I Einstein vs Newton: The concept of inertial vs non inertial frames

[A.T.]

...it's clear that it must be realized by a massive body...

For me it's not clear what "be realized by" means here? Do you mean that a reference frame is defined using massive bodies? Or that it is a massive body itself?

 

[vanhees71]

I think, a reference frame is an abstract mathematical object, to which the calculations refer to, independent of the existence of physical objects. Einstein wrote about moving clocks and rulers to visualize the movement of coordinate systems relative to each other. If you accept calculating with complex numbers, then you can also define a reference frame of a tachyon, which seems not to exist physically.

That's the very point! I don't think that our mutual misunderstanding is about the math. That's standard, how to define a pseudo-Riemannian (Lorentzian) manifold introducing maps and atlasses, the corresponding coordinate bases for the tangent and cotangent spaces and all this.

It's really about the relation of this formalism to real-world physics, and Einstein had a very clear idea about this. The coordinates in a chart of the manifold have no physical significance. They are just "labels" of space-time points given this chart.

I googled a bit last night and got a lot of debates about this. One idea I like most is to interpret the general covariance as a gauge invariance. What's gauged here is the Lorentz subgroup of the proper orthochronous Poincare group as the symmetry group of Minkowski space, and that's what's behind the tetrad formalism, which is obviously what's preferred by @Dale on the formal side, and of course I've no problem with that.

However, in the lab you need to define your reference frame. Usually that's a no-brainer because we just build up our equipment and tacitly use the restframe of the lab as a reference frame. That's why almost no textbooks discuss this issue at all. The only reference I have at the moment is the very beginning of Sommerfeld's vol. 1 of Lectures on theoretical physics, where for Newtonian mechanics he comes to the conclusion that the best realization of an inertial frame there was at his time is the "restframe of the fixed stars".

Today I'd say the most "natural" physical reference frame in the context with GR is the (local!) rest frame of the microwave background radiation within the standard large-scale average spacetime, which is a Robertson-Walker-Friedmann-Lemaitre metric.

A way to establish it is, e.g., to use the data of the cosmic-microwave-background satellite Planck, which meausres the microwave spectrum in all directions around it. Here, the satellite (massive body) is the reference body to define a reference frame, which is to a very good approximation a free-falling reference frame. It's of course not a local rest frame of the CMBR, because the Earth, together with the entire local group our galaxy moves against the local CMBR restframe. Planck is so accurate that even the motion of the Earth around the Sun can be included in the analysis. After correction of this small effect the conclusion is that the local group of our galaxy moves with 627 km/s in a certain direction.

https://doi.org/10.1051/0004-6361/201321556
https://arxiv.org/abs/1303.5087

 

[vanhees71]

Exactly

And when he did so he explicitly used language like “we can imagine”, making it clear that the concept of a reference frame was not intended to be limited to actual physical constructions.

Certainly if you can place a physical clock somewhere then you can also imagine a clock there. So whenever desired we can make a connection between Einstein’s imaginary clocks and some physical clocks, but the reference frame is the abstract side of it, not the physical side of it.

I don't say that it is limited to actual physical constructions but that for any measurement there must be established a reference frame to be able to give positions and time of any (local) observer. Otherwise you couldn't use all your formalism to do physics and to establish that to a high accuracy General Relativity provides the correct spacetime model you use to establish your formalism in the first place. You need both, the formalism, i.e., the mathematical model of spacetime as well as the realization of reference frame you can use to measure distances and times and relate it to this abstract structure to be able to measure to begin with. That's why Einstein indeed is one of the few textbook authors refers to such real-world setups, and be it only in "gedanken experiments".

I think, formally we agree that a reference frame can be defined only a local concept. One way is the tetrad formalism given by a set of time-like worldlines which cover a certain part of spacetime together with an arbitrary tetrad field related to these time-like worldlines. You can use any coordinates you like within the so defined frame, but you don't need to thanks to general covariance.

Hence, this is even a coordinate-independent definition and thus may be realizable by real-world experiments. Each time-like worldline can be physically realized by some (of course massive) body with a clock and giving a tetrad along the worldline, defining a "local rest frame" and a "proper" time. One example is the CMBR satellite Planck, as described in #52.

 

[vanhees71]

For me it's not clear what "be realized by" means here? Do you mean that a reference frame is defined using massive bodies? Or that it is a massive body itself?

Is it more clear what I mean with the example of the Planck satellite I gave in #52?

 

[Dale]

I don't say that it is limited to actual physical constructions

If it is not limited to actual physical constructions then it seems wrong to say that the physical construction is the reference frame.

You need both, the formalism, i.e., the mathematical model of spacetime as well as the realization of reference frame you can use to measure distances and times and relate it to this abstract structure to be able to measure to begin with.

I have only a semantic objection to this. I identify the term “reference frame” with the mathematical model. The physical devices are simply identified as “clocks” “rulers” and so forth. It is not necessary for them to be labeled as part of the reference frame, and in my opinion it is a bad idea to do so.

 

[Sagittarius A-Star]

where for Newtonian mechanics he comes to the conclusion that the best realization of an inertial frame there was at his time is the "restframe of the fixed stars"

In Newton's theory, an infinite number of inertial reference frames can be defined. All are equally "good". Newton speculated also about an "absolute" rest frame (see under "Scholium"), but did not define one. I don't understand, what "realization" means in this context.

Today I'd say the most "natural" physical reference frame in the context with GR is the (local!) rest frame of the microwave background radiation

I think, that you used the correct term "rest frame", to describe the connection between an (abstract) reference frame and a (concrete) physical object, but I find the adjective "physical" for a reference frame incorrect, because it is an abstract mathematical object.

 

[vanhees71]

If it is not limited to actual physical constructions then it seems wrong to say that the physical construction is the reference frame.

I have only a semantic objection to this. I identify the term “reference frame” with the mathematical model. The physical devices are simply identified as “clocks” “rulers” and so forth. It is not necessary for them to be labeled as part of the reference frame, and in my opinion it is a bad idea to do so.

Well, that seems to be the source of our apparent disagreement. For me a reference frame is first a real thing in the lab, a satellite measuring all kinds of astronomical observables, the gravitational-wave detectors of the LIGO/VIRGO collaboration. Then we have a mathematical formalism describing these observables within an assumed spacetime model, in the case of GR a pseudo-Riemannian manifold with a fundamental form of signature (1,3) or (3,1).

Looking in a lot of textbooks and papers, I come to the conclusion that the best treatment in the most simple way is still Einstein's big original review paper (the original papers are in the Proceedings of the Prussian Academy and available online only via very bad scans unfortunately). Since it was quoted in another thread, I dare to post the link here (though I think that may be a copyright violation, and if so, just tell me and I cancel the link immediately):

https://archive.org/details/einstein_relativity/page/n203/mode/2up

The translation of the paper is found beginning from page 204 (electronic) aka page 183 (in print), Sect. A (paragraphs 1-4).

The original reference for this is (in German of course)

A. Einstein, Die Grundlage der Allgemeinen Relativitätstheorie
Ann. Phys. (Leipzig) 14, Supplement, 517 – 571 (2005)
https://doi.org/10.1002/andp.200590044

 

[vanhees71]

In Newton's theory, an infinite number of inertial reference frames can be defined. All are equally "good". Newton speculated also about an "absolute" rest frame (see under "Scholium"), but did not define one. I don't understand, what "realization" means in this context.I think, that you used the correct term "rest frame", to describe the connection between an (abstract) reference frame and a (concrete) physical object, but I find the adjective "physical" for a reference frame incorrect, because it is an abstract mathematical object.

Of course, in Newtonian mechanics and special relativity all inertial reference frames are equivalent, and that's a physical symmetry, formally described as the invariance of the physical laws under Galilei or Lorentz transformations.

To check this assumption experimentally you have to realize a sufficiently accurate approximation to such an inertial reference frame, and that's not as simple as it seems. Usually we start by taking the Earth, which however is rotating (as the experiment with the Foucault pendulum demonstrates in every introductory experimental-physics lecture) and thus for sure no inertial reference frame etc.

Realization of a reference frame means simply to have some instrumental setup in a real lab that enables you to measure "time" and "position". If it were impossible to construct such things, we'd not discuss about theoretical physics, because then all the formal math wouldn't have a relation to observations of Nature. Einstein was very well aware of this, and he has clearly described everywhere, how to physically realize a reference frame in both special and general relativity (see the paper of 1916 quoted in #57).

If the entire debate is about my use of the word "reference frame" in a physical sense, then please tell me how else to call such a physical setup establishing space-time measurements in the real world. In German both is called "Bezugssystem".

 

[A.T.]

... please tell me how else to call such a physical setup establishing space-time measurements in the real world.

Clocks and rulers.

 

[Sagittarius A-Star]

If the entire debate is about my use of the word "reference frame" in a physical sense, then please tell me how else to call such a physical setup establishing space-time measurements in the real world.

Maybe you call it "physical setup establishing space-time measurements". A related reference frame you may call "rest frame of the lab" (in German: "Ruhesystem des Labors").

 

[Dale]

For me a reference frame is first a real thing in the lab, a satellite measuring all kinds of astronomical observables, the gravitational-wave detectors of the LIGO/VIRGO collaboration. Then we have a mathematical formalism describing these observables within an assumed spacetime model, in the case of GR a pseudo-Riemannian manifold with a fundamental form of signature (1,3) or (3,1).

For me, the reference frame is not first one and then the other, it is only the mathematical formalism. Here are the reasons that I think this approach is better:

1) as the famous saying goes “the map is not the territory”. Using the term “reference frame” as you do uses the same word to refer to both the “map” and the “territory”. That is inherently confusing since they are not the same thing.

2) one set of rulers, clocks, etc (your “real thing in the lab”) can be used to define an infinite number of reference frames. So equating the frame with the real things doesn’t make sense. In principle, any set of clocks and rulers that you could use to implement one reference frame could be used to implement any other possible reference frame. Since all possible configurations of clocks and rulers can map to all possible coordinate systems or tetrads it is not at all natural to associate instances of the two as though they were integral parts of the same thing. Either can be swapped out at will without changing anything in principle, either theoretically or experimentally.

3) pedagogically this kind of mixed terminology sends mixed messages to novices who then unnecessarily struggle with questions like that from the other thread’s OP.

4) the principle of relativity states that the laws of physics are the same in all reference frames. The usage in this context refers to the mathematical form of the laws of physics, and therefore a mathematical reference frame is more natural to consider.

Finally, your arguments about why they should be considered together have been primarily either a misrepresentation of my position or an argument by authority. But I think that you dramatically misread the authorities you have presented. Einstein indicates that a frame is a mental construct in his usage of terms like “imagine”, and even the Wikipedia definition talks about “physical reference points” and not “physical reference objects”. Physical points are not material nor do physical points have mass even if they are placed on a massive material object. So I don’t interpret those authorities as in fact supporting your “map and territory” definition. I think you are misinterpreting them (although I am sure you think the opposite)

Now, I have no doubt that if you look you can find an authority who does clearly support your approach. I can also show several that do not. Argument by authority is always rather tenuous, but especially when the authorities disagree. In any case, I think your usage is a bad approach, and using it is akin to using relativistic mass. Yes it is a concept that is out there and perhaps promoted by some, but it has many disadvantages, few advantages, and engenders much confusion.

 

[vanhees71]

For me, the reference frame is not first one and then the other, it is only the mathematical formalism. Here are the reasons that I think this approach is better:

1) as the famous saying goes “the map is not the territory”. Using the term “reference frame” as you do uses the same word to refer to both the “map” and the “territory”. That is inherently confusing since they are not the same thing.

2) one set of rulers, clocks, etc (your “real thing in the lab”) can be used to define an infinite number of reference frames. So equating the frame with the real things doesn’t make sense. In principle, any set of clocks and rulers that you could use to implement one reference frame could be used to implement any other possible reference frame. Since all possible configurations of clocks and rulers can map to all possible coordinate systems or tetrads it is not at all natural to associate instances of the two as though they were integral parts of the same thing. Either can be swapped out at will without changing anything in principle, either theoretically or experimentally.

3) pedagogically this kind of mixed terminology sends mixed messages to novices who then unnecessarily struggle with questions like that from the other thread’s OP.

4) the principle of relativity states that the laws of physics are the same in all reference frames. The usage in this context refers to the mathematical form of the laws of physics, and therefore a mathematical reference frame is more natural to consider.

Finally, your arguments about why they should be considered together have been primarily either a misrepresentation of my position or an argument by authority. But I think that you dramatically misread the authorities you have presented. Einstein indicates that a frame is a mental construct in his usage of terms like “imagine”, and even the Wikipedia definition talks about “physical reference points” and not “physical reference objects”. Physical points are not material nor do physical points have mass even if they are placed on a massive material object. So I don’t interpret those authorities as in fact supporting your “map and territory” definition. I think you are misinterpreting them (although I am sure you think the opposite)

Now, I have no doubt that if you look you can find an authority who does clearly support your approach. I can also show several that do not. Argument by authority is always rather tenuous, but especially when the authorities disagree. In any case, I think your usage is a bad approach, and using it is akin to using relativistic mass. Yes it is a concept that is out there and perhaps promoted by some, but it has many disadvantages, few advantages, and engenders much confusion.

I can agree with almost anything except 2), and this is a very important point.

A physical setup of "clocks and rulers" defines one and only one reference frame. Of course you can use different coordinates and different tetrads to "map it", but the physical observables are independent of this choice. This is guaranteed by general covariance (which best is interpreted as a gauge invariance rather than a physical symmetry). That's what you rightfully stress in 4) yourself!

Also, I only referred to "authorities", because @PeterDonis insisted on this, and for me in this paper of 1916 Einstein precisely takes my point of view, which is not surprising, because physics is about the real world and a physical theory must describe (or "map") the real world in terms of the math it is formulated in.

Last but not least, I have not misinterpreted your standpoint but disagree(d) with it, because physics is more than a mathematical formalism but it has the goal to describe the real world, i.e., quantitative observations and is not merely some mathematical system of axioms and theorems.

I try to avoid using the word "reference frame" in discussions on PF, but it'll be hard to keep this promise if again there's a question about, what the one or the other observer measures in a concrete example.

 

[romsofia]

Yvonne Choquat Bruhat in her book "Introduction to relativity, black holes & cosmology" speaks on this point a few times. Here is an excerpt from her book that is relevant (no pun intended) here

...a model for reality at a macroscopic scale is based on differentiable manifolds, geometric objects whose elements are called points. Each point of, let us say, a 3-manifold is represented by a family of sets of three numbers, each set being the coordinates of the point in a reference frame; different elements of the family are linked by an equivalence relation between reference frames. The physically realistic problem is to link the abstract reference frames with a concrete observable one.

She goes deeper in some other chapters, but, I'll leave those sections for the curious reader.

That being said, I agree with vanhees. That last sentence is very important to me as a physicist. It's very easy to get caught up in all the abstract models. Our job is tie mathematical models with observations.

 

[vanhees71]

That's obvious a very good description. The one thing I learned in this thread is that this point has been a very long heated debate in the physics community as well as the philosophy-of-science community. In the latter context it's related to the hole argument:

https://link.springer.com/article/10.12942/lrr-2014-1 (open access!)

I've not read this paper in much detail.

For me all these issues are resolved with the interpretation of GR as a gauge theory with the diffeomorphism invariance (aka "general covariance" under coordinate transformations) as the gauge transformations. The "gauged global symmetry" is the symmetry under Lorentz transformations of SR.

 

[martinbn]

@romsofia On page 5 she writes

An arbitary set of $n$ linearly independent tangent vectors at $x$ constitute a frame at $x$.

What page is your quote from?

 

[martinbn]

@romsofia On page 5 she writes

What page is your quote from?

Never mind, I found it on page 24. It is just a comment. Cannot be used as a precise definition.

 

[martinbn]

And from page 8

A moving frame, often called simply a frame in what follows, in a subset U of a differentiable n-dimensional manifold V is a set of n vector fields on U that are linearly independent in the tangent space TxV at each point x∈U. A manifold that admits global frames (not necessarily global coordinates) is called parallelizable

 

[Sagittarius A-Star]

A physical setup of "clocks and rulers" defines one and only one reference frame.

No.

One counter-example:

In 1728, James Bradley found the yearly stellar abberation. He looked with his telescope at a star, when the Earth was moving transversal to the star-light. Then he left the telescope un-touched for 6 months and then looked again through it. He did not see the star, unless he changed the angle of the telescop. This angle was 2 * α . The reason for this is, that the Earth ist moving further, while the light-particle, which he assumed, moves from the top end of the telescope to the bottom-end of the telescope.

He developed the formula for the abberation angle, which he used for measurements of c from the speed of the Earth (moving with v = 30 km/s around the sun) and the measured abberation angle α:
tan ( α ) = v/c.

Therefore, he made a Galileo-transformation from a frame, moving with 30 km/s transversal to the star-light to another frame, moving with -30 km/s transversal to the star-light.

So the rest-frame of the telescope changed from the one frame to the other frame, which he could then define for example as reference frame. The first frame moves in this reference-frame with 60 km/s.

Of course, he could also have defined the first mentioned frame as the reference frame.

Second counter-example:

You can calculate the frame-independent age difference of the "twins" in any inertial reference frame. It does not need to be a frame, in which one of the twins or any physical object is at rest.

 

[romsofia]

Never mind, I found it on page 24. It is just a comment. Cannot be used as a precise definition.

It wasn't meant to be a precise definition and I didn't present it as one. Hopefully no one is taking it as that!

And from page 8

Which is... what? I don't see the point of this. No one is disputing the mathematical model of what a frame is. The distinction is how we address them in physics. Which is why she dubs them "Physical comments" because she is clarifying how all of this is *physical*. And here is another physical comment from her on page 45 that I enjoy:

Like all mathematical models, Special relativity aims at providing as accurate as possible a picture of physical reality. No mathematical model can replace reality, but the first problem is to be able to compare the results given by equations with observed facts. In Special Relativity, one should physically identify spacetime inertial reference frames for Minkowski spacetime. The choice posed a puzzle for Einstein, since Minkowski spacetime considered as a global model is empty

So, let me stress this point. I don't disagree with the mathematical model of what a frame is. That wasn't the issue. The issue I have is that is all a reference frame is to me, as a physicist. I don't agree with that, and I never will. You have to tie the reference frame to physical reality, which is the whole point of physics. I'm not a mathematician. My work is based off empirical data that I model using mathematics.

 

[Motore]

The issue for me is this:

A physical setup of "clocks and rulers" defines one and only one reference frame.

Can we or can't we use different (infinite) reference frames for one specific physical setup of "clocks and rulers"? To me it seems we can, therefore there is not one and only one reference frame.
The calculation complexity of using different reference frames is beside the point.

 

[Dale]

A physical setup of "clocks and rulers" defines one and only one reference frame. Of course you can use different coordinates and different tetrads to "map it"

Remember, for me the reference frame is the coordinates or tetrad. So if you agree that you can use different coordinates and tetrads then you do in fact agree with the content of 2, just not the semantics.

In any case, perhaps I should have avoided the term reference frame since that it the term under contention. In neutral terms, the fact I was highlighting is that you can use any possible sufficient set of clocks and rulers to realize any possible coordinate system or tetrad. I assume, worded like that, you have no objection. My opinion (which I assume you disagree with) following from that fact (which I assume you agree with) is that since the two halves are so independent of each other, to me it makes little sense to try to lump them together in one term.

To understand my stance on this issue, consider “the mass of object A”. This could refer to either a mathematical variable $m$ in some equation or to the physical property of A that can be measured by placing A on a balance scale. This uses the same term for the map and the territory, but there is a unique physical quantity and a unique mathematical quantity assigned to represent it. The math and the measurements are isomorphic, so there is no ambiguity.

Now, consider the Coulomb gauge. That is purely mathematical. You can change to a different gauge without changing any of your voltmeters or grounding wires. The mathematical quantity is not isomorphic to any physical object or measurement. Because of this we do not try to assign the label “the Coulomb gauge” to any collection of measurement devices or physical objects. We keep the map and the territory distinct because there is no isomorphism between them.

As you yourself have mentioned, the freedom to select a coordinate system or tetrad is similar to choosing an EM gauge. Certainly a coordinate system has more in common mathematically and theoretically with an EM gauge than with mass. It makes sense that the language should reflect that similarity. Given the lack of an isomorphism between the math and the physics, we should not try to use the same word to refer to both.

but the physical observables are independent of this choice. This is guaranteed by general covariance

Precisely.

because physics is more than a mathematical formalism but it has the goal to describe the real world

I already told you multiple times that this is not a point of disagreement. I do not appreciate this repeated and completely asinine mischaracterization of my opinion. Simply because I do not wish to call a collection of clocks and rulers a “reference frame” does not mean that I cannot use rulers and clocks to measure and describe the real world!

In the end this is the only argument you have put forth for your position and it is a pure misrepresentation. This is intellectually dishonest.

 

[Dale]

Temporarily closed for moderation

 

[PeterDonis]

I have not misinterpreted your standpoint but disagree(d) with it, because physics is more than a mathematical formalism but it has the goal to describe the real world, i.e., quantitative observations and is not merely some mathematical system of axioms and theorems.

@Dale never denied what you are saying here, and indeed has explicitly said he agrees with it, so you are not disagreeing with him, you are misrepresenting his position, as he has said. The only thing you disagree with him on is the usage of the term "reference frame", which is just about words, not about physics.

In any case, this thread has gone far beyond answering the OP's question, which was answered many posts ago, so the thread will remain closed.