Calculating the Lorentz Transformations
With every point of reference there is a coordinate system which 3 axis come out from which can be seen in the first figure above. The X,Y and Z are the three axis’s that belong to the point at the center K. Be sure to look at the first picture above for a reference, K would be where the lines would meet. Now an object observed within the K reference frame would be designated K’ and it’s sub axis’s would be X’ , Y’ and Z’ respectively as observed from K.
As discussed in the recent blog posts on relativity, time and distance change as they observed from a point that is moving in motion from the reference frame. You cannot simply just add up vectors to determine K’ as classical mechanics shows, there are slight deviations in the values to satisfy the law of transmition of light in a vacuum. The calculations for this are shown in the second picture and those are the Lorentz Equations.
As you can see the X’ value (distance) must be modified while the Y’ and Z’ values remain the same as their counterpart if the object is moving in a linear direction in the X axis. Also from what we know of relativity, the time of events that take place are different so t’ is also modified.
In the final picture is the Galilei Transformations which are essentially the equations if you put an infinitely large value for c in the Lorentz Transformations. These, from my knowledge, is the equations if law of transmition of light in a vacuum while assuming distance and time does not change with relative velocity