Definition of General Lorentz Transformations

We are now in a position to determine those coordinate transformations between two inertial frames within the four-dimensional space-time which leave the space-time interval ds invariant. The new coordinates x (in IS ) have to be functions of the old coordinates X (in IS), i.e., x = x (x). Due to the homogeneity of space and time, however, the relationship between the old and new coordinates has to be linear, i.e.. 

The entries of the (4 x 4)-matrix A and the 4-vector a have to be constant, i.e., independent of the coordinates x, since otherwise the transformation would be different at different positions in space-time. Furthermore, the entries have to be real-valued, since space-time coordinates cannot be complex numbers. The 4-vector a simply represents trivial temporal or spatial shifts of the origin of IS with reference to the origin of IS, such that the space-time coordinate differential is given by 

In order to avoid any misunderstanding we explicitly write down the matrix for A, 

we can now determine the defining condition for A by exploiting the invariance constraint for ds as given by Eq. (3.6), 

17) is the fundamental and most important property of Lorentz transformations which completely specifies these transformations. 

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