Consider the effect of the Lorentz transformations on the spacetime interval
Substituting for x and t from the above Lorentz transformations one obtains
Generalizing to four dimensions we see that the spacetime interval
is invariant under the Lorentz transformations.
The most general transformation between and will be more complicated but it must be linear. We can write it as:
where . This linear transformation is called the generalized Lorentz transformations. It contains ten parameters: four correspond to an origin shift , three correspond to a Lorentz boost [ which depends on ] and three to the rotation which aligns the axes of and . The last six are contained in the matrix [ six because is symmetric i.e. .
Note that these parameters for a group galled the Poincaré group.
Later we will show that the Poincaré transformations preserve Maxwells equations as well as light paths.