Whether a non-inertial frame is absolute

In summary, according to the speaker, proper acceleration is invariant under a general Galilean boost. This means that inertial frames are always inertial, and that the magnitude of the acceleration is relative to some frame.
  • #1
feynman1
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If a frame is a non inertial frame, then it must have an acceleration. Then which reference frame is this acceleration with respect to? If this acceleration varies with the reference frame this acceleration is calculated with respect to, is this non inertial frame absolute?
 
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  • #2
This acceleration is most naturally quantified with respect to inertial frames. Exercise, for you: prove that the acceleration of a particle is invariant under a general Galilean boost.
 
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  • #3
feynman1 said:
Then which reference frame is this acceleration with respect to?
Proper acceleration can be measured by a spring balance with no reference to any frame. A reference frame is inertial or not depending on whether objects defined to be at rest in the frame measure proper acceleration.
 
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  • #4
feynman1 said:
Then which reference frame is this acceleration with respect to?
The proper acceleration of an observer is an invariant. It is not "with respect to" anything.

The acceleration of a non-inertial frame is given as the proper acceleration of observers at rest throughout the non-inertial frame. Therefore, it is also not "with respect to" anything. It is invariant, and depending on the details of the specific non-inertial frame, it can vary from location to location throughout the frame.

Edit: thus also an inertial frame is inertial in an invariant sense and not "with respect to" anything, since inertial means roughly that the proper acceleration of a stationary object is 0 everywhere
 
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  • #5
Here is a question. How is the 'proper acceleration' of a point in classical mechanics to be defined, if not as the acceleration of that point with respect to an inertial co-ordinate chart (one of a class of co-ordinate charts defined by the restriction that a Galilean transformation of the co-ordinate space applied to all points in a mechanical system gives world-lines of the same system except with new initial conditions)?
 
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  • #6
etotheipi said:
Here is a question. How is the 'proper acceleration' of a point in classical mechanics to be defined, if not as the acceleration of that point with respect to an inertial co-ordinate chart (one of a class of co-ordinate charts defined by the restriction that a Galilean transformation of the co-ordinate space applied to all points in a mechanical system gives world-lines of the same system except with new initial conditions)?
I prefer to define it as the quantity that is measured by an accelerometer. But the two definitions are equivalent.
 
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  • #7
etotheipi said:
This acceleration is most naturally quantified with respect to inertial frames. Exercise, for you: prove that the acceleration of a particle is invariant under a general Galilean boost.
If This acceleration is most naturally quantified with respect to inertial frames, then are inertial frames absolute and are they always inertial?
 
  • #8
I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?
 
  • #9
feynman1 said:
If This acceleration is most naturally quantified with respect to inertial frames, then are inertial frames absolute and are they always inertial?
Um, yes, inertial frames are always inertial.

feynman1 said:
I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?
Not all kinematic quantities are relative. Proper time is not, nor is proper acceleration.

In fact, it is necessary that proper acceleration be invariant. Otherwise you could not unambiguously define an inertial frame.
 
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  • #10
Dale said:
Um, yes, inertial frames are always inertial. Rainy afternoons are always rainy. And magical fairies are always magical.
Then this reference frame has an acceleration=0, then how is this acceleration measured, is it measured with respect to a reference frame?
 
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  • #11
feynman1 said:
Then this reference frame has an acceleration=0, then how is this acceleration measured, is it measured with respect to a reference frame?
It is measured with an accelerometer. As I said above.
 
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  • #12
Dale said:
Um, yes, inertial frames are always inertial.

Not all kinematic quantities are relative. Proper time is not, nor is proper acceleration.

In fact, it is necessary that proper acceleration be invariant. Otherwise you could not unambiguously define an inertial frame.
Okay thanks but can we speak only Newtonian mechanics and forget about special relativity?
 
  • #13
feynman1 said:
Okay thanks but can we speak only Newtonian mechanics and forget about special relativity?
No, I don’t think so. Newton didn’t have a very clear concept of reference frames at all, and some of his key ideas on the matter were wrong. The modern clarity on reference frames was largely due to relativity forcing us to confront and fix some Newtonian errors.

In any case, Newtonian physics has more invariants, not more relative quantities
 
  • #14
Dale said:
No, I don’t think so. Newton didn’t have a very clear concept of reference frames at all, and some of his key ideas on the matter were wrong. The modern clarity on reference frames was largely due to relativity forcing us to confront and fix some Newtonian errors.

In any case, Newtonian physics has more invariants, not more relative quantities
Got it thanks. So is my question unanswerable at all within the Newtonian framework?
 
  • #15
The question is quite answerable completely within the classical framework. Did you try to prove the invariance of acceleration under a general Galilean transformation, as in post #2?
 
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  • #16
feynman1 said:
Got it thanks. So is my question unanswerable at all within the Newtonian framework?
Which of your many questions are you specifically referring to?
 
  • #17
Dale said:
Which of your many questions are you specifically referring to?
the original post
 
  • #18
etotheipi said:
The question is quite answerable completely within the classical framework. Did you try to prove the invariance of acceleration under a general Galilean transformation, as in post #2?
As per the galilean transformation, two systems need to move relative to each other with a constant speed. Then what if there’s a relative acceleration between them and how to apply the galilean transformation?
 
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  • #19
feynman1 said:
the original post
I already answered it in post 4, which is valid in Newtonian physics too
 
  • #20
Dale said:
I already answered it in post 4, which is valid in Newtonian physics too
Is proper acceleration even defined in classical physics/introductory physics before relativity?
 
  • #21
feynman1 said:
As per the galilean transformation, two systems need to move relative to each other with a constant speed. Then what if there’s a relative acceleration between them and how to apply the galilean transformation?
The point is that in classical mechanics there are certain distinguished frames of reference, so-called inertial frames, where Newton's law of inertia holds (free particles move at constant velocities). A Galilean transformation maps from one inertial frame to a different inertial frame. (And of course a transformation between two frames undergoing non-zero relative acceleration is not Galilean...!)

Take, for instance, a particle with a trajectory ##\mathbf{x} = \mathbf{X}(t)## in one inertial frame. It's acceleration in this frame is nothing but ##\mathbf{a}(t) = \ddot{\mathbf{X}}(t)##. Under a Galilean transformation, for example to a second inertial frame moving at a constant velocity ##\mathbf{v}## with respect to the first and with initial displacement ##\mathbf{s}##, the trajectory in this new inertial frame is ##\mathbf{x}' = \mathbf{X}'(t) = \mathbf{X}(t) - \mathbf{v}t - \mathbf{s}##, which again implies ##\mathbf{a}'(t) = \ddot{\mathbf{X}}(t)##. The acceleration of the particle as measured by any inertial frame is then the same, i.e. ##\mathbf{a}(t) = \mathbf{a}'(t)##.

That's partly why inertial frames are distinguished, because the accelerations of particles as measured by any frame in that class are invariant. Whilst, by a suitable choice of non-inertial frame, you can make the acceleration whatever you want :smile:
 
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  • #22
feynman1 said:
Okay thanks but can we speak only Newtonian mechanics and forget about special relativity?
An odd complaint, since the person who dragged in SR was...you.

I count five answers to your original question. Clearly these answers don't satisfy you. It might help if you explained how and why they don't satisfy you. Otherwise we will keep going around in circles.
 
  • #23
feynman1 said:
Is proper acceleration even defined in classical physics/introductory physics before relativity?
Proper acceleration is just the acceleration measured by an accelerometer. It is perfectly compatible with Newtonian physics.
 
  • #24
feynman1 said:
Aren't all kinematic quantities measured w.r.t a reference frame?
No.
 
  • #25
feynman1 said:
I don't understand why the acceleration can be invariant. Aren't all kinematic quantities measured w.r.t a reference frame? Then the magnitude of the acceleration should also be relative to some frame?
Proper acceleration can be thought of as relative to a local free falling frame. But this boils down to what an accelerometer measures, so you don't need to introduce that frame.
 
  • #26
feynman1 said:
Is proper acceleration even defined in classical physics/introductory physics before relativity?
Yes. It’s essential to understanding what’s going on with centripetal and centrifugal forces when you’re swinging a weight on a string in a circle, for example.

Introductory physics texts often simplify things by choosing coordinates in which the frame-dependent coordinate acceleration (which is “relative”) happens to be equal to the proper acceleration. This simplification is one of the things that you’ll unlearn when you come to a more complete treatment of the subject.
 
  • #27
Nugatory said:
Yes. It’s essential to understanding what’s going on with centripetal and centrifugal forces when you’re swinging a weight on a string in a circle.
How so? You can do all the analysis simply with the definition of acceleration with respect to a frame ##F## ##\mathbf{a} \big{|}_{F} = \frac{\mathrm{d^2} \boldsymbol{x}}{\mathrm{d}t^2} \big{|}_F## and if you're using a non-inertial frame then the additional relationship ##\frac{\mathrm{d} \mathbf{u}}{\mathrm{d}t} \big{|}_{Oxyz} = \frac{\mathrm{d} \mathbf{u}}{\mathrm{d}t} \big{|}_{O'x'y'z'} + \boldsymbol{\omega} \times \mathbf{u}## for any arbitrary vector ##\mathbf{u}##, which is required in order to obtain the equation of motion in the non-inertial frame (either by finding the Lagrangian in the rotating frame, or by deriving the relationship between ##\mathbf{a} \big{|}_{Oxyz}## and ##\mathbf{a} \big{|}_{O'x'y'z'}## and then substituting for ##\mathbf{a} \big{|}_{Oxyz}## in Newton's equation, where ##Oxyz## constitute inertial co-ordinates).

The point I'm making is that I haven't really noticed any classical mechanics textbooks use 'proper acceleration', and there isn't particularly a need to introduce it.
 
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  • #28
etotheipi said:
The point I'm making is that I haven't really noticed any classical mechanics textbooks use 'proper acceleration', and there isn't particularly a need to introduce it.
The term may not be used, but accelerometers are classical devices and their readings are perfectly compatible with classical mechanics. There is nothing inherently non-classical about proper acceleration, and borrowing the term aids greatly in clarity.
 
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  • #29
etotheipi said:
The point I'm making is that I haven't really noticed any classical mechanics textbooks use 'proper acceleration', and there isn't particularly a need to introduce it.
The ##\left.\mathbf{a}\right|_F## you defined is coordinate acceleration, and physical accelerometers will not measure this in general. So you clearly have two concepts here - coordinate acceleration and whatever it is the accelerometers are measuring. The latter is proper acceleration, whether it's called that or not, and I would say that it's actually the truly important concept because it's where the maths corresponds to something directly measurable. (Edit: there's a gag about using the proper terms for things here somewhere, but it'd add more confusion than it'd be worth.)
 
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  • #30
Ibix said:
The ##\left.\mathbf{a}\right|_F## you defined is coordinate acceleration, and physical accelerometers will not measure this in general. So you clearly have two concepts here - coordinate acceleration and whatever it is the accelerometers are measuring. The latter is proper acceleration, whether it's called that or not, and is actually the truly important concept because it's where the maths corresponds to something directly measurable.
But that can just be stated: let ##K## be any inertial reference system, then what you call the proper acceleration is what I would denote ##\mathbf{a} \big{|}_K## - accelerometers are a nice heuristic, but you can't really define an accelerometer mathematically. And I don't see the point of introducing extraneous terminology.

It's nicer to start from the principle of Galilean relativity, which implies a Galilean structure defined by the property that Newton's equation is invariant with respect to the group of Galilean transformations, and from that derive that a mechanical system of one free particle has zero acceleration in any inertial reference system in this class*. Then it's already clear that non-zero accelerations with respect to inertial frames have physical significance, and not least because the values of these accelerations are invariant under Galilean transformations.

Relativity is different, because there the four-acceleration ##a^{\mu} = \mathrm{d}^2 x^{\mu} / \mathrm{d}s^2## can be defined without even acknowledging any reference system, and is an entirely different quantity to co-ordinate acceleration. Whilst in classical mechanics, 'proper acceleration' is just a special case of co-ordinate acceleration!

* [i.e. use that in an inertial reference system, the acceleration ##\ddot{\boldsymbol{x}} = \mathbf{\mathcal{F}}(\boldsymbol{x}, \dot{\boldsymbol{x}}, t)## of a mechanical system can only depend on relative positions and velocities ##\ddot{\boldsymbol{x}} = \varphi( \{ \boldsymbol{x}_i - \boldsymbol{x}_j, \dot{\boldsymbol{x}}_i - \dot{\boldsymbol{x}}_j \})## for some ##\varphi##, and is also rotationally invariant ##\mathcal{F}(G\boldsymbol{x}, G\dot{\boldsymbol{x}}) = G\mathcal{F}(\boldsymbol{x}, \dot{\boldsymbol{x}})## any for orthogonal matrix ##G##]
 
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  • #31
etotheipi said:
Accelerometers are a nice heuristic, but you can't really define an accelerometer mathematically.
Why would you need to define it mathematically? We are doing science, not math.
 
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  • #32
Classical mechanics, especially, is very much a mathematical structure. There are many applied mathematics texts on the subject, e.g. Abraham/Marsden, Spivak, Arnold, et cetera. I'm not a mathematician myself, but I still don't think one should be using mythical objects such as accelerometers in definitions of physical quantities :wink:
 
  • #33
etotheipi said:
accelerometers are a nice heuristic, but you can't really define an accelerometer mathematically.
Then you have a problem. Ultimately, all the maths we do is supposed to support predictions like "if the dial on this machine reads ##x\pm \delta x## units then the dial on that machine will read ##y\pm \delta y## units". If it's not capable of doing that, what's it got to do with the real world?

Don't get me wrong - all of the mathematical abstraction is incredibly powerful and can lead to remarkable insights. But if it can't relate to direct observables then it's either unfinished or useless for physics.
 
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  • #34
etotheipi said:
I still don't think one should be using mythical objects such as accelerometers in definitions of physical quantities
I 100% disagree with this. In my opinion not only are operational definitions necessary for science, they are the most important ones.
 
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  • #35
Dale said:
Why would you need to define it mathematically?
Agreed. Why do you need to define it mathematically if you can just buy one?
 
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<h2>What is a non-inertial frame?</h2><p>A non-inertial frame is a reference frame in which Newton's laws of motion do not hold true. This can be due to the frame being accelerated or rotating.</p><h2>What is an inertial frame?</h2><p>An inertial frame is a reference frame in which Newton's laws of motion hold true. This means that objects in an inertial frame will remain at rest or in motion at a constant speed unless acted upon by an external force.</p><h2>Is a non-inertial frame absolute?</h2><p>No, a non-inertial frame is not absolute. The laws of physics, including the laws of motion, can vary depending on the frame of reference. In a non-inertial frame, these laws may appear to be different due to the acceleration or rotation of the frame.</p><h2>Can a non-inertial frame be used to accurately describe motion?</h2><p>Yes, a non-inertial frame can still be used to accurately describe motion. However, it may require additional equations or corrections to account for the non-inertial effects.</p><h2>How does the concept of a non-inertial frame relate to general relativity?</h2><p>The concept of a non-inertial frame is important in general relativity, as it is a theory of gravity that takes into account the effects of acceleration and rotation on the fabric of spacetime. In general relativity, all frames of reference are considered to be non-inertial due to the presence of gravity.</p>

What is a non-inertial frame?

A non-inertial frame is a reference frame in which Newton's laws of motion do not hold true. This can be due to the frame being accelerated or rotating.

What is an inertial frame?

An inertial frame is a reference frame in which Newton's laws of motion hold true. This means that objects in an inertial frame will remain at rest or in motion at a constant speed unless acted upon by an external force.

Is a non-inertial frame absolute?

No, a non-inertial frame is not absolute. The laws of physics, including the laws of motion, can vary depending on the frame of reference. In a non-inertial frame, these laws may appear to be different due to the acceleration or rotation of the frame.

Can a non-inertial frame be used to accurately describe motion?

Yes, a non-inertial frame can still be used to accurately describe motion. However, it may require additional equations or corrections to account for the non-inertial effects.

How does the concept of a non-inertial frame relate to general relativity?

The concept of a non-inertial frame is important in general relativity, as it is a theory of gravity that takes into account the effects of acceleration and rotation on the fabric of spacetime. In general relativity, all frames of reference are considered to be non-inertial due to the presence of gravity.

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