Derivation of the Lorentz Transformation
A body of lengthmoving with speedalong 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 lengthand two events separated by a time interval of lengthin 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 coordinatesin the inertial frame O andin the inertial frame O'.
andare related bySince O' is moving along the x axis of O,andhence
Since light travels in straight lines, the relationship betweenandwill be linear so we can write(1) and(2)
Sincewhen light ray is emitted, from (1)
Substitute (2) and (3) these expressions intoto 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 thatandflow in the same direction).
The transformation is symmetrical, soand