Deriving the Lorentz Transformations

Deriving the Lorentz Transformations - Deriving the Lorentz...

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Deriving the Lorentz Transformations Let us now suppose that O′ and her crew observe a small bomb to explode in S′ at ( x′, 0, 0, t′ ). In this section, we shall find the space coordinates and time ( x , y , z , t ) of this event as observed by O in the frame S . (As above, S′ moves relative to S at speed v along the x -axis). In other words, we shall derive the Lorentz transformations—which are just the equations giving the four coordinates of an event in one inertial frame in terms of the coordinates of the same event in another inertial frame. We take y′ ,z′ zero because they transform trivially—there is no Lorentz contraction perpendicular to the motion, so y = y′ and z = z′ . First, we consider at what time the bomb explodes as measured by O . O′’ s crew found the bomb to explode at time t′ as measured by a local clock, that is, one located at the site of the explosion, x′ . Now, as observed by O from frame S , O′’ s clock at x′ is not synchronized with O′’ s clock at the origin S′.
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