Light invariance principle

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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 :

LaTeX: x'=\gamma\left(x-vt\right)
LaTeX: t'=\beta\left(t+\alpha x\right)

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

LaTeX: ct'=\gamma\left(c-v\right)t
LaTeX: t'=\beta\left(1+\alpha c\right)t

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

LaTeX: c\beta\left(1+\alpha c\right)t=\gamma\left(c-v\right)t

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

LaTeX: 1+\alpha c=\frac{\gamma}{\beta}(1-\frac{v}{c})

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

LaTeX: x'=\gamma\left(x-vt\right)
LaTeX: t'=\beta\left(t+\alpha x\right)

is :

LaTeX: x=\frac{1}{1-\alpha v}\left(\frac{x'}{\gamma}-\frac{vt'}{\beta}\right)
LaTeX: t=\frac{1}{1-\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)

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

LaTeX: x=\gamma\left(x'+vt'\right)
LaTeX: t=\left(t'+\alpha x'\right)

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

LaTeX: x=\frac{1}{1+\alpha v}\left(\frac{x'}{\gamma}+\frac{vt'}{\beta}\right)
LaTeX: t=\frac{1}{1+\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)

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

LaTeX: x=\gamma\left(x'+vt'\right)=\frac{1}{1+\alpha v}\left(\frac{x'}{\gamma}+\frac{vt'}{\beta}\right)
LaTeX: t=\left(t'+\alpha x'\right)=\frac{1}{1+\alpha v}\left(\frac{t'}{\beta}-\frac{\alpha x'}{\gamma}\right)

This gives the following equalities :

LaTeX: \beta =\gamma=\frac{1}{\sqrt{1+\alpha v}}

Expression of the Lorentz transformation

Using the above relationship

LaTeX: 1+\alpha c=\frac{\gamma}{\beta}(1-\frac{v}{c})

we get :

LaTeX: \alpha =-\frac{v}{c^2}

and, finally:

LaTeX: \beta =\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

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

LaTeX:  x=\frac{x' + vt'}{ \sqrt[]{1 -\frac{v^2}{c^2}} }
LaTeX: t=  \frac{t' + \frac{vx'}{c^2}}{ \sqrt[]{1 -\frac{v^2}{c^2}} }

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

LaTeX: x'=  \gamma\left(x - vt\right)
LaTeX: t'=\gamma\left(t - \frac{vx}{c^2}\right)

These four equations are used according to the needs.