Inertial Frames of Refereence

Question

I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another.

So am I right to understand that "inertness of coordinate systems" is an equivalence relation on all the coordinate systems in a space. Furthermore, one should not talk of an inertial coordinate system on its own. In order to talk about inertness one has to choose two coordinate systems and compare them. Finally no equivalence class is superior to another, whatever superior means in its usage in the first paragraph, to which meaning I am not knowledgeable.

If any of this is not true, please include an example as well.

Answer 1

This is true for both Newtonian mechanics and special relativity. The transformation between inertial frames is different in the two. One inertial frame can be much more convenient than others for calculation, but that does not make it superior in theory.

Answer 2

The statement that no coordinate system is superior to any other would appear to be the source of confusion. I'm not altogether sure what is meant by that statement; it does not reflect how Newtonian mechanics has historically been understood and practiced. Most likely it is intended to reflect a more recent understanding that reflects the development of special and general relativity, but even in the more modern context, I'm not sure what it is trying to say.

The word "inertial" in the term "inertial frame of reference" indicates that the frame is one in which the law of inertia, also known as Newton's first law, holds. That is, it is a frame in which a body not acted on by any forces will remain in the same state of motion, experiencing no acceleration.

One therefore does not speak of one frame being "inertial" with respect to another. Rather, there is a set of preferred frames, the inertial frames, which are all moving with constant rectilinear motion with respect to one another. In all other frames of reference, a body not acted on by any forces will undergo accelerations. These can be understood in terms of centrifugal and Coriolis forces, but such forces are considered "fictitious" since they are not caused by interactions with other physical bodies, but instead are due to the choice of coordinate system.

It might seem that Newton's first law is a special case of Newton's second law ($F=ma$) in which the force on a body is zero, but I don't believe that this is how Newton viewed things. Rather, the first law is making a nontrivial statement about the physical world, namely that inertial frames exist. Assuming that forces are due to other bodies and that force falls off with distance, a body sufficiently far from all other bodies could be used as a test body to determine whether a frame is inertial. Once such an inertial frame is found, all frames moving with constant velocity with respect to the first will also be inertial, and all frames that are accelerating with respect to the first will not be inertial. A big question is whether one can find any inertial frames at all: will all test bodies agree on whether a frame is inertial? Newton's first law is asserting that this is so.