**Note**: the presentation of the Lorentz transformation is done by Einstein in chapter 11 of his book *Relativity*

Remember that since the beginning, we are trying to find the relationships between the coordinates of an event in two inertial frames *R* and *R'*, with *R'* moving with a velocity* v* with respect to* R*.

After having spent time giving some insight into the strange nature of the spacetime and of his contruitive effects, it is now time to give a more precise algebraic formulation of how coordinates change for different inertial observers.

This set of equations are called the **Lorentz transformations**, named after the Dutch physicist Hendrik Lorentz (Nobel Prize 1902)

Given the assumptions that *R* and *R'* are in standard configuration, i.e:

- - An observer in frame of reference
*R*defines events with coordinates*t*,*x*,*y*,*z* - - Another frame
*R'*moves with velocity*v*relative to*R*, with an observer in this moving frame*R'*defining events using coordinates*t'*,*x*',*y'*,*z'* - - The coordinate axes in each frame of reference are parallel
- - The relative motion is along the coincident
*xx'*axes - - At time t = t' =0, the origins of both coordinate systems are the same

If an observer in R records an event *t*,* x*, *y*, *z *, then an observer in R' records the ** same **event with coordinates

*t'*,

*x'*,

*y'*,

*z'*defined as below:

With y = Lorentz factor having been already defined in the previous article Constant Speed of light - Introduction to Time Dilation and Lorentz factor

According to principle of relativity, there is no privileged frame of reference, so that the transformations from R to R' must take exactly the same form as the transformation from *R'* to *R*.

The only difference is *R'* moves with velocity* -v* relative to* R* (same magnitude but is oppositely directed). So that if an observer in *R'* records an event in *t'*, *x*', *y'*,*z'* then an observer in *R* records the* same* event with coordinates

The reciprocal transformations from each referential relative to the other coud be visualised in a synthetic manner as follow:

* Remark 1*: Compared to the classical Galilean physics with absolute time, we begin to understand what Minkowski meant by the union of the two concepts.

* Remark 2: *When v<<c, i.e when the relativistic effects fade away, we can verify that the Galilean transformation can be derived from the Lorentz transormations.

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