Derivation of Coordinate Transformations

To derive an expression for Lorentz Contraction, begin with the following postulates:

• The laws of physics are the same in all inertial frames of reference
• The speed has the same value, c, in all inertial frames

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

x-ct = 0

and analogously in the second coordinate system

x'-ct'=0

That is, a constant of proportionality relates the equations in the two systems

(x'-ct') = L (x-ct)

(For the Newtonian case, this constant would be 1.)
Similarly, for light propagating along the negative x axis, except the constant may be different:

(x'-ct') = M (x-ct)

Adding and subtracting these equations, respectively gives

2x' = (L+M)x - (M-L)ct
2ct' = -(L-M)x + (L+M)ct

Defining new constants, a and b, these equations simplify to

x' = ax - ct
ct' = act - bx

So now, if we know a and b, we will be able to translate between one coordinate system and the other.