to the traveling twin adopting a different simultaneity convention from the Einstein one for his rest frame. (It is still the Einstein convention for a different inertial frame, the home twin's, in which the traveling twin is not at rest--more on that below.)

None of what I said contradicted what you said in your OP. I was simply trying to make an additional point: that simultaneity is a convention. You have repeatedly responded by (apparently, to me) maintaining that the Einstein definition of simultaneity is somehow privileged--that there is something in physics that requires, at the very least, that someone who is moving inertially must adopt the Einstein simultaneity convention for the inertial frame in which he is at rest (or at any rate that he is somehow missing out on some physics if he does not do so). I have pointed out two ways in which that view is mistaken: first, that there is nothing stopping an inertial observer from adopting the Einstein simultaneity convention for a different inertial frame (such as the traveling twin adopting the home twin's convention, as described in your OP); and second, that there is nothing even requiring any observer to use the Einstein simultaneity convention in the first place, whether he is moving inertially or not--he could just as well use some other definition of simultaneity, as long as it meets the basic requirements I gave.

Most of the time, nobody does this for inertial observers; they use standard inertial frames in which those observers (at least one of them) are at rest. And most of the time, people using non-inertial frames use one of a few well-known simultaneity conventions that work, at least over a limited range of space (and possibly time), for non-inertial frames (such as the radar convention). But there's nothing in physics that requires any of this. You can make up any wacky simultaneity convention you like, and as long as it meets the basic requirements I gave, you can describe all of the same physics that you can describe using a standard inertial frame with the standard Einstein simultaneity convention.

If you agree with what I said in the above paragraphs, then my purpose in entering this thread is accomplished (though I'm certainly willing to answer other questions, and I'll respond to your question about the EP in a separate post). If you don't, then I think that's where our discussion needs to focus. That will make it clear that I'm not disputing what you said in the OP, and I'm not disputing any of what you say about how inertial frames work, given that you've already chosen to use an inertial frame and its associated simultaneity convention.

I'll respond to other things in your post separately, because I want to keep the issue I just described separate from the rest of the discussion.

 

[PeterDonis]

I hope you also see that "objects at rest stay at rest..." is equivalent to the geometric expression "zero spacetime curvature". The former is a physical statement, the latter is an abstract mathematical statement that needs some "translation" to become physically meaningful.

Yes (with the added clarification that "at rest" means "at rest relative to each other" and that "objects" means "objects that are close together and are moving inertially").

In the terminology used in physics, an "inertial frame" has global extent in time and space.

More precisely, in the terminology used in SR and in Newtonian physics. In GR, there is no such thing as a global inertial frame (except in the idealized case of perfectly flat spacetime), so nobody uses the term to refer to it since it doesn't exist.

The concept (if there is one so-called) of "inertial motion" is not global.

No, but it's not quite "local" either. It applies to a worldline, or a segment of a worldline. Mathematically, an inertial worldline, or segment of one, has zero path curvature ("path curvature" is the mathematical representation of proper acceleration in differential geometry). Physically, an inertial worldline is one such that an object whose motion is described by that worldline feels zero acceleration (an accelerometer attached to the object reads zero). The terms "free fall" and "weightless" are often used to describe this state of motion.

If "local inertial frame" is the correct terminology for "a local cartesian coordinate system in which proper acceleration is zero"

Not quite, because your phrasing invites the question "proper acceleration of what"? Also, it leaves out the main reason for using local inertial frames in GR: the presence of spacetime curvature/tidal gravity. Since you prefer physical descriptions, consider: suppose we have two free-falling objects (I prefer that terminology) that are close together and, at some instant, are at rest relative to each other. We pick one object (call it A) and set up an inertial frame with its worldline as the "time axis", and the origin (the point t = 0, x = 0, y = 0, z = 0) as the event on its worldline that occurs at the instant at which the other object (call it B) is at rest relative to it.

Now, if there is no tidal gravity, both objects will remain at rest forever in this inertial frame--it's a perfectly normal global inertial frame such as we use all the time in SR. But suppose tidal gravity is present. Then we have a problem. The two objects do not remain at rest relative to each other. That means that, if object A is at rest in the frame (which it is, because we've constructed the frame that way), object B is not, except at the instant t = 0. This fact immediately forces us to admit that, whatever this thing is that we've constructed, it can't work the same as an ordinary inertial frame in SR.

However, even though the thing we've constructed can't be exactly the same as an SR inertial frame, it can still work approximately the same, for a small region of spacetime around the origin. How small a region depends on two things: how strong tidal gravity is (i.e., how much object B's motion deviates, over a given interval of time, from what it would be if no tidal gravity were present), and how accurate we need our measurements to be (i.e., how big the deviation of object B's motion needs to be before it affects whatever we are trying to calculate). Within that small region of spacetime, we can work with our local inertial frame just as if it were a corresponding small patch of an ordinary SR inertial frame.

 

[CKH]

One arrives at a theory via heuristics, guesses, etc. (hopefully good ones!). Once you have a theory, you derive consequences, and the initial heuristics become irrelevant - unless they are also formally derived consequences.

These "heuristics made formal statements of fact" are what I referred to as "assumptions" (about gravity) in my previous post. Surely you don't mean that you can abandon your assumptions once you formulate a theory? That sounds like building a castle on the ground, then claiming the ground is no longer needed to support the castle.

To make another analogy, in mathematics we prove a theorem from predicates. When we are done, we cannot say that those predicates can now be abandoned and that we have a statement of truth independent from them.

The mathematical statement of GR nowhere has the principle of equivalence. However, it is a derivable approximate, local, consequence. Given the caveats, some highly esteemed GR experts (e.g. J. L. Synge), argued it should be abandoned, because as formal, mathematical statement it is simply false everywhere in GR (due the the approximate, local nature).

I'll take that claim with a grain of salt since calculus itself based on derivation from local approximations. I don't see how we can abandon that method, but if we can, I'm interested.

Of course it depends on exactly what his complaint is about the statement of EP. If it needs to be stated as a local (infinitesimal relationship) to make sense, then it ought to be stated that way rather than complaining that it is always false. It is always false that a segment of a circle is straight, but that has not prevented us from finding pi that way.

Most physicists find the principle of equivalence remains a highly useful guide to intuition in GR, but one that always must be used with care, and no derivation based on it can be relied on without some additional formal validation.

Sure. But I doubt anyone seriously finds the derivation from EP (and other assumptions) wrong when properly applied. EP is certainly not sufficient since we do not have uniform gravitational fields. Other assumptions must be brought into play in addition to EP to get the job done.

BTW, I think we can agree that there is no assumption that can be properly used without care. Such a warning is always applicable and goes without saying. We all learn that sooner or later.

 

[PeterDonis]

as soon as you introduce the notion of "tangent space" you are not sticking with your rule that "tangent vectors only 'cover' a single point". You are using them as a basis in this tangent space which extends from that point to points off of the curve.

No, it doesn't. The tangent space is not a space of points that correspond to points in spacetime. It's an abstract space in which vectors, tensors, and other objects "live" that are attached to a particular single point in spacetime. Please learn differential geometry before making further statements in this area, or at least ask questions instead of making statements (but at this point it's going to be hard to answer further questions in this area without writing a book about differential geometry--it's better for you just to take some time out to study it).

 

[PeterDonis]

There is no prefered definition of the angle between the trees, but that angle is perfectly meaningful given it's conditions of measurement.

Sure, I've never said otherwise. I've never said the Einstein simultaneity convention is meaningless. I've simply said it's not required.

There is one problem with the analogy you're making here: the angle between the trees is a direct observable, while simultaneity is not. I have never said that direct observables are meaningless; indeed, as you will see me say below, they are the actual content of physics.

We are forced to conclude (prior to experiment) by symmetry that the transits times are equal (whether in proper time or coordinate time in that local frame). But using these clocks they are not.

Sure, because you've implicitly defined a "frame" that is not a standard inertial frame, and your symmetry argument only applies if measurements are taken with respect to a standard inertial frame.

But now ask the question: is this "transit time" you speak of a direct observable? No, it isn't. (It can't possibly be, since I can make it take any value I like simply by adjusting my clock synchronization convention.) The direct observable is the proper time elapsed on the car's clock between the two markers. If you compute that using your non-standard frame with its non-standard clock synchronization, you will get the same answer as the answer you get using a standard inertial frame to do the computation. The same will be true for any other direct observable. And the actual content of physics is in the direct observables, not in the coordinate values we use to compute them.

So if you really want to claim that Einstein's simultaneity definition is preferred, you're going to have to show me a direct observable that I can't compute correctly using a non-standard frame, but which you can compute correctly using a standard inertial frame.

Would you call it a "correct physical prediction" that the car in the above example moves at different speeds depending upon the direction it is going?

You are assuming that "distance traveled divided by time", as given in the non-standard frame with its non-standard simultaneity convention, is the correct way to represent the direct physical observable "speed". It isn't. To correctly calculate the speed of the car (for example, to predict the Doppler shift a particular observer would see in light emitted by the car), you need to figure out how that observable is represented in your non-standard frame. You can't just assume it must be represented the same way as it is in a standard inertial frame. Of course you can always calculate "distance traveled divided by time"; that's not the issue. The issue is figuring out which calculation you need to do to correctly compute a particular physical observable. In general it will be a different calculation in different frames.

there is a convenient family of charts defined by nature herself, without need of reference to some other chart, a chart in which all proper acceleration vanishes.

More precisely, in which objects at rest have zero proper acceleration. If no tidal gravity is present, yes, this family of charts has a very nice property: it has a symmetry that matches the symmetry of the actual physics. That makes many calculations much easier.

However, this does not mean that nature "defines" this family of charts. Charts do not exist in nature; they exist in our minds. Nature does not define them; we do. There are no "grid lines" in nature that mark off the coordinate lines of any particular chart. (The closest approach to that is a family of inertial objects, whose worldlines can mark off the "time lines" of a local inertial frame--but we still have the freedom to choose whether or not to interpret those worldlines in that fashion.) If we choose to use a particular chart or family of charts, it isn't because nature "requires" us to; it's because the particular problem we are trying to solve has properties that make a particular chart or family of charts more suitable for solving the problem. Different problems have different properties that can make different charts more suitable; not all problems are best solved by using inertial frames.

 

[PeterDonis]

Zero proper acceleration is particularly easy to measure. We need only free an object at a point and see if it moves from that point (to eliminate rotations we actually need more than one point)

This is circular; how do you pick out a "point" with respect to which you make the measurement? "Points" aren't marked out in spacetime. Only objects are. So a "point" in your measurement has to be some object that you are using as a reference, and how do you know it has zero proper acceleration?

 

[PeterDonis]

You don't "like" the non-uniqueness of coordinates, but it nevertheless exists.

Huh? Where have I said that? I think you are still mistaken about which non-uniqueness I am talking about. Consider (again) this particular paragraph in post #41:

That's not the kind of non-uniqueness I'm talking about. I know that you know that different inertial frames have different simultaneity conventions, based on the Einstein simultaneity definition. What I'm not sure you understand is that there is nothing that requires an observer, even if he is moving inertially, to use the Einstein simultaneity definition. Using that definition is a choice--a convention. There is nothing in physics that requires it. Two observers both moving inertially, and both at rest relative to each other, and both in flat spacetime, could perfectly well choose different simultaneity definitions; and as long as they both constructed valid coordinate charts based on their respective definitions, they could both make correct physical predictions.

This paragraph pretty much sums up what I've been trying to get across about simultaneity. I've made some amplifying remarks in recent posts.

 

[PeterDonis]

1) How do you compute the rate of the clock moving in a circle (relative to the "stationary" clock at the center)? (I suspect you apply a MCIF and the Lorentz transformation)

Once again: first you have to define what "rate" means. What physical observable corresponds to "rate"? How would you measure it?

As for how you would compute it, similar remarks apply to this and the next item: you don't need to use MCIFs or Lorentz transformations. See below.

2) How can you integrate those rates? (I suspect you integrate over these tangent MCIFs (or, in case you complain, "tangent local MCIFs"), and yet you claim that is not valid because they cannot be combined)

You suspect incorrectly. You integrate along the worldline of the clock moving in a circle. All you need to do that is a valid coordinate chart; it doesn't have to be one that is in any way related to any of the MCIFs or that requires combining them.

(Also, the MCIF is not the same as the tangent space, and it's important not to get them confused. The MCIF is an ordinary inertial frame; the tangent space, as I said in a previous post, is an abstract space that requires some differential geometry to understand.)

 

[CKH]

Yes (with the added clarification that "at rest" means "at rest relative to each other" and that "objects" means "objects that are close together and are moving inertially").

Thanks Peter. It's enough for today. I'm drained. I don't think I'll find anything to disagree with in this post. I'll study it some more later and try to identify any missteps I've made.

As far as "the conventionality of simultaneity", I'm just not getting your point. I do not know what those word mean in your mind nor what purpose arbitrary choices can possibly serve. Particularly choices that are inconsistent in a given context (e.g. car experiment above).

Perhaps you will get it across if you continue, but I think you need to take another approach because repetition is not getting the job done.

Also, is distance not also "conventional", not to mention position? All depends on how you want to do it? But when you put something at rest in an inertial frame and you measure its length in that frame and then you claim it's length is "just conventional" I don't know what to think. I see your claim that measuring time under the same circumstances is "just conventional" the same way.

 

[PeterDonis]

do you believe that Einstein did not develop his theory around EP and therefore his use of pseudo-gravitational fields was indeed non-circular in the context of his own derivation?

I think the EP was what originally started Einstein on the road to GR. In 1907, he had what he called "the happiest thought of my life", which was basically the core of the EP: that, as he put it, "if a person falls freely, he will not feel his own weight". This is what made him realize that, locally speaking, we can "emulate" the effects of gravity using acceleration: that, for example, if we are standing in an "elevator" that is accelerating at 1 g in free space, locally we will see all the same physical phenomena as if we were standing in a room at rest on the surface of the Earth--for example, if we release a rock, it will accelerate downward, relative to us, at 1 g.

However, as central as the above reasoning was in helping Einstein to work towards GR, it had no place in the actual derivation of the Einstein Field Equation, which is what the actual "derivation of GR" was. Einstein did not use the EP to derive that equation (as you can see from my previous post where I described how he did it); he only used it as a sanity check, so to speak, to verify that the equation he derived produced the EP as a consequence.

EEP speaks of "uniform acceleration". Objects with rigidity (total resistance to mechanical stress) such as a rocket ship cannot accelerate uniformly.

Yes, and this fact means that the concept of "uniform acceleration", as it was apparently originally conceived by Einstein and others, can't actually be consistently formulated. What most sources now actually mean by "uniform acceleration" is what you are describing here: an object in which all parts remain at rest relative to each other as it accelerates, which then requires that the proper acceleration of the parts varies with "height" in the object (it's larger at the bottom and smaller at the top).

Note also that, in relativity, there is no such thing as a "perfectly rigid object", i.e., there is no such thing as an object in which all parts respond instantaneously to mechanical stress. That's impossible because any change caused by a stress applied at a certain point in the object can only be propagated through the rest of the object at the speed of light. This sets a finite limit to the "rigidity" of materials in relativity. The kind of motion described above is called "rigid motion" (or sometimes "Born rigid motion", after Max Born, who did important original work in this area), but it is an idealization, only realizable if precisely timed forces are applied to all parts of the object, which is impossible in practice. It's a very useful idealization, though, which is why it's used a lot.

 

[PAllen]

These "heuristics made formal statements of fact" are what I referred to as "assumptions" (about gravity) in my previous post. Surely you don't mean that you can abandon your assumptions once you formulate a theory? That sounds like building a castle on the ground, then claiming the ground is no longer needed to support the castle.

Heuristics leading to guess at a theory are not assumptions (in general). They could be, in a some case, but on other cases not. The equivalence principle is not contained in any exact way in GR, and it is not present in most formal derivations of it.

To make another analogy, in mathematics we prove a theorem from predicates. When we are done, we cannot say that those predicates can now be abandoned and that we have a statement of truth independent from them.

Prior to proving a theorem, you have to get the idea that it may be true. Any heuristic you might use to do that need not have any representation in the derivation of the theorem, nor need it be a consequence of the theorm.

I'll take that claim with a grain of salt since calculus itself based on derivation from local approximations. I don't see how we can abandon that method, but if we can, I'm interested.

Of course it depends on exactly what his complaint is about the statement of EP. If it needs to be stated as a local (infinitesimal relationship) to make sense, then it ought to be stated that way rather than complaining that it is always false. It is always false that a segment of a circle is straight, but that has not prevented us from finding pi that way.

Well, on this world class experts differ with each other. You provide a good analogy. You can get pi using polygons of ever increasing number of sides. Further, the maximum distance between a sequence of such polygons and the circle approaches zero. However, you might argue that throughout the limiting process, you have zero curvature except in a measure zero set of point. Even in the limit, with countably infinite vertices, you still have only a countably infinite - therefore measure zero - set of vertices. Therefore a circle has zero curvature except at measure zero set of points (because the polygon 'becomes' the circle in the limit). This is obviously nonsense, and shows that not everything locally true becomes true in the limit. The divergence between a chord and arc approaches zero, but a chord is always distinguishable from the arc.

In the case of GR, Synge's point was that curvature has a finite non-zero value at a point; therefore gravity is distinguishable from acceleration of a rocket even at one point. Synge's argument was mathematical (not surprising - he was primarily a mathematician, but he wrote one of the seminal textbooks on GR). However, another physicist (Ohanian) put experimental meat on Synge's point by devising an 'in principle' device that can distinguish acceleration in flat space-time from gravitation due to a mass even in the limit of zero size for the instrument. Its reading approaches a constant times one of the curvature scalars which have a fixed value even at one point. A different line of dispute with principle of equivalence is that it doesn't necessarily hold for charged bodies (though the deviation, in practice, is too small to be measured in the foreseeable future).

Thus, quite objectively, one can say that Einstein's formulation of the principle of equivalence is false in GR. Most physicists (myself included) feel this is still (very clever) nitpicking. Almost all measurements, to any desired precision, are consistent with the principle of equivalence, intelligently applied, so it remains very useful. But, it clearly can't be either an axiom or a consequence of GR if it is technically false in GR.

 

[PeterDonis]

As far as "the conventionality of simultaneity", I'm just not getting your point. I do not know what those word mean in your mind nor what purpose arbitrary choices can possibly serve. Particularly choices that are inconsistent in a given context (e.g. car experiment above).

Some of my latest posts might help with the car experiment. But let me give another illustration, one which is given early on in MTW, to illustrate the sense in which coordinates themselves are "conventional".

MTW describe a thought experiment which they call "the centrifuge and the photon". Suppose we have a "centrifuge", which we idealize as a ring rotating in its own plane (i.e., about an axis perpendicular to its own plane and running through the geometric center of the ring) at a constant angular velocity. At some instant, we fire a photon across the ring, so it is emitted at some point on the ring and received (some time later) at some other point on the ring. The photon has a known fixed frequency when it is emitted. What is its frequency when it is detected? I.e., is it redshifted, blueshifted, or neither?

You might try to solve this using frames--perhaps an inertial frame in which the geometric center of the ring is at rest, or even using MCIFs at the events of emission and reception. But MTW point out that you can solve this problem without using coordinates at all. All you need is the following:

(1) The photon is described by a 4-momentum vector which does not change along its trajectory. By "does not change" we mean that it behaves the same as the 4-velocity of an observer moving inertially--i.e., the "proper acceleration" of the photon (I put the term in quotes because it's not quite the same as an ordinary object's proper acceleration) is zero, so it moves along a geodesic, and its 4-momentum vector at a given point is just the tangent vector to that geodesic at that point. (Physically, this means that the photon is moving freely through vacuum, with no interactions with anything between emission and absorption, and that there is no tidal gravity or anything else that might affect its energy or momentum.)

(2) The frequency of the photon, as measured by a given observer, is just the inner product of the observer's 4-velocity with the photon's 4-momentum. (Strictly speaking, this gives the photon's energy, but quantum theory gives a direct relationship between energy and frequency--the former is just the latter times Planck's constant. For this problem that is sufficient.)

Therefore, if we write $\mathbf{e}$ for the 4-velocity of the point on the ring where the photon is emitted, at the instant it is emitted, $\mathbf{p}$ for the 4-momentum of the photon, and $\mathbf{r}$ for the 4-velocity of the point on the ring where the photon is received, at the instant it is received, we have (using $E$ for the frequency since the difference is just Planck's constant, as above) $E_e = \mathbf{e} \cdot \mathbf{p}$ and $E_r = \mathbf{r} \cdot \mathbf{p}$. Note that $\mathbf{p}$ is the same in both formulas.

Now we just need to evaluate the two inner products. The inner product, as you can see from the way I wrote it, is just the generalization of the ordinary "dot product" in vector analysis to the case of 4-d spacetime; in other words, it is just the (cosine of the) "angle" in spacetime between two vectors. The only complication in spacetime is that "angle" contains the time dimension as well as the space dimensions; but this turns out to just be relative velocity (or at least that's a good enough way of looking at it for this problem--see below for how it works).

So what are the respective angles here? First, since the ring is rotating at a constant angular velocity, and since there is no tidal gravity (so we can directly compare velocities at different points in space), the relative velocity of the point of emission and the point of absorption is the same at the instant of emission as it is at the instant of absorption. This means that the "time" portion of the "angle in spacetime" between the photon and the ring's 4-velocity is the same at both emission and reception (since the photon itself always moves at $c$).

That only leaves the "space" portion of the angle, but that is just the ordinary angle between the spatial vectors tangent to the ring and to the photon's motion in space. If you draw a diagram, you will see that these two angles are the same. So both the "space" and "time" portions of the angle in spacetime are the same at emission and reception. That means there is zero frequency shift!

The fact that we can obtain this answer without ever using coordinates at all (let alone any particular kind of coordinates, such as inertial ones) shows the sense in which coordinates are "conventional": you can do physics without them. And since a definition of simultaneity is just part of a definition of coordinates, since you can do physics without coordinates, you can do physics without simultaneity, so simultaneity is "conventional" in the same sense as coordinates are.

Furthermore, even if you choose to use coordinates, nothing requires you to use a particular kind of coordinates. For example, we could choose to solve the above problem using ordinary inertial coordinates (the obvious ones to use are ones in which the geometric center of the ring is at rest). But we could also choose to solve it using non-inertial coordinates in which the ring is at rest (these are called "Langevin coordinates" after Paul Langevin, who introduced them). These non-inertial coordinates use a different simultaneity definition from that of any of the MCIFs of points on the ring--in fact they use the same definition (i.e., the same sets of simultaneous events) as the inertial frame I just described. (But they are not the same coordinates, because, as I said, in these coordinates the ring is at rest, whereas it isn't in the inertial frame.) Or we could solve it using two MCIFs, those of the emission and absorption events, and use a Lorentz transformation to convert quantities from one to the other. Or we could use any other valid coordinate chart.

So none of these coordinates can be said to be necessary for solving the problem, nor can any particular definition of simultaneity. (In this particular case, all of the definitions turned out to be based on the Einstein convention somehow; but in other problems that would not be the case.) That is the sense in which these things are "conventional".

is distance not also "conventional", not to mention position?

If "conventional" means "depends on your choice of coordinates", then sure. Or if it means "depends on your method of measuring them", then again, sure. I'm not sure this is quite the same sense of "conventional" as the one I was using; see above.

Let me expand on this a bit more by using a simpler example. I can measure the distance between New York and Los Angeles along the great circle that connects the two, by laying a ruler end to end repeatedly starting at New York and ending at Los Angeles, taking care to make sure I lay the ruler "straight" along itself. (Or I can do it by other methods that give equivalent results.) The result of such a measurement is a geometric fact about the shape of the Earth's surface; it is only "conventional" in the sense that I picked which geometric fact I wanted to measure.

However, I can also compute this geometric fact using different coordinate charts. For example, I could use a standard Mercator chart, or I could use a stereographic projection centered on the North Pole. These charts are "conventional" in a sense in which the measurement itself, the geometric fact, is not.

In the case of spacetime, direct observables, like the elapsed time on a particular clock between two events on its worldline, or the length of a particular spacelike geodesic between two events (which could be interpreted as a "distance" between two points of interest), are geometric facts like the distance from New York to Los Angeles. They are only "conventional" in the sense that I can pick which ones I'm interested in. But coordinate charts are "conventional" in a different, stronger sense, the sense in which the charts used to describe the Earth's surface are "conventional". This is true even of "natural" charts like inertial coordinates in flat spacetime or latitude and longitude on the Earth; they happen to match particular symmetries of the objects being described, but that doesn't stop them from being conventional; it just means they're more suitable for certain problems.

 

[harrylin]

But in the link that you posted above, it seemed clear that Einstein thought that there was only one simultaneity perspective for the traveler. He didn't seem to think that some kind of choice of simultaneity "convention" had to be made.

Only for the particular point of view that the traveller is always in absolute rest, as you clarified yourself in your summary! That implies simultaneity conventions. But as I mentioned earlier*, it doesn't really work anyway.
* https://www.physicsforums.com/threa...paradox-as-paradox.780185/page-7#post-4921049

 

[harrylin]

You are missing a subtlety of the history of GR. In the modern view, GR is the theory of curved spacetime, SR the theory of flat spacetime that (only) is locally true in GR. However, Einstein had a different way of looking at it, which (to the best of my knowledge) he never abandoned. In his view, the derivation of the properties of non-inertial frames in SR was part of GR, which also extended this to cover significant mass with curved spacetime. Thus, he is using the features of accelerated coordinates in SR displaying a position dependent potential (which you can see in the Rindler metric - to which I believe I referred you earlier), all derived for flat spacetime (no Einstein field equations of GR involved). Einstein just viewed this physics as special case of GR rather than part of SR.

Thus, with Einstein's packaging, there is no circularity.

That's quite correct except on one point: Einstein derived the properties of accelerated frames in SR as part of SR. From that he next derived, by means of the EEP (which was in his view part of GR), the properties for an equivalent non-accelerated reference frame in a gravitational field.

And with that correction, your explanation gains clarity:

[..] in the non-inertial coordinates in which the metric shows a potential, the traveling twin is not accelerating. It is the home twin that is accelerating in these coordinate, and that acceleration plays no role in the clock rate of the home twin.

Yes, exactly :)

 

[Dale]

If indeed GR is an independent theory (from SR), then you should not require the equivalence principle to arrive at GR. Is it true the the equivalence principle is unnecessary for GR? I'm asking because I haven't actually followed any derivation of GR (the math is still difficult for me).

GR cannot be logically derived from SR, nor can it even be derived from SR + the equivalence principle. In fact, Einstein between 1905 and 1915 had several false-starts. Other theories that looked like they would be suitable generalizations of SR with the equivalence principle. That is the key problem with trying to go in that direction. There is usually more than one possible theory, as there was in this case.

The equivalence principle was what we call a "desideratum". In other words, any candidate theory should obey the equivalence principle. It allows you to eliminate any candidate theories that do not follow it, but it does not allow you to derive the theory.

What do you think? Is GR justified (derived) is some completely independent way (how?)

"Justified" and "derived" are two different things. Like all fundamental laws of physics, GR is not derived at all. The fundamental physical laws are always simply assumed. They are then justified by experimental data.

 

[PAllen]

One debate running through here (and many other threads) is whether acceleration affects (ideal) clock rates (as distinct from enables different paths between events). Within the formalism and both SR and GR, in all generality, it is trivially impossible for acceleration to have an effect on passage of proper time (readings of ideal clocks). The proper time along a world line in any coordinates at all (even ones, such as Dirac used, for which there are two lightllike and two spatial coordinates with no time coordinate at all) is the integral of the square root of the contraction of the metric with coordinate derivatives by a parameter. Derivatives, not second derivatives. Thus, there is no place in the mathematics for acceleration (second derivative by some time coordinate) to play a role. Only velocity and position can play a role, even in the most arbitrary coordinates in either SR or GR.

 

[CKH]

Regarding conventions of simultaneity that are not the same as the standard (Einstein) convention in an inertial frame, I posed this problem:

Consider the consequences is using a different convention than Einstein's in a local inertial frame. Let's imagine two mile markers along the x-axis within the boundaries within this local inertial frame. We will have one clock at each marker, but they will be de-synchronized wrt to Einstein's convention by 1 hour to satisfy some other "convention of simultaneity". We perform two experiments in which a car (starting from rest in the frame) undergoes a certain proper acceleration for a certain amount of proper time and then stops accelerating (all within the local inertial frame). After accelerating, in each experiment the car cruises past the two milestones (but in the opposite direction).

The measurement system (the milestones/clocks) have no proper acceleration. We are forced to conclude (prior to experiment) by symmetry that the transits times are equal (whether in proper time or coordinate time in that local frame). But using these clocks they are not.

I received this response:

Sure, because you've implicitly defined a "frame" that is not a standard inertial frame, and your symmetry argument only applies if measurements are taken with respect to a standard inertial frame.

It does not matter. (Measurements above that are said to be in a "local inertial frame" are the same in a "standard inertial frame"). However, to remove your objection, you may remove all references to "local" from the experiment and then agree that the argument is valid.

 

[PAllen]

Regarding conventions of simultaneity that are not the same as the standard (Einstein) convention in an inertial frame, I posed this problem:I received this response:It does not matter. (Measurements above that are said to be in a "local inertial frame" are the same in a "standard inertial frame"). However, to remove your objection, you may remove all references to "local" from the experiment and then agree that the argument is valid.

Nonstandard coordinates do not (generally) have the property of isotropy, and the laws of motion are required to have extra terms (in such coordinates). Thus, there is no contradiction. Your symmetry argument does not apply when you choose to use non-standard simultaneity. But that doesn't mean you can't do it, just that computations become more complicated.

 

[PhoebeLasa]

When I read (and re-read) Einstein's wonderful little book "Relativity", It seems clear to me that he DID believe that simultaneity at a distance (for a GIVEN inertial observer) IS meaningful in special relativity. And it seems clear that the simultaneity given by the Lorentz equations IS the simultaneity that he believed had meaning in special relativity. And that those special coordinates worked throughout all (assumed flat) spacetime, not just locally.

I'm sure Einstein realized that a given inertial observer can choose to adopt some other observer's (rest) inertial reference frame instead of his own, but I think that Einstein would say that that choice would usually be undesirable, because those alternative coordinates wouldn't be meaningful to the given observer. And I'm sure that Einstein realized that an inertial observer is even free to adopt coordinates in an almost completely arbitrary manner, but I think Einstein would have considered that to be a very stupid thing to do in special relativity ... why chose meaningless coordinates when you can have meaningful coordinates? Why choose complexity over simplicity?

Einstein didn't use, or need, differential geometry in his development of special relativity. It was only when he had finished developing special relativity, and was trying to understand how to develop general relativity, that he realized that those meaningful coordinates he used in special relativity wouldn't work in general relativity. He expressed that by saying that the "reference frame" of an observer in special relativity (which he regarded as a rigid and meaningful (mental) construction) must be replaced by a set of rather mushy, non-rigid "reference mollusks" in general relativity, with coordinates that are arbitrary and basically meaningless. And, by using the equivalence principle, applied to the rotating disk example of special relativity, he realized that Euclidean geometry doesn't work in general relativity: the ratio of the circumference of a circle to its diameter ISN'T pi in general relativity, and the sum of the three interior angles of a triangle ISN'T 180 degrees. THAT was when he realized that he needed to learn differential geometry, in order to develop general relativity.

 

[PeterDonis]

Nonstandard coordinates do not (generally) have the property of isotropy, and the laws of motion are required to have extra terms (in such coordinates)

Not just the laws of motion: all physical laws will, in general, have different terms in non-standard coordinates, if you expand them out from their covariant tensor formulations. (OTOH, if you write all physical laws in their covariant tensor formulations, they look exactly the same in any valid coordinate chart, whether standard or non-standard.)

There's also another key distinction here (which I know you understand but which I'm stating explicitly for the benefit of other readers of this thread). A "local inertial frame" is not just a small patch of spacetime around a chosen event: it's a small patch of spacetime around a chosen event, plus a standard inertial coordinate chart on that patch of spacetime. Using a non-standard coordinate chart (such as the non-standard simultaneity convention CKH described) on the same small patch of spacetime (such as the one in which the car scenario takes place) means you are not using a local inertial frame, even though the same small patch of spacetime can be described by a local inertial frame (by using a standard inertial coordinate chart on it). This is just a special case of the general rule that it's important to keep in mind the distinction between spacetime (or a small patch of it), the geometric object, and coordinate charts that we can use to describe it.

 

[PAllen]

When I read (and re-read) Einstein's wonderful little book "Relativity", It seems clear to me that he DID believe that simultaneity at a distance (for a GIVEN inertial observer) IS meaningful in special relativity. And it seems clear that the simultaneity given by the Lorentz equations IS the simultaneity that he believed had meaning in special relativity. And that those special coordinates worked throughout all (assumed flat) spacetime, not just locally.

I'm sure Einstein realized that a given inertial observer can choose to adopt some other observer's (rest) inertial reference frame instead of his own, but I think that Einstein would say that that choice would usually be undesirable, because those alternative coordinates wouldn't be meaningful to the given observer. And I'm sure that Einstein realized that an inertial observer is even free to adopt coordinates in an almost completely arbitrary manner, but I think Einstein would have considered that to be a very stupid thing to do in special relativity ... why chose meaningless coordinates when you can have meaningful coordinates? Why choose complexity over simplicity?

Einstein didn't use, or need, differential geometry in his development of special relativity. It was only when he had finished developing special relativity, and was trying to understand how to develop general relativity, that he realized that those meaningful coordinates he used in special relativity wouldn't work in general relativity. He expressed that by saying that the "reference frame" of an observer in special relativity (which he regarded as a rigid and meaningful (mental) construction) must be replaced by a set of rather mushy, non-rigid "reference mollusks" in general relativity, with coordinates that are arbitrary and basically meaningless. And, by using the equivalence principle, applied to the rotating disk example of special relativity, he realized that Euclidean geometry doesn't work in general relativity: the ratio of the circumference of a circle to its diameter ISN'T pi in general relativity, and the sum of the three interior angles of a triangle ISN'T 180 degrees. THAT was when he realized that he needed to learn differential geometry, in order to develop general relativity.

This is mostly true, historically (with a caveat below). However, other physicists came to disagree on interpretation of what is SR and what is GR. No physical predictions are affected by this disagreement - it is yet another philosophy debate. The disagreement began early: already Eddington in his 1922 treatise adopted the 'modern' point of view that analyzing flat spacetime with general coordinates was SR not GR. Bergmann's 1942 book introduced the whole machinery of tensor calculus in its presentation of SR.

The caveat is that, in SR, Einstein analyzed non-inertial motion only in a single inertial frame. Analyzed as a special case of GR, one notes the importance he attached to general covariance - any coordinates are good. I have never seen him use (and have looked) the concept 'planes or lines of simultaneity' for a non-inertial observer. He actually did write a paper using radar simultaneity for a non-inertial observer, but I am not able to find reference for it right now.

 

[Dale]

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