What is an inertial frame? A conflict of two definitions

[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 height 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 height 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 calculate 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).

 

[vanhees71]

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.

Maybe it's again a language problem. For you it seems as if a reference frame is only a coordinate system in the theory but for me it has to be an object realized by the meausurement apparatus. A massless reference frame doesn't exist as you say yourself. In that sense light-cone coordinates in relativity are mathematical calculational tools that cannot be realized in experiment.

In the video for me the rod and the clock establish a reference frame. That you can calculate the coordinates of the falling ball in any other coordinate system, be it realized by another observer or not, is of course not a problem in any sense.

 

[A.T.]

For you it seems as if a reference frame is only a coordinate system in the theory...

More like a set of coordinate systems with constant relative transformations.

... for me it has to be an object realized by the meausurement apparatus

That's not the standard use of the term, as far as I know.

 

[Dale]

I have no objection to the importance of experiment in physics. What I object to is identifying the reference frame with the measurement devices.

In the video for me the rod and the clock establish a reference frame.

Without adding or changing any physical material, is a reference frame where that same rod and clock are accelerating also a valid reference frame? Is a reference frame where that same rod and clock are moving horizontally at a constant velocity also a valid reference frame? How about one where it is rotating?

Please do not avoid answering this question directly this time. State clearly your opinion about the validity of multiple reference frames for a single physical setup.

 

[A.T.]

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

The cross product was just an example, that 3 points can be sufficient. If you include time in your definition, 2 points can be sufficient as well.

 

[etotheipi]

The cross product was just an example, that 3 points can be sufficient. If you include time in your definition, 2 points can be sufficient as well.

So N+1 points are always sufficient, but not necessary. Less than or equal to N points might be sufficient depending on the context.

 

[vanhees71]

I have no objection to the importance of experiment in physics. What I object to is identifying the reference frame with the measurement devices.

Without adding or changing any physical material, is a reference frame where that same rod and clock are accelerating also a valid reference frame? Is a reference frame where that same rod and clock are moving horizontally at a constant velocity also a valid reference frame? How about one where it is rotating?

Please do not avoid answering this question directly this time. State clearly your opinion about the validity of multiple reference frames for a single physical setup.

Of course all these are legitimate frames of rererence. Why shouldn't they be?

 

[Dale]

Of course all these are legitimate frames of rererence. Why shouldn't they be?

Then it is logically impossible to claim “A reference frame is something very concrete”. Nothing concrete changed. If you can change reference frames without changing anything concrete then the reference frame cannot be “something very concrete”

You cannot have it both ways. If it is something concrete then changing it requires making concrete changes. If you can change it without making concrete changes (as you agree) then it cannot be something concrete.

 

[vanhees71]

I think I don't understand your objection against the fact that any measurement must define a reference frame by its very setup, and why do you think that any of the in my opinion possible physical reference frame is less concrete than any other. You can realize very different reference frames observing the same phenomena. I can obseve the moon on Earth at the same time as do the astronauts at thd ISS. I'm in a different reference frame than the astronauts, but neither frame is more or less legitimate than the other, and we'll agree on what we observe, because we can transform from the results of the moon's coordinates of position. velocity, etc. wrt. each of these reference frames.

 

[A.T.]

I'm in a different reference frame than the astronauts,

So now reference frames are boxes that contain some things, but not others?

 

[vanhees71]

No to the contrary. As I said all of Dale's examples can well be provide well-defined reference frames, though some are not inertial frames.

 

[A.T.]

As I said all of Dale's examples can well be provide well-defined reference frames,

If you can simply define as many reference frames as you wish, then they are not physical objects themselves, even if physical objects are being referenced in their definition.

 

[jbriggs444]

I think I don't understand your objection against the fact that any measurement must define a reference frame by its very setup

I can make and report a measurement: "the needle on my accelerometer dial hit 10.5" without ever defining a reference frame.

 

[etotheipi]

I don't think the terminology "concrete" being used here means "something you can hold in your hand". It just pertains to how you establish the reference frame.

We can speak of the rest frame of a moving car, or the rest frame of the space station, or the rest frame of the moon. Those are all quite concrete notions in my view. Each defines a state of motion, i.e. an infinite grid of hypothetical point-like observers who share the same rigid body motion (this part only applies to classical physics, with SR we can only get Born rigidity!). The measuring apparatus in the "finding-g" experiment defines a perfectly good (approximately) inertial reference frame.

On the other hand, some choices of reference frames are more abstract. We might speak of a frame rotating at some arbitrary angular velocity $\vec{\omega}$ about a fixed axis. Or a reference frame accelerating at $\vec{a}$ w.r.t. the thing being studied. These are not as "concrete" as the previous examples, in a sense, because now we're not "attaching" our reference frame to anything as obvious per se. and we're abstracting away some practical details.

All of them allow you to construct a coordinate system. Is it worth losing sleep over this? The term "reference frame" is so ill-defined from what I've come across anyways (apart from maybe some way more complex explanations to do with fiber-bundles, which I don't understand at all ).

 

[Dale]

You can realize very different reference frames observing the same phenomena.

Yes. Because the reference frame is not physical. It is not concrete. If it were then you could only realize one reference frame.

I think your terminology here is absurd. Do you know what concrete is? Maybe this is a language barrier. Concrete is a mixture of cement and rock which is hardened to form roads, buildings, foundations, dams, and even military fortifications.

Its connotation in English is something solid, strong, inflexible, definite. The connotation of concrete is completely incompatible with the flexibility of “You can realize very different reference frames observing the same phenomena”. Maybe you don’t intend this self contradiction, but in English you are simply contradicting yourself.

I'm in a different reference frame than the astronauts,

No, you are not in a different reference frame. This is terminology I expect to hear from novices, not an expert like you. Both you and the astronauts are in all global reference frames.

because we can transform from the results of the moon's coordinates of position. velocity, etc. wrt. each of these reference frames.

If a mathematical operation allows you to change between reference frames then reference frames are mathematical. You cannot have it both ways.

 

[vanhees71]

If you can simply define as many reference frames as you wish, then they are not physical objects themselves, even if physical objects are being referenced in their definition.

The ruler and the stop watch in the quoted youtube movie about the measurement of $g$ is a very concrete realization of a reference frame by physical objects. It's one of many possible realizations of the abstract concept of a reference frame.

Mathematically in Newtonian mechanics any reference frame that provides (at least locally) a one-to-one mapping to the coordinates defined in any inertial frame is a valid reference frame. Whether or not you can realize it by a measurement setup in reality is a different question.

 

[vanhees71]

I can make and report a measurement: "the needle on my accelerometer dial hit 10.5" without ever defining a reference frame.

The rest frame of the accelerometer is already the used reference frame (e.g., a smartphone or iphone). I only hope that it's calibrated with useful units ;-)). Do you mean $10.5 g$? ;-)).

 

[vanhees71]

Yes. Because the reference frame is not physical. It is not concrete. If it were then you could only realize one reference frame.

This is an utter misunderstanding, because in everyday life we have a lot of example of different reference frames realized in a very concrete way. Einstein always discussed such real-world realizations using the example with the reference frame set up in a train vs. one set up at the embarkment of a station or the elevator fixed at rest relative to Earth and one free falling etc. etc. There are a plethora of legitimate reference frames realized by real-world objects. If this were not the case we'd not be able to make sense even of the very beginning of kinematics in Newtonian (or relativistic) mechanics, and there'd be no use of all the abstract definitions you have in mind.

 

[etotheipi]

The rest frame of the accelerometer is already the used reference frame (e.g., a smartphone or iphone). I only hope that it's calibrated with useful units ;-)). Do you mean $10.5 g$? ;-)).

$10g$: "Maximum permitted g-force in Mikoyan MiG-35 plane and maximum permitted g-force turn in Red Bull Air Race planes"

Does @jbriggs444 commute to work in a fighter jet?

 

[A.T.]

The ruler and the stop watch in the quoted youtube movie about the measurement of $g$ is a very concrete realization of a reference frame by physical objects. It's one of many possible realizations of the abstract concept of a reference frame.

Mathematically in Newtonian mechanics any reference frame that provides (at least locally) a one-to-one mapping to the coordinates defined in any inertial frame is a valid reference frame. Whether or not you can realize it by a measurement setup in reality is a different question.

Okay, as long as we agree that reference frames themselves are abstract, we are on the same page. I would avoid using the same term for the physical measurement setup, to avoid confusion.

 

[A.T.]

$10g$: "Maximum permitted g-force in Mikoyan MiG-35 plane and maximum permitted g-force turn in Red Bull Air Race planes"

Does @jbriggs444 commute to work in a fighter jet?

Just drop your phone on the ground to beat the MIG.

 

[vanhees71]

Okay, as long as we agree that reference frames themselves are abstract, we are on the same page. I would avoid using the same term for the physical measurement setup, to avoid confusion.

Then, how do you call concrete realizations of that abstract concept? For me it's absurd to think about physics starting from abstract concepts. The very possibility to realize at least some valid reference frames is the prerequesite to make an abstract mathematical theory a relevant theory for physics. If it can't make contact with real-world observations/measurements it's simply not a physical theory though it may be interesting mathemaics (e.g., string theory).

 

[A.T.]

Then, how do you call concrete realizations of that abstract concept? For me it's absurd to think about physics starting from abstract concepts. The very possibility to realize at least some valid reference frames is the prerequesite to make an abstract mathematical theory a relevant theory for physics. If it can't make contact with real-world observations/measurements it's simply not a physical theory though it may be interesting mathemaics (e.g., string theory).

Nobody said you shouldn't connect the abstract concepts with measurements. The suggestion was to use different terms for the abstract concepts and the measurement tools.

 

[Dale]

There are a plethora of legitimate reference frames realized by real-world objects.

And the fact that you can realize a plethora of legitimate frames with the same real world objects is why it is wrong to identify the frame with the real world objects.

You cannot have simultaneously $Frame_A=Objects_X$ and $Frame_B=Objects_X$ and $Frame_A\ne Frame_B$. It is a logical impossibility. The only resolution that is consistent with the math and the principle of relativity is $Frame \ne Objects$

The very possibility to realize at least some valid reference frames is the prerequesite to make an abstract mathematical theory a relevant theory for physics. If it can't make contact with real-world observations/measurements it's simply not a physical theory

I have no disagreement with this. That isn’t the argument.

The problem is identifying the mathematical objects with the physical objects. The map is not the territory. And it is particularly problematic here since there is a one-to-many relationship between the physical objects and the mathematical objects.

 

[etotheipi]

And the fact that you can realize a plethora of legitimate frames with the same real world objects is why it is wrong to identify the frame with the real world objects.

But we can speak of the rest frame of the object. In that sense an object can define a reference frame (e.g. the rest frame of a car). That is not to say that the reference frame is the car; instead, it's a set of coordinate systems related by time independent translations and rotations all of which are fixed w.r.t. the car. I don't see a problem with saying that a moving car defines a reference (rest) frame. It's just a physical concept that pertains to a state of motion.

Maybe there is some additional insight to be gained with the GR description of a reference frame. I remember @vanhees71 once gave a definition something to do with to a set of timelike geodesics and a hypersurface, but I can't remember it exactly.

 

[Dale]

I don't see a problem with saying that a moving car defines a reference (rest) frame.

Nor do I (provided the car is moving inertially).

The problem isn’t using material objects to define a reference frame. The problem is identifying the reference frame with the physical objects. The reference frame itself is a mathematical construct that is defined based on its relationship to the physical objects. For a given physical setup many different mathematical constructs may be used, each with a different relationship to the physical objects and to each other.