Derivation Of The Lorentz Transformation

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Lorentz transformation Un article de Quantic.

Sommaire 1 Concise derivation of the Lorentz transformation 1.1 Galilean reference frames 1.2 Speed of light independent of the velocity of the source 1.3 Principle of relativity 1.4 Expression of the Lorentz transformation

Concise derivation of the Lorentz transformation The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. It is similar to that of Einstein Albert Einstein, Relativity: The Special and General Theory (http://web.mit.edu/birge/Public/books/Einstein-Relativity.pdf) . More details may be found in Bernard Schae!er, Relativités et quanta clarifiés (http://www.publibook.com/boutique2006/detail-3102-0-0-1-PB.html) As in the Galilean transformation, the Lorentz transformation is linear : the relative velocity of the of the reference frames is constant. They are called inertial or Galilean reference frames. According to relativity no Galilean reference frame is priviledged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light source.

Galilean reference frames In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x' in frame R' and of the displacement x in frame R. If v is the relative velocity of R' relative to R, we have v : x = x’+vt or x’=x-vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames of reference. In Einstein's relativity, the main di!erence is that space is a function of time and vice-versa: t " t’. The most general linear relationship is obtained with four constant coe#cients, $, %, & and v:

The Lorentz transformation becomes the Galilan transformation when % = & = 1 et $ = 0.

2/08/09 6:21

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Speed of light independent of the velocity of the source Light being independent of the reference frame as was shown by Michelson, we need to have x = ct if x’ = ct’. Replacing x and x' in the preceding equations, one has:

Replacing t’ with the help of the second equation, the first one writes:

After simplification by t and dividing by c%, one obtains:

Principle of relativity According to the principle of relativity, there is no priveledged galilean frame of reference. One has to find the same Lorentz transformation from frame R to R' or from R' to R. As in the Galilean transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other. The following derivation uses only the principle of relativity which is independent of light velocity. The inverse transformation of

is :

In accordance with the principle of relativity, the expressions of x and t are:

They have to be identical to those obtained by inverting the transformation except for the sign of the velocity of transport v:

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http://deonto-ethics.org/mediawiki/index.php?title=Lorentz_transformation

We thus have the identities, verified for any x’ and t’ :

Finally we have the equalities :

Expression of the Lorentz transformation Using the relation

obtained earlier, one has :

and, finally:

We have now all the coe#cients needed and, therefore, the Lorentz transformation :

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

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2/08/09 6:21