First about the Lorentz Transformations....

The Lorentz Transformations are mathematical equations that allow us to transform from one coordinate system to another. Why would we want to do this? Because special relativity deals with frames of reference. When you analyze properties from one frame to another, it is necessary to first transform from one coordinate system to another. Thus, we can utilize the Lorentz Transforms to convert length and time from one frame of reference to another. For example, if you are flying in an airplane and I am standing still on the ground, you could apply the transformations to transform my frame of reference into your frame of reference and I could do the same for you in my frame of reference. The previous statements imply that lengths and times are not the same for objects that are in motion with respect to each other. As unbelievable as this may seem, it is a result of SR. Einstein utilized the transformations because they provide a method of translating the properties from one frame of reference to another when the speed of light is held constant in both.