Light invariance principle

Galilean reference frames

In classical kinematics, the total displacement x in reference frame R is the sum of the relative displacement x’ in R’ and of the displacement vt of R’ relative to R at a velocity v : x = x’+vt or, equivalently, x’=x-vt. This relation is linear when the velocity v is constant, that is when the frames R and R' are galilean. Time t is the same in R and R’, which is no more valid in special relativity, where t ≠ t’. The more general relationship, with four constants α, β, γ and v is :

The Lorentz transformation becomes the Galilean one for β = γ = 1 et α = 0.

Light invariance principle

The velocity of light is independent of the velocity of the source, as was shown by Michelson. We thus need to have x = ct if x’ = ct’. Replacing x and x' in these two equations, we have

Replacing t' from the second equation, the first one becames

After simplification by t and dividing by cβ, one obtains :

Relativity principle

This derivation does not use the speed of light and allows therefore to separate it from the principle of relativity. The inverse transformation of

is :

In accord with the principle of relativity, the expressions of x and t should write :

They should be identical to the original expressions except for the sign of the velocity :

We should then have the following identities, verified independently of x’ and t’ :

This gives the following equalities :

Expression of the Lorentz transformation

Using the above relationship

we get :

and, finally:

We have now all the four coefficients needed for the Lorentz transformation which writes in two dimensions :

The inverse Lorentz transformation writes, using the Lorentz factor γ :

These four equations are used according to the needs.