# Relativistic Frames of Reference

From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that

Figure 2.1: Primed reference frame moving in the z direction with velocity v.

Using Einstein's Theory of Relativity, we know that the speed of light is a constant, c, for all reference frames. In the unprimed frame, from the definition of speed:

where

Using the distance equation in a Cartesian coordinate system, the equation for the speed of light becomes

Following the postulate of Special Relativity, this implies for the primed frame

We can rewrite this as

This is possible since the ratios of distance to time are multiples of the same base, i.e. the square of the speed of light . Therefore for the relative change in the time in one frame, the distance must change by the same factor to maintain the same constant. With this we can write

This quantity is known as the time space interval when the change is infinitesimal

Since the speed of light is a constant for all frames of reference, this allows the space time interval to also be invariant for inertial frames.

From the rest frame of v'=0

Assuming motion is only along the z direction

Substituting these changes into the Galilean transformations