Whether a non-inertial frame is absolute

[feynman1]

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?

 

[etotheipi]

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.

 

[Ibix]

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.

 

[Dale]

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

 

[etotheipi]

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)?

 

[Dale]

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.

 

[feynman1]

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?

 

[feynman1]

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?

 

[Dale]

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.

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.

 

[feynman1]

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?

 

[Dale]

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.

 

[feynman1]

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?

 

[Dale]

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

 

[feynman1]

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?

 

[etotheipi]

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?

 

[Dale]

Got it thanks. So is my question unanswerable at all within the Newtonian framework?

Which of your many questions are you specifically referring to?

 

[feynman1]

Which of your many questions are you specifically referring to?

the original post

 

[feynman1]

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?

 

[Dale]

the original post

I already answered it in post 4, which is valid in Newtonian physics too

 

[feynman1]

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?

 

[etotheipi]

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

 

[Vanadium 50]

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.

 

[Dale]

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.

 

[PeterDonis]

Aren't all kinematic quantities measured w.r.t a reference frame?

No.

 

[A.T.]

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.

 

[Nugatory]

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.

 

[etotheipi]

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.

 

[Dale]

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.

 

[Ibix]

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.)

 

[etotheipi]

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 $G$]

 

[Dale]

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.

 

[etotheipi]

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

 

[Ibix]

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.

 

[Dale]

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.

 

[Vanadium 50]

Why would you need to define it mathematically?

Agreed. Why do you need to define it mathematically if you can just buy one?

 

[etotheipi]

Mhm. This is turning into something philosophical and uninteresting, but I might as well say that in my opinion you are looking at things in the wrong way. Classical mechanics is fundamentally a mathematical structure, a study of certain differential equations, differential and symplectic geometry, et cetera.

In the context of this discussion, the mathematical concept is one of a Galilean co-ordinate chart $\varphi: A^4 \longrightarrow \mathbf{R} \times \mathbf{R}^3$, where $A^4 \cong \mathbf{R}^4$ is an four-dimensional affine space. In this chart one can describe motions of a system of $n = N/3$ particles by a function $\boldsymbol{x} : I \longrightarrow \mathbf{R}^N$ where $I \subseteq \mathbf{R}$. The acceleration of the system is nothing but the second derivative $\ddot{\boldsymbol{x}}(t_0) = \mathrm{d}^2 \boldsymbol{x} / \mathrm{d} t^2 \big{|}_{t_0}$. And finally Newton's principle of determinacy ensures the existence of a unique function $\mathbf{F} : \mathbf{R}^{N} \times \mathbf{R}^N \times \mathbf{R} \longrightarrow \mathbf{R}^N$ such that $\ddot{\boldsymbol{x}} = \mathbf{F}(\boldsymbol{x}, \dot{\boldsymbol{x}}, t)$.

The above describes a purely mathematical and self-contained framework. In order to make physical predictions we must somehow map from the mathematical model to the real world; this is the point at which one implements operational procedures. One uses rulers and clocks to realize position and time respectively. One realizes an inertial frame by ensuring that an isolated particle advances at a constant rate in a fixed direction. And through experiment one determines the form of the function $\mathbf{F}$ which reproduces the motions observed in the real word.

For example, for a suitable range of extensions the length of a vertical spring changes is observed to change in proportion to the mass attached to the end; this constitutes an operational definition of force. One then lays the spring horizontally on an ice-rink, and pulls the spring in such a way that its extension is always constant. By calculating the rate at which the velocity of the mass increases the acceleration may be deduced, and by measuring the extension of the spring the force exerted on the mass may also be deduced. It is then confirmed through repeated experiments that, to good accuracy, their ratio is constant and Newton's equation holds good for $\mathbf{F} = \text{constant vector}$.

In short; an accelerometer is something physical which is described by the theory, but must not be part of the theory itself.

 

[PeterDonis]

an accelerometer is something physical which is described by the theory

Which means the theory has to be capable of predicting what the accelerometer will read; which means there has to be some theoretical entity that corresponds to that prediction. So call whatever that theoretical entity is in classical (non-relativistic) mechanics "proper acceleration" and you're done.

 

[etotheipi]

Which means the theory has to be capable of predicting what the accelerometer will read; which means there has to be some theoretical entity that corresponds to that prediction. So call whatever that theoretical entity is in classical (non-relativistic) mechanics "proper acceleration" and you're done.

Ah, but that theoretical entity corresponding to it via the inverse physical map, which I'll light-heartedly call $\mathcal{P}^{-1}$ for fun, is precisely $\mathbf{a} \big{|}_K$! It is nothing but the acceleration with respect to an inertial reference system.

 

[PeterDonis]

that theoretical entity corresponding to it via the inverse physical map, which I'll light-heartedly call $\mathcal{P}^{-1}$ for fun, is precisely $\mathbf{a} \big{|}_K$! It is nothing but the acceleration with respect to an inertial reference system.

Ok, then call that "proper acceleration". What's the problem?

 

[etotheipi]

Ok, then call that "proper acceleration". What's the problem?

That's exactly it - what others here are referring to as 'proper acceleration' is nothing but a special case of co-ordinate acceleration in classical mechanics (as opposed, however, to relativistic theories). There's no need for this extra terminology:

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.

 

[PeterDonis]

There's no need for this extra terminology

I can see at least two good reasons to have it:

(1) "Proper acceleration" is shorter and easier to use than "coordinate acceleration in an inertial frame".

(2) It makes use of the fact that classical mechanics is an approximation to relativistic mechanics, and helps to illustrate an important connection between them.

 

[PeterDonis]

What others here are referring to as 'proper acceleration' is nothing but a special case of co-ordinate acceleration in classical mechanics

Actually, there is one important case where this is not true: in classical mechanics (as opposed to relativity), gravity is a force. So in classical mechanics, the acceleration of a rock dropped from a height above the surface of the Earth is coordinate acceleration in an inertial frame (because the frame in which the surface of the Earth is at rest is an inertial frame), so it would qualify as "proper acceleration" under the correspondence suggested in posts #38 and #39. But in relativity, it isn't.

And note that in this case, the mathematical entity in the theory that corresponds to the actual accelerometer reading is not $\mathbf{a} \big{|}_K$, because that is nonzero, but the accelerometer reading for an accelerometer attached to the rock is zero.

 

[vanhees71]

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

Measurement devices are not mythical, but the real-world equipment we use to observe the world quantitatively and only this operational definition of observables enables us to relate the mathematical descriptions of our beloved theories with the observed phenomena in Nature.

Nowadays most of us have pretty precise accelerometers with us since any Smart or iPhone contains accelerometers, and my colleagues from the physics-didactics department (and many more worldwide) use it to develop nice experiments directly illustrating these sometimes confusing subjects about inertial (global in Newtonian and special-relativistic physics, local in general relativistic physics) frames of reference and how the physical laws look in non-inertial reference frames.

 

[etotheipi]

Actually, there is one important case where this is not true: in classical mechanics (as opposed to relativity), gravity is a force. So in classical mechanics, the acceleration of a rock dropped from a height above the surface of the Earth is proper acceleration (because the frame in which the surface of the Earth is at rest is an inertial frame). But in relativity, it isn't.

And note that in this case, the mathematical entity in the theory that corresponds to the actual accelerometer reading is not $\mathbf{a} \big{|}_K$, because that is nonzero, but the accelerometer reading for an accelerometer attached to the rock is zero.

Right, sure, but everything I've been saying in this thread is justified by the assumption of sticking well within the realm of classical physics, so here gravity well and truly is a force $- \nabla \varphi$

 

[etotheipi]

Having just said that I was restricting myself to classical physics, here's another example from special relativity: what is time?

In some instances it is a scalar field $t : U \longrightarrow \mathbf{R}$ defined on the manifold $M$ where $U \subseteq M$ is some open subset of $M$. In other instances it is a parameter of a curve $\gamma: I \longrightarrow M$ taking values in some interval $I \subseteq \mathbf{R}$.

It's often also said that "time is what a clock measures". For similar reasons to above, to me this is not really a satisfactory definition, at least. It's much better in my opinion to go the other way and define an ideal clock as a set of events $\{ E_i \}$ along a timelike worldline such that the interval of the worldline parameter between two tick events is $\tau(E_{i+N}) - \tau(E_{i}) = \kappa N$ where $\kappa$ is some constant. Realisation of that physically amounts to having a physical object, of negligible spatial size, whose tick events along its worldline more or less approximate the behaviour of this ideal clock defined above.

But the clock, as a physical device, really ought not to factor into the definition of time!

 

[PeterDonis]

here's another example from special relativity: what is time?

There is no single definition for the ordinary language word "time" in relativity. The two most common meanings are "coordinate time" (which requires choosing a coordinate chart, and which at least strongly implies that the coordinate you are labeling as "time" is timelike) and "proper time" (which requires choosing a timelike curve). Your "scalar field" and "curve parameter" definitions correspond to these two cases.

It's often also said that "time is what a clock measures".

Which is true for the "proper time" (curve parameter) definition: the clock measures proper time along its worldline.

the clock, as a physical device, really ought not to factor into the definition of time!

It doesn't; "time is what a clock measures" is what defines a clock, just as you suggest, not what defines time.

 

[atyy]

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?

@etotheipi gave the answer in post #2. Acceleration is measured with respect to an inertial frame, but you can show that the answer is the same in all inertial frames.

There are important distinctions between accelaration in Newtonian mechanics and proper acceleration of relativistic physics that @PeterDonis gives in post #42.

 

[Dale]

'proper acceleration' is nothing but a special case of co-ordinate acceleration in classical mechanics

Emphasis added.

It is very often useful to have special terms for special cases.

Classical mechanics is fundamentally a mathematical structure, a study of certain differential equations, differential and symplectic geometry, et cetera.

No. Classical mechanics is fundamentally a scientific theory that uses those mathematical structures to make accurate predictions of classical physics experiments. You cannot remove the connection to experiment and still claim to be doing classical mechanics. Hence operational definitions are essential to the theory. Those operational definitions are precisely what make it classical mechanics instead of just symplectic geometry etc.

 

[Dale]

in classical mechanics (as opposed to relativity), gravity is a force. So in classical mechanics, the acceleration of a rock dropped from a height above the surface of the Earth is coordinate acceleration in an inertial frame (because the frame in which the surface of the Earth is at rest is an inertial frame), so it would qualify as "proper acceleration" under the correspondence suggested in posts #38 and #39. But in relativity, it isn't.

Of course, there is more than one way to formulate classical mechanics. With the Newton Cartan formulation of Newtonian gravity you get the relativistic definitions of inertial frames, the equivalence principle, and you can consider gravity to be a fictitious force in a local inertial frame. It is a little cumbersome to actually use, but it is nice to know that these specific good features of GR are “backwards compatible”.

 

[atyy]

https://dspace.mit.edu/bitstream/handle/1721.1/60691/16-07-fall-2004/contents/lecture-notes/d14.pdf
Here is a treatment of the accelerometer in Newtonian mechanics.

"The quantity (a−g) is called the specific force and is the actual measurement that we will obtain from the accelerometer."

"The concept is that, since the specific force (a−g) is the only quantity that can be measured, gravity and inertial acceleration cannot be distinguished by measurement only"