Uniqueness of Lorentz transformations

In section 3.1.3 we mentioned that Lorentz transformations of the form as given by Eq. (3.12) are the only nonsingular coordinate transformations from 

IS to IS, i.e., X x (x), that leave the four-dimensional space-time interval ds invariant. Nonsingular in this context means that both x = x (x) and X = x(x ) are sufficiently smooth and well-behaved functions that feature a well-defined inverse. 

Proof We consider an arbitrary nonsingular coordinate transformation x - x (x) and calculate the four-dimensional distance ds between two infinitesimally neighboring events. 

We now differentiate this equation with reference to the arbitrary space-time component x and, by the product rule, arrive at 

In order to solve for the second derivatives we write down Eq. (C.16) with indices y and cr interchanged. 

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