Lorentz Transformation

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In relativity the most important concept is that of an event. An event is something that can be labeled with a time and a position. A firecracker detonating is an example of an event. To specify the spatial location of an event three numbers, i.e. spatial coordinates, are required. The numbers are chosen to uniquely specify the location of an event. To specify the time of an event a single number is required, a temporal coordinate. The location, r = (x, y, z), and time, t, together constitute an event X º (ct, x, y, z). We desire a relationship between the coordinates of two inertial frames of reference. Consider two such frames, S and S’ which are in standard configuration with S’ moving with a velocity v with respect to S in the +x direction. See Fig. 1 

 E is the event whose spatial coordinates we wish to find. The event can be specified in two ways corresponding to two sets of spacetime coordinates. In terms of the coordinates corresponding to frame S frame we have

In terms of the coordinates corresponding to frame S’ frame we have

In the primed coordinate system the distance from the y’z’ plane to the event is x’. Let d represent the distance from the y’z’ plane to the event. Then since the relative velocity of S and S’ is v it follows that

However due to Lorentz contraction d and x’ are not equal but are related by

Substituting this into Eq. (3) and solving for x’ gives

This is our first relation for the set of transformation equations that we are looking for. Due to the principle of relativity there should be no difference between the two frames except for a change of sign for the velocity. Thus we simple exchange x’ with x, t’ with t, and v with –v to give

We now are in the position to find t’.  We substitute x’ from Eq.  (5) and then solve for t’

This is the second relation we need. Due to the fact that distances perpendicular to motion does not contract then we know immediately that y’ = y and z’ = z. We therefore have our complete transformation from S to S’ 

The transformation given by Eq. (8) is called the Lorentz transformation.

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