On the definition of an inertial frame of reference

Question

A rotating frame rotating at constant angular velocity with reference to a stationary one is considered non-inertial. This makes sense if one considers that the velocity vector of any point in the rotating frame is changing direction with respect to the stationary one, and is hence accelerating.

However, if one considers another vector $\vec{\omega}$ (angular velocity vector) of the two frames, the two frames differ by at most a constant velocity (albeit angular). In fact, the angular dynamics transforms the same way ($\vec{\omega}$ = $\vec{w_1}$ - $\vec{w_2}$) for both these frames as it does for any two inertial frames that differ by a constant linear velocity ($\vec{v}$ = $\vec{v_1}$ - $\vec{v_2}$).

So why isn’t this considered inertial?

Answer 1

A rotating frame rotating at constant angular velocity with reference to a stationary one is considered non-inertial

It is not true, it depends on the nature of what you named "stationary" reference frame. If it is inertial, then the "rotating" frame is not inertial. If the "stationary" frame is not inertial, the "rotating" frame may be inertial instead (it is not necessary however).

A reference frame $K$ is rotating with respect to the reference frame $K_0$ if the angular velocity of $K$ with respect to $K_0$ does not vanish.

A reference frame $K$ is accelerating with respect to the reference frame $K_0$ if a point at rest with $K$ has nonvanishing acceleration with respect to $K_0$.

It turns out that if $K_0$ is inertial, then $K$ is inertial as well if and only if it is both nonrotating and it does not accelerate with respect to $K$.

All that is more or less mathematics once the language of kinematics has been translated into mathematical notions.

The physically relevant fact are

(a) the definition of inertial reference frame,

(b) the physical (indirect) evidence of the existence of reference frames.

Regarding (a), there are several definitions more or less equivalent which depend on the level of formalization you assume. A crucial point is the use of the notion of force. I prefer not to use it to avoid logical loops.

From a very operative (though ideal) point of view we can say that

a reference frame $K$ is inertial if a set of $N>1$ material points such that they are sufficiently far from each other and far from all bodies in the universe, then they all simultaneously have constant velocity (depending on the material point and possibly zero velocity) with respect to $K$.

This is a very unexpected and non trivial physical fact since one cannot simultaneously fix the motion of a number $N>1$ of bodies by the choice of a suitable reference frame (*).

When an inertial reference frame is given, then all Newton's construction (2nd, 3rd, and force superposition principles) can be implemented and the distinction between real forces and inertial forces (they do not satisfy the 3rd principle in particular) is effective.

The existence of inertial reference frames is however quite indirect and it is based on the fact that Newton's formulation works very well, far from big masses and in spatial regions also quite large (the Solar system). For instance, a posteriori it is physically strongly tenable that a reference frame with origin at the center of Sun such that the so called "fixed stars" appear at rest there is an inertial reference frame.

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(*) It is clear that this formulation assumes that all types of interactions (a posteriori described by forces) switch off when bodies are sufficiently far to each other. As a consequence the superinteraction that forces the mutually far bodies to move with relative constant velocity cannot be described in terms of force. This observation was one of the starting points of Einstein to build up the theory of general relativity.

Answer 2

It is a matter of logic and the definition of the term "inertial".

Suppose we define the term "elephant" to mean a member of a certain species of animal, a large mammal with a trunk and large ears. Someone may then point out that there are common features with a member of some other species such as mouse. Both are mammal, both have large ears in comparison to their body, for example. But it does not follow that a mouse is an elephant.

A rotating frame is not an inertial frame because it does not have all the properties of an inertial frame.

Answer 3

You said it yourself. If a frame is accelerating then it is not inertial. There isn't anything saying that if you can relate a vector in one frame to a vector in another frame by addition of a constant vector that both frames are inertial / non-inertial. Just because you can make an analogy doesn't mean the analogy is valid at all levels.