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

[Aeronautic Freek]

Is concept of inertial vs non inertial frame inveted in Einsten theory of relativity or Newton knows that we can see on same object from different perspective?
(Newton set 3 laws for inertial frame,so did he knew for realitivtiy when view object form different perspective/frame and did he use/knew fake forces,centifugal,coriolis etc..?)

 

[anuttarasammyak]

Newton's bucket episode
https://en.wikipedia.org/wiki/Bucket_argument
shows that he noticed centrifugal forces.

 

[vanhees71]

The concept of a (global) inertial frame applies to both special relativity and Newtonian physics. In fact you can mathematically prove that any theory that obeys the special principle of relativity and has an Euclidean space for any inertial observer must either be Newtonian or special-relativistic physics with the corresponding spacetime descriptions as Galilei-Newton spacetime (a fiber bundle) or an Einstein-Minkowski spacetime (a pseudo-Euclidean affine space).

Newton was well aware of the fact that acceleration of a non-inertial reference frame against the class of inertial refeference frames is observable. He was, however, pretty much worried about how to physically determine an inertial reference frame. This was a debate among physicists until the 20th century. Particularly Ernst Mach made a proposal with his famous principle how to determine Newton's "absolute space".

As it is understood today, the most comprehensive spacetime model is the pseudo-Riemannian spacetime manifold introduced by Einstein in his theory of general relativity (or maybe a Cartan spacetime to make it compatible with spin). Within GR there are no more any preferred frames of reference, but it is always possible to choose a reference frame that is locally inertial. It is constructed by the choice of a non-rotating Vierbein fixed at a free falling test mass (e.g., the ISS is a good approximation of such a frame).

 

[Dale]

Is concept of inertial vs non inertial frame inveted in Einsten theory of relativity or Newton knows that we can see on same object from different perspective?

I don’t know the history too well, but I believe that Newton was aware of some of the issues but didn’t resolve them himself. I think the concepts were greatly clarified and formalized by other researchers between Newton and Einstein. So by the time Einstein arrived he had access to much better theoretical tools than Newton did.

 

[pervect]

Is concept of inertial vs non inertial frame inveted in Einsten theory of relativity or Newton knows that we can see on same object from different perspective?
(Newton set 3 laws for inertial frame,so did he knew for realitivtiy when view object form different perspective/frame and did he use/knew fake forces,centifugal,coriolis etc..?)

I'm assuming "inveted" means "invented", and not, say, "inverted". Clarify is necessary.

Newtonian mechanics has the notion of non-inertial frames, though I"m not sure historically of when this was realized or who did the work.

I can give a bit of a modern perspective, though. The modern perspective is that for the purposes of Newtonian mechanics, we simply postulate (assume) an inertial frame exists. We don't attempt to justify this in any way physically, we just assume it is true, and use logic to explore the consequences so we can compare our predictions to experiment. Starting with these assumptions, we develop the mathematical machinery, often called "general covairance", that allows us to go from the laws of physics as expressed in one frame (or more generally, set of coordinates) to arbitrary coordinates.

I'm not quite sure of the history of the development of general covariance, to be honest. I suspect Lagrangian mechanics , if you're familiar, was a step in that direction. It was certainly a strong influence on Einstein, though his approach to it differed from the modern approach. Perhaps someone else can give you more details about this - the notion of covariance seems to be at the heart of your question.

There is one important result though that you may not be aware of. In Newtonian mechanics, general covariance (and/or it's earlier predecessors) gives the result that an accelerated frame of reference appears to be basically Newtonian, but with the addition of "fictitious forces". This is no longer true in Special relativity. An accelerated frame has some noticable effect on the behavior of clocks - pseudogravitational time dilation, that are not just a matter of adding in "fictitious forces". This was influential in Einstein's work on incoporating gravity into special relativity, which eventually lead (after many many years of work on his part) to general relativity.

 

[vanhees71]

There is of course no problem to introduce non-inertial reference frames in special relativity. Usually they are only local maps, not covering the entire spacetime. Examples often found in the literature are the "Rindler coordinates" referring to an observer in hyperbolic motion relative to an inertial reference frame or a rotating reference frame. Both sets of coordinates cover only a part of Minkowski spacetime of course.

 

[PeterDonis]

Both sets of coordinates cover only a part of Minkowski spacetime of course.

This is true of Rindler coordinates, but is it true of rotating coordinates such as Langevin coordinates? AFAIK those cover all of Minkowski spacetime; they just don't have a timelike "time" coordinate in all of Minkowski spacetime.

 

[vanhees71]

Interesting. Do you have a definition of these coordinates? I Thought langevin observers move on helical curves, and the map covers only the part of space time with $\omega R<c$.

 

[PeterDonis]

Interesting. Do you have a definition of these coordinates? I Thought langevin observers move on helical curves, and the map covers only the part of space time with $\omega R<c$.

Do you mean the Born chart, as shown in this Wikipedia article?

https://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart

If so, while the article does give limits of $0 < r < 1 / \omega$ for the radial coordinate (it's using units in which $c = 1$), I'm not sure that's actually required for it to be a valid chart. As far as I can tell, the chart is still one-to-one for $r \ge 1 / \omega$; the $t$ coordinate is simply no longer timelike there, so the frame field that is given for the Langevin observers is not a valid frame field there (since it no longer has a timelike vector).

 

[vanhees71]

Yes, that was what I had in mind.

 

[Dale]

This is true of Rindler coordinates, but is it true of rotating coordinates such as Langevin coordinates? AFAIK those cover all of Minkowski spacetime; they just don't have a timelike "time" coordinate in all of Minkowski spacetime.

I think that depends on whether by “reference frame” you mean “coordinate chart” or if you mean “tetrad”/“frame field”. The coordinate chart covers all of Minkowski spacetime, but the tetrad cannot

 

[PeterDonis]

I think that depends on whether by “reference frame” you mean “coordinate chart” or if you mean “tetrad”/“frame field”.

Yes, this is a fair point, but note that the specific quote from @vanhees71 that I originally responded to said "coordinates".

 

[vanhees71]

Now I'm confused. I hope we'll not have again a debate about reference frames, but mathematically the Born coordinates provide a chart which covers only a part of Minkowski space. In the notation of Wikipedia the pseudometric reads
$$\mathrm{d}s^2 = -(1-\omega^2 r^2) \mathrm{d} t^2 + 2 \omega r^2 \mathrm{d} t \mathrm{d} \varphi + \mathrm{d} z^2 + r^2 \mathrm{d} \varphi^2.$$
The ranges of the coordinates are $t,z \in \mathbb{r}$, $0<r<1/\omega$, $\phi \in [-\pi,\pi]$.

I use the usual physicists' sloppy slang and call these coordinates.

 

[Dale]

Now I'm confused. I hope we'll not have again a debate about reference frames, but mathematically the Born coordinates provide a chart which covers only a part of Minkowski space. In the notation of Wikipedia the pseudometric reads
$$\mathrm{d}s^2 = -(1-\omega^2 r^2) \mathrm{d} t^2 + 2 \omega r^2 \mathrm{d} t \mathrm{d} \varphi + \mathrm{d} z^2 + r^2 \mathrm{d} \varphi^2.$$
The ranges of the coordinates are $t,z \in \mathbb{r}$, $0<r<1/\omega$, $\phi \in [-\pi,\pi]$.

I use the usual physicists' sloppy slang and call these coordinates.

It is a perfectly valid coordinate chart for $0 < r < \infty$

 

[vanhees71]

Well, this is not the standard point of view, because obviously $\mathrm{det} g=0$ for $\omega r=1$. I think this is again a matter of semantics. I just refer to a chart in Minkowski space, which must be compatible with the entire structure of the manifold, including the fundamental form. In regular points of the map its components must be a matrix with signature $(3,1)$ which is not the case for $\omega r>1$.

[EDIT: Wikipedia is of the same opinion, i.e., it's a chart covering not the entire Minkowski space]

https://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart

 

[Dale]

That isn’t a requirement for a coordinate chart. All that is required for a coordinate chart is that the chart is a smooth and invertible mapping between an open set of events in the manifold and an open set of points in R4. The metric is not relevant. See p 37 at: https://arxiv.org/abs/gr-qc/9712019

There are many valid coordinate charts which do not have 3 spacelike and 1 timelike coordinate basis vectors.

 

[vanhees71]

What you discribe is a coordinate chart for a differentiable manifold without additional structure. Minkowski space is a pseudo-Euclidean affine manifold and thus you want coordinate charts which are compatible with this structure, but I guess that's again a fruitless discussion about pure semantics rather than any physics or math content. So let's stop here...

 

[Dale]

What you discribe is a coordinate chart for a differentiable manifold without additional structure.

Yes, the additional structure is not necessary for defining coordinate charts.

Btw, I take Carroll over wikipedia. And regarding Born, if he proposed the same restriction as Wikipedia, then it is important to recognize that it is fine to define a chart as being any open subset of another chart. So the restriction on the range of $r$ is valid, but not necessary.

 

[DrGreg]

... obviously $\mathrm{det} g=0$ for $\omega r=1$.

I think you have forgotten the off-diagonal term. By my reckoning$$
\mathrm{det} \, g = \begin{vmatrix}
-(1 - \omega^2 r^2) & 0 & \omega r^2 & 0 \\
0 & 1 & 0 & 0 \\
\omega r^2 & 0 & r^2 & 0 \\
0 & 0 & 0 & 1
\end{vmatrix} = \begin{vmatrix}
-(1 - \omega^2 r^2) & \omega r^2 \\
\omega r^2 & r^2
\end{vmatrix} = -r^2 \, ,
$$ strictly negative except for a coordinate singularity at $r = 0$.

 

[Dale]

except for a coordinate singularity at r=0.

Good point. I had forgotten that. So indeed it doesn’t cover the whole manifold. I believe it cannot cover not just the point at $r=0$ but also $\phi = \pi$. But large $r$ is fine

 

[vanhees71]

Argh. Yes you are right. Calculating the eigenvalues with Mathematica also shows that $g$ has the right signature everywhere, except at $r=0$ (which is just the usual coordinate singularity of spatial cylinder coordinates). So indeed one has a chart covering almost all of Minkowski space.

Now I'm puzzled. How is then the standard result that the non-inertial reference frame of an observer rotating with constant velocity around an axis with distance $r$ to be explained? The Wikipedia article seems correct to me

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

 

[Dale]

How is then the standard result that the non-inertial reference frame of an observer rotating with constant velocity around an axis with distance r to be explained?

What specifically about it are you wanting to explain?

 

[vanhees71]

The fact that it is usually said that the Born coordinates only cover the part of spacetime given by $\omega r<1$.

 

[DrGreg]

How is then the standard result that the non-inertial reference frame of an observer rotating with constant velocity around an axis with distance $r$ to be explained?

Even though it makes sense as a mathematical coordinate system for all $r > 0$, it has the physical meaning of representing observers on a rotating disk only for $r < 1/\omega$. Beyond that radius the $t$ coordinate is either null or spacelike.

 

[Dale]

The fact that it is usually said that the Born coordinates only cover the part of spacetime given by $\omega r<1$.

Probably it is because people usually confuse tetrads/frame-fields and coordinate charts. The corresponding tetrad covers only $0 \le \omega r < c$, but the coordinate chart is valid $0<\omega r < \infty$.

This is one situation where the distinction between a tetrad and a coordinate chart is important. As a result, the typical ambiguous usage of "reference frame" is problematic. Hence my comment in post #11.

 

[vanhees71]

Then we should once and for all define what a "reference frame" is. I don't dare to say it anymore, what I think it is after the last discussion when I was accused to spread "private opinions". Just so much: I think the standard textbook definition is right and the Born coordinates used in the usual sense to define a reference frame of a rotating observer only covers the part $0<\omega r<c$ of space time ($r=0$ is a coordinate singularity anyway).

 

[A.T.]

... once and for all define ...

Will never happen.

 

[martinbn]

Then we should once and for all define what a "reference frame" is. I don't dare to say it anymore, what I think it is after the last discussion when I was accused to spread "private opinions". Just so much: I think the standard textbook definition is right and the Born coordinates used in the usual sense to define a reference frame of a rotating observer only covers the part $0<\omega r<c$ of space time ($r=0$ is a coordinate singularity anyway).

To be fare you are the one who brought it up and you used the terms frame of reference and coordinates interchangeably without saying what you mean by a frame.

 

[Dale]

Then we should once and for all define what a "reference frame" is.

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.

I think the standard textbook definition is right

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

the Born coordinates used in the usual sense to define a reference frame of a rotating observer only covers the part 0<ωr<c of space time (r=0 is a coordinate singularity anyway).

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.

 

[anuttarasammyak]

I think the standard textbook definition is right and the Born coordinates used in the usual sense to define a reference frame of a rotating observer only covers the part $0<\omega r<c$ of space time ($r=0$ is a coordinate singularity anyway).

When the requirement "the coordinates must be realized by REAL body" is applied, we should be limited in that region. Beyond that region no rods and points can stay still to represent r axis.

But this requirement is not shared in general. IMAGINALY r-axis could extend to infinity. For a daily example we, on the spinning Earth frame, observe stars which lie far beyond that region at r ,($\theta$,$\phi$) or in such and such Zodiacs. They rotate with period of 24 hours. No stars can stay still.

 

[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.