OK, so let's verify the main desired consequence of the Lorentz transformation: that everyone observes the same speed of light.
Observers will measure the speed of light by calculating how long it takes the light going towards cross a rod of length laid in the x axis at position .
TODO image.
Each observer will observe two events:
  • : the light touches the left side of the rod
  • : the light touches the right side of the rod
Supposing that the standing observer measures the speed of light as and that light hits the left side of the rod at time , then he observes the coordinates:
Now, if we transform for the moving observer:
and so the moving observer measures the speed of light as:

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