To derive an expression for Lorentz Contraction, begin with the following postulates:
Consider two frames, one which will be assumed to be fixed, and another moving along a common x axis with a constant velocity relative to the first frame. This is shown above. A beam of light is propagating along the common x axis. In the first frame, this satisfies
and analogously in the second coordinate system
That is, a constant of proportionality relates the equations in the two systems
(For the Newtonian case, this constant would be 1.)
Similarly, for light propagating along the negative x axis, except the constant may be different:
Adding and subtracting these equations, respectively gives
Defining new constants, a and b, these equations simplify to
So now, if we know a and b, we will be able to translate between one coordinate system and the other.