How would you define an inertial frame of reference?

[D H]

This sounds very ambiguous to me, and possibly circular. To say that the velocity is constant, we must first use a coordinate system, and which one should we use if not the inertial one that we're trying to define?

You don't need a coordinate system. All you need is a nice combination of very ancient and rather modern measuring devices. A protractor, a laser, and a good clock do the trick. (Adding something to measure Doppler shift is a nice to have, but is not essential.) Upon taking a number of measurements of some object with those devices over time, it's easy to deduce whether those measurements are statistically consistent with the hypothesis that the object has been moving at a constant velocity.

And how is "force" defined here?

Newton's definition of force was perhaps circular. Or even better, an undefined term, which is how I like to look at it. Then again, my table might just have one too many beer mugs placed on it.

 

[Sugdub]

I have never seen anyone here do that. The fact that it is a map is well understood, even if it is not expressly stated in every post which uses math.

Well, I'm not going to fight on that.

More important for me is whether physicists agree or not with my last statement whereby changing the coordinate system does not necessarily capture the essence of a change of the frame of reference, which consists in changing the assignment of which objects are considered at rest.

 

[Dale]

More important for me is whether physicists agree or not with my last statement whereby changing the coordinate system does not necessarily capture the essence of a change of the frame of reference, which consists in changing the assignment of which objects are considered at rest.

I have mixed feelings about that statement:

Overall, changing the space-time coordinate system does not necessarily imply a change of the frame of reference (i.e. changing the set of objects which are considered “at rest”). For me these are two different concepts.

I agree that frame of reference and coordinate system are two different concepts, but not for the reason that you give. There are a few main differences between frame of reference and coordinate system. While understanding that there is a mapping between the physical and mathematical aspects of a theory, the differences between a frame and coordinate systems are easiest to see on the mathematical side.

The mathematical object used to represent a frame of reference is called a tetrad or vierbein. It is a set of four orthonormal vector fields that are used to represent local rods and clocks at each event.

A coordinate system (or coordinate chart) is only a mathematical object and has no physical counterpart. It is simply a mapping (smooth and invertible) between events in spacetime and points in R4.

A tetrad must be orthonormal, whereas there is no such restriction on a coordinate chart. Also, a coordinate chart (usually) includes a notion of simultaneity, whereas a tetrad does not. A coordinate chart must be defined on an open subset of the spacetime, whereas a tetrad can be defined on any subset.

 

[Fredrik]

Although I concur to the need to avoid circularity as pointed out in #47 by Fredrik, I think his suggestion does not make it. The mere reference to an “accelerometer” which measures “acceleration” is introducing circularity in his definition, in the same way as a reference to “forces” does.

There's no circularity there. The term "accelerometer" in that sentence isn't supposed to be defined a "any device that measures acceleration". It's supposed to be defined by instructions on how to build such a device.

I don't consider what I wrote there a "definition". It's just a statement about what inertial coordinate systems in different theories have in common. A definition would specify what coordinate systems are considered "inertial" in one theory.

 

[pervect]

The mathematical object used to represent a frame of reference is called a tetrad or vierbein. It is a set of four orthonormal vector fields that are used to represent local rods and clocks at each event.

That's close to my understanding of the 4d relativistic case, though I thought of the objects as vectors, not vector fields. As vectors, the needed mathematical properties are to be able to be multiplied by scalars and added together. The space spanned by the vectors is in general the tangent space - this is equal to the manifold itself only when the manifold is flat.

I don't quite see how to re-interpret my understanding in terms of vector fields, I was wondering if we really had different views or whether it's just a semantic and word-choice issue. I'm hoping that are views are basically the same.

On another issue:

As near as I can tell, the Newtonian case has 3 basis vectors, time isn't handled explicitly (it is assumed to be separate from space, and handled as what is usually called absolute time, which is the usual Newtonian way of handling time).

Another important difference is the treatment of gravity, as the PSSC film quoted by bcrowell mentions, (I think bcrowell's book does too).

in the Newtonian case a frame fixed to a non-rotating Earth would be considered to be inertial, while in the GR case the frame anchored to the non-rotating Earth would be considered non-inertial, and inertial frame would be free-falling.

 

[Dale]

That's close to my understanding of the 4d relativistic case, though I thought of the objects as vectors, not vector fields. As vectors, the needed mathematical properties are to be able to be multiplied by scalars and added together. The space spanned by the vectors is in general the tangent space - this is equal to the manifold itself only when the manifold is flat.

I don't quite see how to re-interpret my understanding in terms of vector fields, I was wondering if we really had different views or whether it's just a semantic and word-choice issue. I'm hoping that are views are basically the same.

A vector field is simply a set of vectors which are located somewhere in the manifold. Since the rods and clocks represented by a tetrad are somewhere in the universe, they form a vector field.

Note, the vector field doesn't need to be defined everywhere on the manifold, just somewhere. Even along just a single worldline or at a single event (although that would be hard to arrange physically). I suspect that is the thing that was bothering you.

As near as I can tell, the Newtonian case has 3 basis vectors, time isn't handled explicitly (it is assumed to be separate from space, and handled as what is usually called absolute time, which is the usual Newtonian way of handling time).

Another important difference is the treatment of gravity, as the PSSC film quoted by bcrowell mentions, (I think bcrowell's book does too).

in the Newtonian case a frame fixed to a non-rotating Earth would be considered to be inertial, while in the GR case the frame anchored to the non-rotating Earth would be considered non-inertial, and inertial frame would be free-falling.

Yes, all of that is correct according to my understanding also.