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===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 :
:<math>x'=\gamma\left(x-vt\right)</math>
:<math>t'=\beta\left(t+\alpha x\right)</math>
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
:<math>ct'=\gamma\left(c-v\right)t</math>
:<math>t'=\beta\left(1+\alpha c\right)t</math>
Replacing t' from the second equation, the first one becames
:<math>c\beta\left(1+\alpha c\right)t=\gamma\left(c-v\right)t</math>
After simplification by t and dividing by cβ, one obtains :
:<math>1+\alpha c=\frac{\gamma}{\beta}(1-\frac{v}{c}) </math>

====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
:<math>x'=\gamma\left(x-vt\right)</math>
:<math>t'=\beta\left(t+\alpha x\right)</math>
is :
:<math>x=\frac{1}{1-\alpha v}\left(\frac{x'}{\gamma}-\frac{vt'}{\beta}\right)</math>
:<math>t=\frac{1}{1-\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)</math>
In accord with the principle of relativity, the expressions of x and t should write :
:<math>x=\gamma\left(x'+vt'\right)</math>
:<math>t=\left(t'+\alpha x'\right)</math>
They should be identical to the original expressions except for the sign of the velocity :
:<math>x=\frac{1}{1+\alpha v}\left(\frac{x'}{\gamma}+\frac{vt'}{\beta}\right)</math>
:<math>t=\frac{1}{1+\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)</math>

We should then have the following identities, verified independently of x’ and t’ :
:<math>x=\gamma\left(x'+vt'\right)=\frac{1}{1+\alpha v}\left(\frac{x'}{\gamma}+\frac{vt'}{\beta}\right)</math>
:<math>t=\left(t'+\alpha x'\right)=\frac{1}{1+\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)</math>
This gives the following equalities :
:<math>\beta =\gamma=\frac{1}{\sqrt{1+\alpha v}}</math>

===Expression of the Lorentz transformation===
Using the above relationship
:<math>1+\alpha c=\frac{\gamma}{\beta}(1-\frac{v}{c}) </math>
we get :
:<math>\alpha =-\frac{v}{c^2}</math>
and, finally:
:<math>\beta =\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}</math>
We have now all the four coefficients needed for the Lorentz transformation which writes in two dimensions :
:<math> x=\frac{x' + vt'}{ \sqrt[]{1 -\frac{v^2}{c^2}} } </math>
:<math>t= \frac{t' + \frac{vx'}{c^2}}{ \sqrt[]{1 -\frac{v^2}{c^2}} } </math>

The inverse Lorentz transformation writes, using the Lorentz factor γ :
:<math>x'= \gamma\left(x - vt\right) </math>
:<math>t'=\gamma\left(t - \frac{vx}{c^2}\right) </math>

These four equations are used according to the needs.

Light invariance principle - Historique des versions

Jacques Lavau : Page créée avec « ===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 dis... »