Frames of Reference (1)

Amongst the many ideas in classical mechanics that are taken to be axiomatic in nature, we might include the idea of a reference frame. An idea often taken for granted as something that can be intuitively understood and rarely tackled with. The point of this post is to look into this idea and find out the need for Inertial frames of reference.

Definition: A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer. It may also refer to both an observational reference frame and an attached coordinate system as a unit.[1]

A frame of reference may thus be used as a counter to measure dynamical variables pertaining to an event. It may be observed that position, orientation or velocity etc, depend explicitly on the observer's own state of motion. It thus would appear that the definition of a frame of reference is circular in nature because the very coordinate system so defined to measure dynamical variables of an observed quantity depend explicitly on the dynamics of the observer itself.

How could we then reconcile this apparently self implying circular definition with the standards one would expect from an axiom of a  successful theory like Classical Mechanics?

Observing the observer - A point observer cannot observe itself

To resolve the aforementioned, we must ask the question - How is the observer's dynamic state (the position,conjugate momentum, etc) being measured in the first place? The very moment we wish to ask about these characteristic properties of the observer, we have to ask who/what is observing the observer?

An ideal (point) observer cannot observe itself and thus the moment we wish to talk about the dynamical properties of the observer we must employ "observer 2" a secondary observer to document our primary observer's dynamical situation. Our primary observer has thus become the observed rendering the situation invalid.

How then, do we fix a coordinate system in our reference frame if we cannot find out the observer's dynamic state without employing a third party? Here we come to our first axiom.

Axiom 1 : For any reference frame, all of the observer's dynamical characteristic variables are assigned the value zero (0).

Thus the observer has a position vector 0, a momentum vector 0, an acceleration vector 0 and so on, in the specific frame of reference that it is a committed observer to. We can now proceed to set up a coordinate system that would help us in identifying the dynamical variables of the system under observation.

Next time we shall try to formulate a mathematical framework and show that the set of all reference frames is a group under linear coordinate addition and discuss about cosets of reference frames.