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

[vanhees71]

As I mentioned in post 11 and repeatedly since, it is used inconsistently. There is no way to avoid that. The best that we can do is clarify what we mean in a particular instance.

Me too, and I think that you both egregiously misunderstand the standard definitions and misrepresent my position.

Then that is the corresponding tetrad rather than the coordinate system. No problem, it is perfectly legitimate. In fact, the tetrad avoids the problems with the coordinates at $r=0$ and $\phi=\pi$ and also avoids the difficulties with synchronization endemic to rotating coordinate systems. For rotation I think the tetrad is a better tool than the coordinates.

How about identifying (local) reference frames generally with a time-slicings/foliations of spacetime? This would apply even in GR!

 

[PeterDonis]

How about identifying (local) reference frames generally with a time-slicings/foliations of spacetime?

This won't work in general. Born coordinates themselves are a counterexample, because the congruence of Langevin observers is not hypersurface orthogonal.

This would apply even in GR!

Only in the very limited set of spacetimes where you can find a congruence of timelike worldlines that is hypersurface orthogonal.

 

[Dale]

How about identifying (local) reference frames generally with a time-slicings/foliations of spacetime? This would apply even in GR!

In addition to the comments by @PeterDonis this would defeat the purpose of reducing confusion by adding a third possible meaning instead of just the current two meanings.

My personal preference is to use reference frame to mean tetrad, but using it to mean coordinate system is so common that I actually use that meaning more often than my preferred meaning.

 

[Dale]

When the requirement "the coordinates must be realized by REAL body" is applied, we should be limited in that region.

That is not a general requirement, but even if you wanted to add that requirement it still would not prevent you from using coordinates in the $c<r\omega$ region.

The requirement “the coordinates must be realized by REAL body” in no way implies that said real bodies must be at rest in those coordinates. The GPS system is a good example. There is no “real body” part of the GPS system that is at rest in the ECI coordinates.

 

[vanhees71]

In addition to the comments by @PeterDonis this would defeat the purpose of reducing confusion by adding a third possible meaning instead of just the current two meanings.

My personal preference is to use reference frame to mean tetrad, but using it to mean coordinate system is so common that I actually use that meaning more often than my preferred meaning.

So a reference frame is given by a time-like curve and a tetrad with the tangent vector on the curve as the time-like basis vector. Then I don't understand, what we were fighting about two weeks ago. That's precisely what I understood as a (local) reference frame all the time, though I didn't discuss the relativsitic but only the Newtonian case, where it is much simplified by the assumption of absolute time and absolute space.

Also, why must it be a tetrad? Usually you start with holonomic bases in GR, i.e., with tangent vectors along the coordinate lines parametrized with the coordinates. In this sense also coordinates define frames of references.

 

[Dale]

Then I don't understand, what we were fighting about two weeks ago.

We were fighting about the fact that I said a reference frame was a mathematical construct, either a tetrad or a coordinate system, with no mass and you said that you “heartily disagree”.

If you now agree, then you should post an apology or at least a correction in that previous thread. I forewarn you not to distort my position by claiming that calling it a mathematical construct means that it cannot even be related to physical objects. I already covered that mischaracterization, and I am no longer feeling sufficiently generous towards you to allow continued misrepresentation of my position.

 

[vanhees71]

I don't dare to discuss this anymore then. If you a priori don't tolerate any other opinion than your own, it's anyway useless to discuss it.

I'd only like to have a clear definition of what's allowed to be said what a reference frame is and what not in these forums. Obviously the standard definition, given even in Wikipedia is not "tolerated" by you:

================================================================================================
Wikipedia:

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame.

================================================================================================

Are you now accepting the part I've put in italics or not? From your above statement it's not clear to me: In the first paragraph you seem to indicate that you only accept a definition without the parts in italics. Then I still heartily disagree, because then this implies the conclusion that you are not tolerating to be even claimed.

 

[Dale]

I don't dare to discuss this anymore then. If you a priori don't tolerate any other opinion than your own, it's anyway useless to discuss it.

Not tolerating your misrepresentation of my opinion is entirely different from not tolerating other opinions than my own. I welcome your opinion, but not your distortion of mine. Don't pretend that I am a tyrant for being irritated at your repeated mischaracterization of my opinion.

Are you now accepting the part I've put in italics or not?

I do not accept even the non italicized part, let alone the italicized part, as a correct definition of a reference frame.

then this implies the conclusion that you are not tolerating to be even claimed.

It does not imply that at all. If "X" is not part of the definition of term "A" does not in any way imply "not X" nor even "If A then not X".

Here, I disagree that the physical reference points are part of what defines a reference frame. That in no way whatsoever implies that a mapping cannot be done between physical reference points and mathematical points in the coordinate system or tetrad. In fact I completely recognize that the principal value of a reference frame is that such mappings between physical events and the mathematical constructs can be done, but I identify the reference frame itself with the coordinate system or the tetrad, not with the physical reference points. Claiming that it is therefore my position such a mapping cannot be done is a gross misrepresentation of my position.

EDIT: I could even accept the mapping between the coordinate system (or the tetrad) and physical events as part of the definition of the reference frame, but not the physical reference points themselves. If I included the mapping as part of the definition then I would specify that the mapping must be smooth and invertable.

I still heartily disagree,

I made exactly two claims in the disagreeable post:
1) a reference frame is a coordinate system or a tetrad
2) a reference frame has no mass

Which of those two do you heartily disagree with?

 

[vanhees71]

This won't work in general. Born coordinates themselves are a counterexample, because the congruence of Langevin observers is not hypersurface orthogonal.

Ok, for me to understand this issue with the reference frames better, let's discuss Born coordinates and in which sense they can be used to define a reference frame. Let's use Cartesian coordinates. Then there are no coordinate singularities.

Let $(t',x',y',z')$ be "Galilean coordinates" defining a global inertial reference frame in SR. The pseudometric components are there of course $g_{\mu \nu}'=\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$. I set $c=1$ for convenience.

Then the "Cartesian Born coordinates" are defined by
$$t'=t, \quad \begin{pmatrix}x' \\ y' \\z' \end{pmatrix} = \begin{pmatrix} \cos(\omega t) & -\sin(\omega t) &0 &0 \\ \sin(\omega t) & \cos(\omega t) & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
The metric components are most easily given by
$$\mathrm{d} s^2=\mathrm{d} t^{\prime 2}-\mathrm{d} \vec{x}^{\prime 2} = (1-\omega^2 R^2) \mathrm{d} t^2 -\mathrm{d} \vec{x}^2 +2 \mathrm{d} t \mathrm{d} x \omega y - 2 \mathrm{d} t \mathrm{d} y \omega x \quad \text{with} \quad R^2=x^2+y^2.$$
Though the transformation is a global diffeomorphism, seen from the point of view of Minkowski space as a differentiable manifold, the coordinates with the coordinate lines' tangent vectors as basis vectors define only a reference frame for a part of the space. That's easy to see: for 3 coordinates held fixed and one varying one gets the line elements for the coordinate lines,
$$\mathrm{d} s_t^2 = \mathrm{d} t^2 (1- \omega^2 R^2), \quad \mathrm{d} s_{j}^2=-(\mathrm{d} x^j)^2 \quad \text{for} \quad j \in \{1,2,3\}.$$
Thus for defining a reference frame the coordinates cover only the range $0 \leq R<1/\omega$.

 

[vanhees71]

I made exactly two claims in the disagreeable post:
1) a reference frame is a coordinate system or a tetrad
2) a reference frame has no mass

Which of those two do you heartily disagree with?

Please give a clear definition of what a reference frame is in your definition. If you don't accept the standard definition as quoted from Wikipedia, I don't understand 1) at all, and 2) doesn't make any sense to me. Is there a textbook or paper, which describes what you understand as a "reference frame"?

I could agree with 1), if I'd know you use the standard meaning, which defines the reference frame of an arbitrarily moving ("point-like") reference body. Mathematically that's a point particle (i.e., an objects whose extension can be neglected for the physical situation to be described) moving on a time-like world line (in this sense it has mass), described as functions $q^{\mu}=q^{\mu}(\lambda)$ (with $q^{\mu}$ arbitrary coordinates defining a chart of a open subset of the spacetime manifold and $\lambda$ an arbitrary world-line parameter).

To make this a reference frame you can either use the coordinate-line tangent vectors or tetrades to define a reference frame. In the first case you use at any point along the world-line of the reference body the tangent vectors of the coordinate lines as basis vectors for the tangent space.

A tetrade has the advantage that it defines the reference frame as the restframe of the reference body. Then you use the normalized tangent vector to the worldline of the reference body (which is necessarily timelike) together with three space-like (pseudo-)orthogonal normalized vectors at each point along the world line. This is usually needed for more advanced topics like the definition of spinor fields etc.

 

[PeterDonis]

Please give a clear definition of what a reference frame is in your definition.

I thought @Dale made that clear earlier: his preferred definition of "reference frame" is "tetrad field", i.e., a mapping of orthonormal tetrads to points in an open region of spacetime.

the standard definition as quoted from Wikipedia

Wikipedia is not a good source for a "standard definition" of anything. If you can find a physics textbook that defines the term "reference frame", we can discuss that definition.

 

[PeterDonis]

for me to understand this issue with the reference frames better, let's discuss Born coordinates and in which sense they can be used to define a reference frame

Here you are using the Wikipedia definition that you quoted. But my post that you quoted was in response to your suggestion here:

How about identifying (local) reference frames generally with a time-slicings/foliations of spacetime?

What time slicing/foliation are you suggesting for the case of Born coordinates?

 

[vanhees71]

Here you are using the Wikipedia definition that you quoted. But my post that you quoted was in response to your suggestion here:

What time slicing/foliation are you suggesting for the case of Born coordinates?

I already admitted that time slicing/foliation seems to narrow to give a general definition of a reference frame. I only wanted a clear stated what's wrong with my much simpler standard definition I know from all GR textbooks of a reference frame using (local) coordinates as exemplified with Born coordinates of Minkowski spacetime. Is this view wrong? If yes, why?

Can you or @Dale give a reference you accept, where "reference frame" is properly defined?

It's really just to learn what you accept in order not to run always into this debate, when it comes to a discussion about reference frames in this forum.

I thought @Dale made that clear earlier: his preferred definition of "reference frame" is "tetrad field", i.e., a mapping of orthonormal tetrads to points in an open region of spacetime.

That's what I thought too, but then he says again that this contradicts my point of view on the physical interpretation of the formalism. Here is, how I understand it (I cannot point to any specific textbook, where this is formulated in this specific way, but I think it's standard; if not, please tell me):

You have a pseudo-Riemannian spacetime manifold with a pseudometric of signature (1,3) given. This is the spacetime model of GR. By definition in some region of spacetime you can define coordinates $q^{\mu}$, and these define in this region bases of the tangent $\mathrm{d} q^{\mu}$ and cotangent spaces, $\partial_{\mu}$, at any point of the manifold. This is not yet a reference frame.

To define one, one can indeed use the tetrad formalism, which I understand as follows (again given in physical terms): Start with a single observer, which is simply described by a time-like worldline. In the local coordinates given by $q^{\mu}=q^{\mu}(\lambda)$. At any point of his trajectory you can then use a tetrad as a basis of the tangent space, which consists of a unit timelike tangent vector (in the sense of the spacetime pseudometric) along the worldline and three spacelike pseudoorthogonal (in the sense of the spacetime pseudometric) unit vectors (which are also pseudoorthogonal to the timelike tangent vector).

The same you can do for an entire set of such local observers covering some part of the spacetime, defining a tetrade field and thus a field of local reference frames.

Another more elementary way to define such local reference frames is given in Landau Lifshitz vol. 2 or in the following nice AJP article:

https://doi.org/10.1119/1.1607338

 

[martinbn]

Can you or @Dale give a reference you accept, where "reference frame" is properly defined?

Sachs and Wu "General Relativity for Mathematicians" has a definition and is a reliable source.

But why can't you accept that people in different situations use different definitions?

 

[vanhees71]

I can easily accept that, but my problem is that they never give a clear definition to begin with. I'll see whether, I can check the mentioned book as soon as possible.

 

[Dale]

Please give a clear definition of what a reference frame is in your definition

A reference frame is a coordinate system or a tetrad.

I have not been at all unclear about my definition at any point.

2) doesn't make any sense to me

You honestly think that it doesn’t make sense to claim that a reference frame doesn’t have mass? Meaning not only does a reference frame have mass but it is so obviously massive that it doesn’t even make sense to question it? Really? Please clarify your position on the mass of a reference frame.

I could agree with 1), if I'd know you use the standard meaning, which defines the reference frame of an arbitrarily moving ("point-like") reference body.

I use the standard meaning of “coordinate system” and “tetrad”. I also use the standard meanings of “is”, “a”, and “or”.

my problem is that they never give a clear definition to begin with

Total BS

 

[vanhees71]

So do you agree to my statements in #43 or not? Your statement is far from being clear let alone a clear definition of what you accept a frame of reference to be.

As is clear from my statements in #43, a reference point of a reference frame in the there given sense, it's clear that it must be realized by a massive body, because it must have a timelikd worldline.

 

[Dale]

Your statement is far from being clear

Nonsense.

it's clear that it must be realized by a massive body, because it must have a timelikd worldline.

OK, so you disagree with “a reference frame has no mass”.

Do you also disagree with “a reference frame is a coordinate system or a tetrad”? With both coordinate system and tetrad having their standard definitions. In other words, do we have one hearty disagreement or two?

That's what I thought too, but then he says again that this contradicts my point of view on the physical interpretation of the formalism.

Regarding your statements in 43, this started because you “heartily disagree” disagree with either or both of the statements: 1) a reference frame is a coordinate system or a tetrad or 2) a reference frame has no mass. You are the one who asserts a contradiction between your view and my statements.

Regarding your definition of a tetrad, it is overly involved. A tetrad is a set of four orthonormal vector fields on spacetime, one timelike and three spacelike. That is it. An observer is not a necessary part of the definition of a generic tetrad.

Of course, we often speak of “the observer’s tetrad”. And clearly “the observer” is an essential part of the definition of “the observer’s tetrad”. But not all tetrads are some observers tetrad and so an observer is not part of the definition of the generic concept of a tetrad.

 

[Sagittarius A-Star]

... a reference point of a reference frame in the there given sense, it's clear that it must be realized by a massive body, because it must have a timelikd worldline.

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.

 

[Dale]

I think, a reference frame is an abstract mathematical object, to which the calculations refer to, independent of the existence of physical objects.

Exactly

Einstein wrote about moving clocks and rulers to visualize the movement of coordinate systems relative to each other.

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.

 

[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").