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

In summary, Newton was aware of the centrifugal forces and developed the theory of general covariance to allow for the use of any inertial reference frame.
  • #1
Aeronautic Freek
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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..?)
 
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  • #3
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).
 
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  • #4
Aeronautic Freek said:
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.
 
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  • #5
Aeronautic Freek said:
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.
 
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  • #6
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.
 
  • #7
vanhees71 said:
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.
 
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  • #8
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##.
 
  • #9
vanhees71 said:
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).
 
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  • #10
Yes, that was what I had in mind.
 
  • #11
PeterDonis said:
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
 
  • #12
Dale said:
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". :wink:
 
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  • #13
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.
 
  • #14
vanhees71 said:
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##
 
  • #15
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
 
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  • #16
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.
 
  • #17
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...
 
  • #18
vanhees71 said:
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.
 
  • #19
vanhees71 said:
... 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##.
 
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  • #20
DrGreg said:
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
 
  • #21
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
 
  • #22
vanhees71 said:
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?
 
  • #23
The fact that it is usually said that the Born coordinates only cover the part of spacetime given by ##\omega r<1##.
 
  • #24
vanhees71 said:
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.
 
  • #25
vanhees71 said:
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.
 
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  • #26
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).
 
  • #27
vanhees71 said:
... once and for all define ...
Will never happen.
 
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  • #28
vanhees71 said:
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.
 
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  • #29
vanhees71 said:
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.

vanhees71 said:
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.

vanhees71 said:
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.
 
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  • #30
vanhees71 said:
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.
 
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  • #31
Dale said:
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!
 
  • #32
vanhees71 said:
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.

vanhees71 said:
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.
 
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  • #33
vanhees71 said:
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.
 
  • #34
anuttarasammyak said:
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.
 
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  • #35
Dale said:
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.
 
<h2>1. What is the difference between inertial and non-inertial frames?</h2><p>Inertial frames are reference frames in which Newton's first law of motion holds true, meaning that an object at rest will remain at rest and an object in motion will continue to move at a constant velocity unless acted upon by an external force. Non-inertial frames, on the other hand, do not follow this law and may have forces acting on objects even when they appear to be at rest.</p><h2>2. How did Einstein's theory of relativity challenge Newton's concept of inertial frames?</h2><p>Einstein's theory of relativity introduced the concept of spacetime, where space and time are intertwined and not separate entities. This challenged Newton's idea of absolute space and time, and proposed that the laws of physics should be the same in all inertial frames, regardless of their relative motion.</p><h2>3. Can you give an example of a non-inertial frame?</h2><p>An example of a non-inertial frame is a rotating frame, such as a merry-go-round. On the merry-go-round, objects appear to be moving in a curved path, even though they are not being acted upon by an external force. This is because the frame itself is accelerating, causing objects to appear to move in a non-inertial manner.</p><h2>4. How do inertial and non-inertial frames affect our understanding of gravity?</h2><p>Inertial frames do not take into account the effects of gravity, while non-inertial frames do. Einstein's theory of relativity showed that gravity is not a force between masses, but rather a curvature of spacetime caused by the presence of mass. This means that the laws of physics in a non-inertial frame, where gravity is present, may appear different from those in an inertial frame.</p><h2>5. Which frame of reference is more accurate for describing the motion of objects in everyday life?</h2><p>In most cases, inertial frames are more accurate for describing the motion of objects in everyday life. This is because the effects of gravity are negligible in most situations and can be ignored. However, for objects moving at high speeds or in the presence of strong gravitational fields, non-inertial frames may be necessary for accurate predictions.</p>

1. What is the difference between inertial and non-inertial frames?

Inertial frames are reference frames in which Newton's first law of motion holds true, meaning that an object at rest will remain at rest and an object in motion will continue to move at a constant velocity unless acted upon by an external force. Non-inertial frames, on the other hand, do not follow this law and may have forces acting on objects even when they appear to be at rest.

2. How did Einstein's theory of relativity challenge Newton's concept of inertial frames?

Einstein's theory of relativity introduced the concept of spacetime, where space and time are intertwined and not separate entities. This challenged Newton's idea of absolute space and time, and proposed that the laws of physics should be the same in all inertial frames, regardless of their relative motion.

3. Can you give an example of a non-inertial frame?

An example of a non-inertial frame is a rotating frame, such as a merry-go-round. On the merry-go-round, objects appear to be moving in a curved path, even though they are not being acted upon by an external force. This is because the frame itself is accelerating, causing objects to appear to move in a non-inertial manner.

4. How do inertial and non-inertial frames affect our understanding of gravity?

Inertial frames do not take into account the effects of gravity, while non-inertial frames do. Einstein's theory of relativity showed that gravity is not a force between masses, but rather a curvature of spacetime caused by the presence of mass. This means that the laws of physics in a non-inertial frame, where gravity is present, may appear different from those in an inertial frame.

5. Which frame of reference is more accurate for describing the motion of objects in everyday life?

In most cases, inertial frames are more accurate for describing the motion of objects in everyday life. This is because the effects of gravity are negligible in most situations and can be ignored. However, for objects moving at high speeds or in the presence of strong gravitational fields, non-inertial frames may be necessary for accurate predictions.

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