What is an inertial frame? A conflict of two definitions

[Trying2Learn]

Good Morning

(I see this has been discussed here, but I am more interested in two specific definitions and whether the conflict.)

1. I had always thought that the definition of an inertial frame was: "A frame in which Newton's Laws are valid."
2. A person has been arguing with me that the definition is: "one on which no forces act."

I am not entirely sure I agree with this second definition. On the one hand, it does sort of derive from the first, but is it an "if and only if" situation?

(I would like to exclude from this the case where there ARE forces acting, but they are equal and opposite so that the effective force is zero, while still being forces.)

For example, I could have a frame where forces DO act, but they are sufficiently negligible so that I can approximate the frame as inertial. But I am not entirely sure that invalidates the second definition

Could someone please help me out?

 

[A.T.]

Summary:: Define an inertial frame

(I see this has been discussed here, but I am more interested in two specific definitions and whether the conflict.)

1. I had always thought that the definition of an inertial frame was: "A frame in which Newton's Laws are valid."
2. A person has been arguing with me that the definition is: "one on which no forces act."

Forces cannot act on a reference frame, so definition 2 makes no sense.

 

[anuttarasammyak]

I had always thought that the definition of an inertial frame was: "A frame in which Newton's Laws are valid."

Not all the three laws in all but only the first one, the law of inertial motion.

At first we admit there exist FREE BODIES. IFR is the frame of reference where free bodies move at a constant speed including zero speed case (at rest). We find a IFR itself is moving at a constant speed in another IFR. Force is defined as generator of acceleration in IFR, that is Newton's second law of motion.

 

[Trying2Learn]

Not all the three laws in all but only the first one, the law of inertial motion.

At first we admit there exist FREE BODIES. IFR is the frame of reference where free bodies move at a constant speed including zero speed case (at rest). We find a IFR itself is moving at a constant speed in another IFR. Force is defined as generator of acceleration in IFR, that is Newton's second law of motion.

Thank you, and to A.T.

For A.T.: I think this person would suggest there is an isomorphic map between a body and the frame layered upon it (I am trying to put myself in their mind).

However, while anuttarasammyak gets to teh core of the issue, I would actually say this as an example.

Consider a Merry-Go-Round operating a full speed (no longer any driving moments to increase the angular velocity; ignoring gravity). On such a device, Newton's F=ma does not hold due to the introduction of terms historically deemed "fictitious." So, here is a frame with no forces acting on it, yet one which is not inertial.

Finally for anuttarasammyak . Why do you harp on the first law and not the second. I can understand your referencing the first law, but the actual operation involves the second law. In fact, as I read your response, I almost want to remove the first law of motion and elevate to a definition. Could you share your thoughts?

 

[anuttarasammyak]

Finally for @anuttarasammyak . Why do you harp on the first law and not the second. I can understand your referencing the first law, but the actual operation involves the second law. In fact, as I read your response, I almost want to remove the first law of motion and elevate to a definition. Could you share your thoughts?

The second law is, wiki says for an example,
------------------
Second law: In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma.
-----------------
You see force is defined by acceleration in IFR, NOT the reverse, i.e.
IFR is NOT defined by (zero) force on a body.

IFR is defined by force, force is defined by IFR, IFR is defined by force, force is ..., it is a chicken or egg sequence.

 

[Ibix]

For A.T.: I think this person would suggest there is an isomorphic ma

A single object does not define a frame, not without auxiliary assumptions. What you could say is that an inertial frame is one in which any object whose spatial coordinates are independent of time is one on which no net force acts. That disposes of the rotating frame attached to your merry-go-round - just put a ball on top of it and watch its coordinates change.

 

[Trying2Learn]

Thank you everyone!

 

[Ibix]

What you could say is that an inertial frame is one in which any object whose spatial coordinates are independent of time is one on which no net force acts.

That isn't enough, actually. You need to insist on the more general rule that any object with constant coordinate speed be subject to no net force. Otherwise you could define an anisotropic coordinate grid and things moving at constant $d\vec x/dt$ would need to be accelerated.

 

[jbriggs444]

Not all the three laws in all but only the first one, the law of inertial motion.

I do not agree with this. If we restrict our attention to the first law only then rotating frames are inertial.

The only object subject to zero net force is one continuously at rest at the center. It is subject to no net force and remains at rest. The first law is upheld for that object and does not apply to any other.

The third law is useful to exclude such frames from consideration.

 

[Dale]

Summary:: Define an inertial frame

I am not entirely sure I agree with this second definition.

I heartily disagree with it. A reference frame is a mathematical construct, either a coordinate system or a tetrad field depending on the context. Either way, it has no mass and therefore never has any forces acting on it.

 

[anuttarasammyak]

If we restrict our attention to the first law only then rotating frames are inertial.

Ref.　Wiki
-------------------
First law　In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.[2][3]
------------------

You are right if "an object remains at rest or.. " here is meaning " there exist at least one object that remains at rest or ... ". I would like to take it as "any object at rest or ... ".

 

[jbriggs444]

Ref.　Wiki
-------------------
First law　In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.[2][3]
------------------

You are right if "an object remains at rest or.. " here is meaning " there exist at least one object that remains at rest or ... ". I would like to take it as "any object at rest or ... ".

In a rotating frame, the First law as stated above holds. The relevant clause is "unless acted upon by a force". In a uniformly rotating frame there are Coriolis and centrifugal forces. The only object not acted upon by a force is an object stationary in the center. That object remains at rest.

The third law is not upheld. The Coriolis and centrifugal forces have no third law partner.

 

[anuttarasammyak]

The relevant clause is "unless acted upon by a force".

I would like to take "unless acted upon by a force" here as "free motion" or "motion of no constraint" without using the word "force" that also appears in the 2nd law after the definition of IFR. Mach and other scientists have made serious thoughts about what is force that I cannot follow fully.

In a rotating system particles of free motion do not move with constant speeds, except it stay at rest at center, so we know that rotating system is not a IFR.

 

[jbriggs444]

I would like to take "unless acted upon by a force" here as "free motion" or "motion of no constraint" without using the word "force" that also appears in the 2nd law after the definition of IFR. Mach and other scientists have made serious thoughts about what is force that I cannot follow fully.

That is where Newton's second and third laws come in. To define what you mean by force.

In a rotating system particles of free motion do not move with constant speeds, except it stay at rest at center, so we know that rotating system is not a IFR.

The only particle free of force is one at rest in the center. It stays at rest. Hence the rotating frame is inertial by your definition.

 

[Trying2Learn]

I love this web site. I love how I can ask questions to make up for what I did not learn properly.

And how people jump in and disagree with each other and it is all polite.

It is a rest from a weary world.

And thanks again.

 

[vanhees71]

I heartily disagree with it. A reference frame is a mathematical construct, either a coordinate system or a tetrad field depending on the context. Either way, it has no mass and therefore never has any forces acting on it.

I heartily disagree with that ;-)).

A reference frame is something very concrete in the lab. In Newtonian mechanics it's easy to define by e.g., putting up three rigid rods fixed at a point defining a right-handed Cartesian coordinate system in the lab. Whether or not these three rods define an inertial frame must be decided experimentally. For that to be true any free particle must run with constant velocity given by its initial velocity, no matter in which direction this velocity points (a particle at rest relative to this reference frame is a special case, i.e., if $\vec{v}=0$).

 

[Dale]

A reference frame is something very concrete in the lab.

If that were true then you couldn’t change reference frames through mathematical operations. Changing reference frames would require changing those very concrete things in the lab.

 

[vanhees71]

Why? You can take the active standpoint for transformations (which makes the mathematical formalism a description of a physical description btw.): The transformations describe how the same physical situation changes when using two different frames of reference.

E.g., you can ask, how the motion of a free particle looks from the point of view of Alice sitting at rest wrt. an inertial reference frame (for her it's simple, because it just keeps moving with a constant velocity) and how it looks from the point of view of Bob, who's using a "rigid-rod-reference frame" sitting in a merry-go-around.

For Alice the equation of motion simply reads
$$m \ddot{\vec{x}}_{\text{A}}=0,$$
while for Bob it reads
$$m \ddot{\vec{x}}_{\text{B}} + 2 m \vec{\omega}_{\text{B}} \times \dot{\vec{x}}_{\text{B}} + m \vec{\omega}_{\text{B}} \times (\vec{\omega}_{\text{B}} \times \vec{x}_{\text{B}})=0.$$
So it's much more complicated for Bob to describe the motion in his non-inertial (rotating) frame.

Nevertheless the math tells how it's described for two very concretely chosen physical frames of reference. Of course it's not the math that changes the reference frames but the concrete realization using three rigid rods either by Alice or Bob, who is rotating relative to Alice.

 

[A.T.]

...it's easy to define by e.g., putting up three rigid rods fixed at a point defining a right-handed Cartesian coordinate system in the lab.

You know what's even easier? Defining your coordinate systems without building a stick model for each of them.

 

[Dale]

The transformations describe how the same physical situation changes when using two different frames of reference

Since it is “the same physical system” that means that nothing physical changed. Therefore the reference frames are not physical.

You can build a bunch of sticks and I am free to use a reference frame where those sticks are moving. I am free to do that because the sticks are not the reference frame.

Another good example of how a reference frame is mathematical and not concrete is the GPS. The GPS system uses Earth centered inertial coordinates (among others) even though no part of the system is at rest in that frame. There is nothing “very concrete” you can point to and say “that very concrete thing is the Earth centered inertial frame”.

 

[vanhees71]

You know what's even easier? Defining your coordinate systems without building a stick model for each of them.

As I said, that's only the most direct and simple example, but what do you have in mind?

 

[vanhees71]

Since it is “the same physical system” that means that nothing physical changed. Therefore the reference frames are not physical.

You can build a bunch of sticks and I am free to use a reference frame where those sticks are moving. I am free to do that because the sticks are not the reference frame.

Another good example of how a reference frame is mathematical and not concrete is the GPS. The GPS system uses Earth centered inertial coordinates (among others) even though no part of the system is at rest in that frame. There is nothing “very concrete” you can point to and say “that very concrete thing is the Earth centered inertial frame”.

What physically changed was that Bob sits in an accelerated frame. The investigated system of course doesn't change (as long as the interaction between the measurement devices and the system of the different observers can be neglected, but that's no problem within classical physics of macroscopic bodies).

What I don't understand is that you don't want to define a reference frame as a physical object but a mere mathematical fiction. Without a realization of this mathematical fiction in terms of real-world measurement devices, it's not clear how to relate the mathematical fiction (choosing a reference point and a basis in an affine space) with real-world measurements, which need to be done in a well-defined physical reference frame. It's always defined by the experimental setup. Usually it's so self-evident that we don't need to think much about it, but still it's defined by real-world physical objects.

The GPS is a good example. First it is defined by setting a bunch of satellites into orbit defining with there relative positions and their position to Earth the reference frame, which of course is "reconstructed" in terms of earth-centered reference frame, and that's very concrete, at least concrete enough that you find your way within a resolution of a few meters!

 

[etotheipi]

The frame of reference corresponds to a physical state of motion of observers at rest in the frame. You define an affine base space (e.g. a Euclidian space for classical physics), and to define a coordinate system you choose within it an origin and a set of basis vectors. You require $N+1$ physical reference points in order to position and orient the coordinate system.

So the coordinate system is abstract, but you must still establish it physically . E.g. "a right-handed coordinate system with the origin in the corner of the lab, and the three axes along the three edges". Otherwise you can't make measurements.

The choice of reference frame and coordinate system does not affect natural phenomena (e.g. my TV still works if I run around it) but it does affect whether I need to include inertial forces in NII and what values I calculate for certain quantities.

 

[A.T.]

The frame of reference corresponds to a physical state of motion of observers at rest in the frame.

If you mean actual physical observers, then no: You don't need to have any actual physical observers at rest in a reference frame.

You define an affine base space (e.g. a Euclidian space for classical physics), and to define a coordinate system you choose within it an origin and a set of basis vectors. You require $N+1$ physical reference points in order to position and orient the coordinate system.

Why N+1? In 3D you require only 3 points, if you are using points only.

So the coordinate system is abstract, but you must still establish it physically.

You must relate the reference frame to physical objects, but none of these objects has to be at rest in that reference frame.

 

[etotheipi]

If you mean actual physical observers, then no: You don't need to have any actual physical observers at rest in a reference frame.

Right, sure. But a reference frame is associated with a state of motion, which goes hand-in-hand with the idea of a team of hypothetical "point-like observers" at rest in the frame.

Why $N+1$? In 3D you require only 3 points, if you are using points only.

In N-dimensional space we require an origin + N basis vectors. E.g. in 1D we need two points; one to specify the origin, the other to orient the positive $\hat{x}$ direction.

You must relate the reference frame to physical objects, but none of these objects has to be at rest in that reference frame.

Sure, they don't need to be. The key part is that you can position and orient the coordinate system physically.

 

[A.T.]

Right, sure. But a reference frame is associated with a state of motion, which goes hand-in-hand with the idea of a team of hypothetical "point-like observers" at rest in the frame.

The key word being "hypothetical".

In N-dimensional space we require an origin + N basis vectors. E.g. in 1D we need two points; one to specify the origin, the other to orient the positive $\hat{x}$ direction.

You don't need a point for each basis vector. For the third basis vector you can use the cross product of the other two basis vectors.

 

[etotheipi]

You don't need a point for each basis vector. For the third basis vector you can use the cross product of the other two basis vectors.

Point taken .

I suppose we need not constrain our third basis vector to be orthogonal to the other two.

 

[hutchphd]

Might I point out that much of this conversation conjures the Platonic notion of a coordinate system? This seems an odd thing to discuss in a physics forum.

 

[Dale]

What I don't understand is that you don't want to define a reference frame as a physical object but a mere mathematical fiction.

Because that is what it is.

Without a realization of this mathematical fiction in terms of real-world measurement devices, it's not clear how to relate the mathematical fiction (choosing a reference point and a basis in an affine space) with real-world measurements, which need to be done in a well-defined physical reference frame. It's always defined by the experimental setup. Usually it's so self-evident that we don't need to think much about it, but still it's defined by real-world physical objects.

Just because the reference frame itself is a mathematical construct does not imply that you cannot define a reference frame based on physical objects. It just means that the reference frame is not the same as those physical objects. That is what I object to.

If the reference frame were a physical concrete object then changing reference frames would imply changing a physical concrete object. For example, the (invariant) mass of an object is a concrete physical quantity. I cannot change the mass without changing the physical concrete object, for example by adding more concrete. A reference frame is not like mass. It is not something concrete in the lab. It is a mathematical quantity and may be changed without changing anything physical.

Please answer the following: do you agree or disagree that you may change reference frames without changing any concrete physical object?

The GPS is a good example.

Ok, so which physical concrete portion of the GPS system is the Earth centered inertial frame, and which physical concrete portion of the GPS system is the Earth centered Earth fixed frame? And what physical part of the GPS system is the sun centered inertial frame? And which concrete physical part is the frame where the Earth is traveling at 0.6 c in the direction of celestial north? And which concrete physical part of the GPS system is the non inertial frame where the Earth is accelerating in the direction of celestial south at a rate of 2 m/s^2? And which concrete physical part of the GPS system is the frame ...

And since each of those different reference frames are different physical concrete objects then what is each different reference frame’s mass and charge and so forth? After all, that is what this thread is actually about.

 

[etotheipi]

Because that is what it is. Just because the reference frame itself is a mathematical construct does not imply that you cannot define a reference frame based on physical objects. It just means that the reference frame is not the same as those physical objects. That is what I object to.

I agree. Reading through the thread I'm not too sure where the disagreement is.

Reference frames, coordinate systems, whatever, are abstract mathematical notions. They obviously do not exist in reality, have no "charge" or "mass", cannot "feel forces", I can't pick it up and throw it across the room, etc.

For it to be of use in physics, the abstract coordinate system must be established using physical references. We might choose the inertial "space frame" of reference and in it establish Earth-Centred Inertial (ECI) coordinates. We might choose the rotating frame of the Earth and in it establish Earth-Centred Earth-Fixed (ECEF) coordinates. In any case, the coordinate system has been positioned and oriented within the frame by physical references, namely the Earth (and in the case of ECI, also the 'fixed backdrop of the stars').

The set of 3 orthonormal rods mentioned earlier are not the coordinate system, I think they're just a pedagogical means of establishing it physically.

 

[Dale]

You must relate the reference frame to physical objects, but none of these objects has to be at rest in that reference frame.

And none of the physical objects IS the reference frame because you can change the reference frame without changing the physical objects.

 

[vanhees71]

Ok, we agree to disagree, because I don't know, how you can make sense of an abstract frame of reference in the sense that you can compare it to measurements in the lab. To do so you need to realize an appropriate reference frame.

Take the most simple example of an experiment in high school to demonstrate the rules of free fall, e.g., take this youtube video:



So there's clearly a frame of reference set up in the sense I mentioned. Here of course you need only one "rigid rod" to meausure the hight in direction of $\vec{g}$ and a clock measuring the time of fall, but it's a realization of a reference frame in the lab with physical objects. Without that you couldn't do this simple meausurement.

It's also only to good approximation an inertial reference frame, but not exactly because of the rotation of the Earth, i.e., at higher accuracies you could measure the effect of the Coriolis force etc.

 

[etotheipi]

It is a good question. Is there a difference between a coordinate system and a measuring apparatus?

Does the measuring apparatus only tell you how to obtain the tuple $(x^1, x^2, x^3)$, whilst the coordinate system is a more abstract notion? Or is a coordinate system just more generally a means of labelling points in space, in which case a set of rulers is the coordinate system?

I personally like the former case more, but I don't know. I suspect it's more of a philosophical issue .

 

[Dale]

Ok, we agree to disagree, because I don't know, how you can make sense of an abstract frame of reference in the sense that you can compare it to measurements in the lab. To do so you need to realize an appropriate reference frame.

Take the most simple example of an experiment in high school to demonstrate the rules of free fall, e.g., take this youtube video:



So there's clearly a frame of reference set up in the sense I mentioned. Here of course you need only one "rigid rod" to meausure the hight in direction of $\vec{g}$ and a clock measuring the time of fall, but it's a realization of a reference frame in the lab with physical objects. Without that you couldn't do this simple meausurement.

It's also only to good approximation an inertial reference frame, but not exactly because of the rotation of the Earth, i.e., at higher accuracies you could measure the effect of the Coriolis force etc.

I notice that you failed to address any of the questions I asked you to address.

The point that you seem to be missing is that clocks and rods are concrete physical objects but a reference frame is something different. I can take one single set of physical objects and define an infinite number of reference frames based on that one set of physical objects. Therefore the reference frame is not the same as the physical objects used to define it.

In particular for the purpose of this thread while clocks and rods have mass and can have forces acting on them reference frames do not have mass and can accelerate without force.

The rod and clock in your video are not a reference frame. They are a rod and a clock and they can be used to define an infinite number of reference frames.

 

[vanhees71]

That I suspect too ;-). I think, it's important to keep the physics in mind. Of course you can do theoretical physics by just talking about mathematical abstract structures, but that's not the entire picture about physics as a natural science, which is about observations of the real world and experiments providing as well as possible idealized conditions to measure certain aspects as accurately as possible with the given means etc.

Newtonian mechanics wouldn't be as successful a theory, describing a lot of everyday experience quite accurately, if it weren't possible to realize "reference frames" which correspond in good approximation with the abstract "coordinate systems". Of course, this is something you explain only in an introductory chapter of a textbook or in a lecture briefly and then just use the formalism to formulate the theory, as long as you learn theoretical physics. It should, however, also be complemented by experimental physics, where you see on a lot of examples, how the laws are checked (or maybe even new laws found) by experiment.

Of course, when you think about a simple experiment like shown in the youtube video about the free fall, nobody starts to deeply think philosophically about the fact that everything starts with establishing a reference frame, because that's so obvious that it seem to be a no-brainer.

But is it really a no-brainer? Newton didn't think so to begin with. He was pretty puzzled by the question, how to establish his absolute space and time, i.e., in modern language, how to realize an inertial reference frame physically. There is the famous discussion of the rotating bucket, with which he wanted to demonstrate that you can distinguish a non-inertial frame from the inertial frame apparently realized as a frame at rest relative to Earth.

Of course he was well aware that it is unlikely that a rest frame on Earth really establishes an exact inertial reference frame, because the Earth rotates around its axis and moves on an elliptic orbit around the Sun. By the way also an example that you need an adequate reference frame, as demonstrated by the history of the issue with the heliocentric vs. the geocentric point of view of how to choose a reference frame. What seems obvious today Copernicus's ingenious idea to use a heliocentric frame of reference was a revolution in the literal sense. It's considered a turning point from ancient to modern natural sciences and it was used by Kepler with success to establish his famous laws of planetary motion (starting from a tedious analysis of the Mars orbit) with a lot of mathematical effort to calcualte the heliocentric coordinates from Brahe's data (of course taken on Earth).

Back to the question about the physical establishment of Newton's absolute space and time. The issue has been an enigma even well until the 20th century. A famous piece in the puzzle is of course Mach's principle, where he conjectured that an inertial reference frame is established by the rest frame of all the fixed stars and that inertia is indeed somehow related to the interactions of the object under consideration with all masses in the universe.

The status today, I'd say, is the point of view provided by Einsteins General Relativity, according to which an inertial frame can only be established locally by a reference frame at rest wrt. a freely falling small non-rotating volume (small compared to typical distances across which you can still neglect inhomogeneities of the gravitational field and thus tidal forces, which of course always again is a question of the accuracy you look at this field). An example is the ISS, which is the best microgravity lab there is today (one speaks today rather about "microgravity" than "free of gravity" or "weightlessness"). For me that's quite contrary to Mach's principle, because it's rather a local resolution than the idea that inertia is due to the interactions of the objects under consideration with all the masses in the universe. It's rather the reinterpretation of the gravitational field or rather potential in terms of a pseudo-Riemanian fundamental form of spacetime, which let's you "construct" a locally inertial reference frame by letting a very small non-rotating volume fall freely (i.e., moving along a geodesic in pseudo-Riemannian spacetime).