Reference Frames and Principle of Relativity: UNIZOR.COM - Relativity4All - Principles
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-05-16

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Reference Frames and Principle of Relativity

In Physics we use numbers to describe the motion of objects, like distance, speed, time period etc.
These numbers are derived from certain system of coordinates in space to measure space-related characteristics and some devices to measure time, like clocks.
In this course we will use three-dimensional Cartesian system of space coordinates and some time origin to measure time intervals.

We will not choose any particular Cartesian system of coordinates with fixed origin and directions of axes (like originated in the center of the Sun and the X-axis directed toward a star Sirius).

Instead, we assume the existence of some system of Cartesian coordinates where any object with zero cumulative force acting upon it is moving along a straight line with a constant speed (speed might be zero, in which case the trajectory is not a straight line but a point).
We will call such a system of coordinates an inertial reference frame and will refer to it as "primary".

Obviously, there is no absolutely inertial system with absolutely no external forces, this is an abstraction. But many systems, including the one mentioned above related to stars are as good an approximation to an inertial reference frame as it can be.

Notice that any other Cartesian system of coordinates that moves in such a way that its origin moves in the "primary" system along a straight line with constant linear speed, and its axes are always parallel to axes of the "primary" system would also qualify for being called inertial reference frame.

Also, we will not choose any particular moment in time as the beginning of time (like midnight on January 1st, 1900). Any moment will do, as long as it's fixed and called the "beginning of time" or "zero-time".

With an introduction of inertial reference frames we can state an important principle of classical Physics - all the Physics laws must be the same if expressed quantitatively, using corresponding coordinates and time, in all inertial reference frames.

This principle is called Galilean Invariance or Galilean Relativity Principle and, in other words, it states that identical experiments conducted within two different inertial reference frames, that move relative to each other, produce identical results.

The parallel (no rotation) and uniform (regarding velocity) movement of inertial reference frame α relative to another one β (uniform translation of α relative to B) cannot be experimentally detected from inside frame α or from inside frame β.
It needs an external observer, who sees both α and β moving relative to each other, to come up with quantitative judgement about this relative motion.

Consequently, there is no system of coordinates that can be considered as absolutely "at rest", while others are moving relative to it. For any reference frame α there is another β moving relatively to it, while, for the same token, frame α can be considered as moving relatively to frame β.
This is the essence of Principle of Relativity.

Applied to inertial reference frames, this principle can be formulated as all laws of Physics are expressed in the same form in all inertial reference frames.
This is called the Principle of Relativity.