## Derivation of the Lorentz Transformation

A body of length moving with speed along the x – axis of an inertial frame O, but at rest in its own rest frame O' will be observed by a stationary observer to be apparently shortened to a length and two events separated by a time interval of length in the rest frame O' will be observed by the stationary observer in rest frame O to be separated by a longer interval of time Suppose the two inertial frames O and O' coincide at t=t'=0. At this instant a light pulse is emitted and produces an event E at coordinates in the inertial frame O and in the inertial frame O'.  and are related by Since O' is moving along the x axis of O, and hence Since light travels in straight lines, the relationship between and will be linear so we can write (1) and (2)

Since when light ray is emitted, from (1) (1) becomes (3)

Substitute (2) and (3) these expressions into to get Equating coefficients of  (4)

Equating coefficients of  (5)

Equating coefficients of  (6)

Square (5) to give (7)

From (4) and from (6) Substitute these into (7) to obtain Expanding and simplifying gives Then (Take the positive square root so that and flow in the same direction).

From (5) Then and The transformation is symmetrical, so and  