The Lorentz - Transformation was used by Einstein in the developement of the Special Theory of Relativity, only that he used it with a very particular point of view and with very particular ways of thinking. The relativity of the Lorentz - FitzGerald transformation at some points lead to believe that relativity only leads to a pile of condradictions and absurdes. This is why it is important to keep an opened-mind at all times and to always think about everything being relative to something else, specially when we're dealing with motion-states, velocities, accelerations, etc.
Let's suppose we're observers on a street where our trailer-lab is passing by. We can observe that the trailer has a speed of 50[km/h]. We also observe that the trailer seems to be a bit smaller than what it is when it is not moving. Now our group divides into two parts: there are those who say that the trailer is the same size at all times and motion states, and that what we saw is an optical illusion of some sort and the other groups affirms that the trailer is in fact smaller when it is in motion than when it is not moving. Classical physics takes the position of the first group: the distance has not to deal with the motion state of the object, it remains the same at all motion states. However, the Dutch physicist Hendrik Antoon Lorentz and the Irish physicist George Francis FitzGerald proved that the object's size is in fact contracted.
To deal with the mathematics Lorentz and FitzGerald ame up with the following equation to determine the factor of contraction of objects:
In this equation the Greek letter beta is the factor, v is the velocity of the particle and c is the speed of light. If we replaced the values for a velocity of 50[km/h] we will obtain a factor of contraction which is rather small. First we must either convert the speed of the trailer to [km/s] or the speed of light to [km/h]. Converting the speed of the trailer we obtain that it is 0.000013[km/s], now we can replace in the formula:
This result is so small that some calculator machines round it up and show 1 instead (happened to one of mine!). Let's suppose our trailer is 50 [m] long, if we multiply it by this factor we would obtain its contracted size and it would be:
As you can see the difference between 50 and the result we obtained is
imperceptible, so in our example we have to imagine we have some sort of super-sences
which allow us to percieve the difference. Now we became fasinated with our experience
observing the trailer and we will try to find more about it.