Special Theory of Relativity:
Galilean Transformations,. Newtonian Relativity.
This was a lecture delivered to physics-elective class of a 3 year non-physics degree students on 10th April 2017. This is also a good exposition to honors students and anyone at an introductory level of the special theory of relativity, with requisite mathematical background.
Let us consider an inertial frame of reference S. The space and time coordinates of any event occurring in frame S are given by x, y, z, t.
Now let us consider another frame of reference S’ which is inertial but moves wrt frame S at speed v, along +x direction.
The coordinates of the same event in the S’ frame are given as: x’, y’, z’, t’. The relationship among the coordinates of any event in two different frames of reference both of which are inertial frames, is known as Galilean Coordinate Transformation or Galilean Transformation.
If we assume that time passes by at the same rate in both S and S’ frames, the resulting laws satisfy Newtonian Relativity. We say time is an absolute quantity in an infinitude of equivalent inertial frames of references as the rate of time change is independent of the particular inertial frame of reference we have chosen. Consequently: t = t’.
The above equation is known as velocity addition rule in Newtonian Relativity. This is valid only for classical mechanics in the sense of speed of objects and speed of frame of reference, which are quite insignificant with respect to the speed-of-light value.
Velocity addition is nothing but a relation of velocities of objects in different frames among each other. So its exactly what we call “relative velocities” in elementary mechanics. Relative velocity, velocity addition and velocity transformation are the exact same thing. Read more about these here and here. The second link also expounds on what happens when speeds approach that of light.