I have recenly registered to a physics forum website https://www.physicsforums.com/, where i was paying special interest to Special and General Relativity category.

A few days ago, there was a discussion about the interval transformations under Lorentz transformations.

Some contributors/moderators there don't seem to believe that the space and time coordinate differences Δx or Δt do transform under Lorentz transformations (which is obvious due to the** linearity** of the transformation) or that these intervals could even be used to demonstrate the length contraction or time dilation phenomena, simply showing **that they don't understand even the basis and real signification of the Lorentz transformation!**

The proof of it can be found in any basic course on relativity, and is used by Einstein himself in chapter 12 of his book *Relativity* (1920) to derive the time dilation and space contraction between inertial referentials in relative uniform movement.

Below a copy of my contribution in the thread, which has been questionned by a moderator: i would not change a simple word of it!

This discussion is linked to this article The Lorentz transformations Part III - Measuring-Rods and Clocks in motion, in which we derive the equations of the length contraction and time dilation from the Lorentz transformation (even though presented in a slightly different manner).