What is an inertial frame? A conflict of two definitions

[etotheipi]

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.

I agree with this, yes.

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

Or even if the car is not moving inertially, then it defines a non-inertial reference frame, the rest frame of the car.

 

[Dale]

Or even if the car is not moving inertially, then it defines a non-inertial reference frame, the rest frame of the car.

The problem is that there is no unique frame associated with a non inertial object. So merely having a non inertial object is not enough to specify a reference frame.

 

[etotheipi]

The problem is that there is no unique frame associated with a non inertial object. So merely having a non inertial object is not enough to specify a reference frame.

Good point, rotating frames do add a complication!

For instance, the rest frame of a particle that is at rest in a rotating frame of reference is not well defined. The particle is at rest in many different frames, e.g. the rotating frame, or instead one that is translating with the particle but not rotating, etc. In that sense, the reference frame is not uniquely defined.

For extended bodies undergoing general planar motion, I think they do uniquely define a rest frame.

 

[jbriggs444]

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 $10.5 g$? ;-)).

No. I mean that the number on the dial where the needle points is halfway between the 10 and the 11. It is still a measurement. That we might need calibration against some other standard to make it meaningful is beside the point.

I do not need some imagined reference frame to look at my instrument and copy a reading into a lab notebook. A reference frame is something that I can invoke later when I am analyzing the data from my lab notebook.

 

[vanhees71]

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.

So how else would you call a "reference frame"? For me a reference frame always was a concrete realization in the lab. What's abstract are the coordinates defined with respect to such a concrete realization.

 

[vanhees71]

No. I mean that the number on the dial where the needle points is halfway between the 10 and the 11. It is still a measurement. That we might need calibration against some other standard to make it meaningful is beside the point.

I do not need some imagined reference frame to look at my instrument and copy a reading into a lab notebook. A reference frame is something that I can invoke later when I am analyzing the data from my lab notebook.

That's my very point! You have not an imagined but a concrete and well defined reference frame given by your accelerometer (e.g., a smartphone using its built-in accelerometer). Experimental setups always define an reference frame against which you measure your observables. Otherwise it's not a measurement. It's also obvious that a measurement doesn't provide any information it you don't know the units in which you express your result, but that's indeed not the point here.

 

[jbriggs444]

So how else would you call a "reference frame"? For me a reference frame always was a concrete realization in the lab. What's abstract are the coordinates defined with respect to such a concrete realization.

I am more on the Platonic side. For me, the reference frames exist whether I am contemplating them or not.

When we analyze an experiment, I think in term of picking out one or more of those pre-existing reference frames and expressing or converting our physical measurements in terms of numbers relative to those frames.

 

[vanhees71]

Good point, rotating frames do add a complication!

For instance, the rest frame of a particle that is at rest in a rotating frame of reference is not well defined. The particle is at rest in many different frames, e.g. the rotating frame, or instead one that is translating with the particle but not rotating, etc. In that sense, the reference frame is not uniquely defined.

For extended bodies undergoing general planar motion, I think they do uniquely define a rest frame.

There is also not a unique inertial reference frame. That's why we think about Galilei transformations which tell us how to convert the coordinates wrt. to one inertial frame to another. Of course there's also always (at least locally) a diffeomorphism between the observables as measured in one (inertial or accelerated) to any other (inertial or accelerated) frame of reference.

 

[jbriggs444]

That's my very point! You have not an imagined but a concrete and well defined reference frame given by your accelerometer (e.g., a smartphone using its built-in accelerometer). Experimental setups always define an reference frame against which you measure your observables. Otherwise it's not a measurement. It's also obvious that a measurement doesn't provide any information it you don't know the units in which you express your result, but that's indeed not the point here.

Nonsense. I can calibrate an accelerometer without pontificating about a lab frame.

 

[Dale]

There is also not a unique inertial reference frame.

Therefore even for inertial devices my above argument holds. 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 $Frame \ne Objects$

 

[vanhees71]

That's indeed nonsense: You have to define acceleration against what. Otherwise your results are completely meaningless. Of course we cannot discuss this in detail if you don't give the details about your accelerometer.

 

[etotheipi]

There is also not a unique inertial reference frame. That's why we think about Galilei transformations which tell us how to convert the coordinates wrt. to one inertial frame to another. Of course there's also always (at least locally) a diffeomorphism between the observables as measured in one (inertial or accelerated) to any other (inertial or accelerated) frame of reference.

Sure but there is a unique inertial rest frame for a body moving inertially, which corresponds to a class of coordinate systems related by constant translations and rotations .

 

[A.T.]

Of course we cannot discuss this in detail if you don't give the details about your accelerometer.

How about a really cheap one, that only gives the magnitude of proper acceleration. Do we need a reference frame for that? What about a thermometer?

 

[vanhees71]

Therefore even for inertial devices my above argument holds. 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 $Frame \ne Objects$

I guess, we should end this discussion, because it's obvious that I cannot make my point clear.

If somebody is interested in these very fundamental questions, I recommend to read the 2nd paragraph in Sommerfeld, Lectures on Theoretical Physics, vol. 1 (mechanics) or, more detailed: Einstein, Relativity: the special and the general theory), where in the first chapter he carefully summarizes the important difference between geometry as an abstract mathematical entity and its meaning in connection with the observable physical world. It's an age-old subject starting with Leibniz and Newton with important contributions by Poincare, Mach, H. Hertz, and finally Einstein (to mention only a few).

 

[A.T.]

So how else would you call a "reference frame"?

You mean the physical objects used to define a reference frame? Since these objects could be almost anything, I would just name the objects used.

 

[vanhees71]

Then you cannot talk about physics in general, and that's what theoretical physics is about... I think Sommerfeld and more so Einstein have it put into the best words, and they simply use the word "reference frame", and that's how this word has been used since Newton's times!

 

[Dale]

Then you cannot talk about physics in general,

I can easily talk about physics while using the word "clock" to refer to clocks and the word "rod" to refer to rods, and the word "reference frame" to refer to any of the many corresponding mathematical objects that could derived from the measurements of clocks and rods.

What you cannot do is claim that the clocks and rods themselves are a reference frame because you cannot logically have simultaneously $Frame_A=Objects_X$ and $Frame_B=Objects_X$ and $Frame_A\ne Frame_B$.

 

[jbriggs444]

That's indeed nonsense: You have to define acceleration against what. Otherwise your results are completely meaningless. Of course we cannot discuss this in detail if you don't give the details about your accelerometer.

I am taking a reading from an instrument. That reading is an invariant fact of the matter that will hold regardless of what reference frame is used. One does not need a reference frame to read a number from an instrument. Nor does one need a specific reference frame in order to compute an invariant quantity.

 

[vanhees71]

No, your accelerometer will show a different reading whether you hold it fixed relative to the Earth or let it fall. These are two different reference frames. The issue, in which frame you define physical quantities to be measured becomes very important in relativistic physics, e.g., in thermodynamics this has been clarified only in the mid 1960ies after half a century confusion and debates: the thermodynamical potentials (like internal energy, entropy, enthalpy, free energy) and observables like temperature, chemical potentials, etc. are defined to be measured in the (local) rest frame of the matter.

@Dale Of course two different reference frames from which you observe the same "events" consist of two different measurement apparati which realize these two different reference frames. In your language in #77: If $\text{Frame}_A \neq \text{Frame}_B$ you have $\text{Frame}_A = \text{Objects}_X$ and $\text{Frame}_B = \text{Objects}_Y$.

 

[A.T.]

Then you cannot talk about physics in general, and that's what theoretical physics is about.

I have no idea why you think there is a problem in using different words for abstract concepts and the physical objects that are used for measurements, which can be related to those abstract concepts.

Do you also have a problem with "time" & "clock" being different words? What about "length" & "ruler"?

 

[Dale]

No, your accelerometer will show a different reading whether you hold it fixed relative to the Earth or let it fall. These are two different reference frames.

No, they are two different experiments. Each experiment can be analyzed using an infinite number of reference frames

Of course two different reference frames from which you observe the same "events" consist of two different measurement apparati which realize these two different reference frames. In your language in #77: If FrameA≠FrameB you have FrameA=ObjectsX and FrameB=ObjectsY.

This violates the principle of relativity. The principle of relativity specifically says that you can use any frame for analyzing an experiment. It does not require you to use a specific frame. According to relativity you are indeed free to use both $Frame_A$ and $Frame_B$ to describe the exact same $Objects_X$

Besides, if you make this argument then the GPS system becomes a counter example as no part of the GPS system is at rest in the ECI frame.

 

[Dale]

Closed temporarily for moderation

 

[PeterDonis]

I think a key issue needs to be clarified for this discussion: the term "reference frame" can be used to mean multiple things in physics. In particular, it can be used to mean any of the following three things:

(1) A coordinate chart;

(2) A frame field (i.e., a mapping of points in spacetime to 4-tuples of orthonormal vectors, one timelike and three spacelike);

(3) A concrete measuring apparatus that physically realizes #1 or #2.

These are three distinct things that should not be conflated.

In terms of the above three distinct things (which I'll denote by their numbers above), I'll try to summarize what I think are the key points relevant to the original thread topic (what counts as "inertial"):

We can make physical measurements with #3 without having #1 or #2 at all. We can use our mathematical theories to make predictions using #1 or #2 without having #3 at all, but the predictions for physical measurement results should be independent of any particular choice of #1 or #2. To compare physical measurements with predictions, of course we need both, and we also need a way to set up a correspondence between #3 and either #1 or #2, so that the comparison between the predicted and actual measurement results is well defined.

In terms of #1, a chart is inertial if objects that are not subject to any forces (or subject to multiple forces whose vector sum is zero) have zero coordinate acceleration in the chart. The definition of "not subject to any forces", however, is different in Newtonian mechanics than in GR. In Newtonian mechanics, gravity counts as a force, so, for example, a non-rotating coordinate chart centered on the Earth and covering the entire Earth and its vicinity counts as inertial. (However, "fictitious" forces like centrifugal force do not count as forces for this purpose.) In GR, gravity is not a force (or, if you like, it is considered a "fictitious" force), and the chart just described is not inertial, since, for example, a rock dropped off a cliff and free-falling downward does not have zero coordinate acceleration in this frame.

In terms of #2, a frame field by itself can't really be called "inertial" or "non-inertial"; we have to add to the frame field a family of timelike worldlines, defined such that at each event in spacetime, the timelike vector of the 4-tuple assigned to that event is the tangent vector to the unique worldline passing through that event. Then the frame field is inertial if the worldlines are all worldlines of objects subject to no forces (with all the caveats given above).

In terms of #3, a measuring apparatus is inertial if, as a physical object, it is subject to no forces (with all the caveats given above).

 

[PeterDonis]

After discussion among the moderators, this thread will remain closed.