5.5 The lorentz transformation  (Page 2/12)

The lorentz transformation equations

The Galilean transformation nevertheless violates Einstein’s postulates, because the velocity equations state that a pulse of light moving with speed c along the x -axis would travel at speed $c-v$ in the other inertial frame. Specifically, the spherical pulse has radius $r=ct$ at time t in the unprimed frame, and also has radius $r\prime =ct\prime$ at time $t\prime$ in the primed frame. Expressing these relations in Cartesian coordinates gives

The left-hand sides of the two expressions can be set equal because both are zero. Because $y=y\prime$ and $z=z\prime ,$ we obtain

This cannot be satisfied for nonzero relative velocity v of the two frames if we assume the Galilean transformation results in $t=t\prime$ with $x=x\prime +vt\prime .$

To find the correct set of transformation equations, assume the two coordinate systems S and $S\prime$ in [link] . First suppose that an event occurs at $(x\prime ,0,0,t\prime )$ in $S\prime$ and at $(x,0,0,t)$ in S , as depicted in the figure.

Suppose that at the instant that the origins of the coordinate systems in S and $S\prime$ coincide, a flash bulb emits a spherically spreading pulse of light starting from the origin. At time t , an observer in S finds the origin of $S\prime$ to be at $x=vt.$ With the help of a friend in S , the $S\prime$ observer also measures the distance from the event to the origin of $S\prime$ and finds it to be $x\prime \sqrt{1-v^2\text{/}{c}^2}.$ This follows because we have already shown the postulates of relativity to imply length contraction. Thus the position of the event in S is

and

The postulates of relativity imply that the equation relating distance and time of the spherical wave front:

must apply both in terms of primed and unprimed coordinates, which was shown above to lead to [link] :

We combine this with the equation relating x and $x\prime$ to obtain the relation between t and $t\prime :$

The equations relating the time and position of the events as seen in S are then

This set of equations, relating the position and time in the two inertial frames, is known as the Lorentz transformation    . They are named in honor of H.A. Lorentz (1853–1928), who first proposed them. Interestingly, he justified the transformation on what was eventually discovered to be a fallacious hypothesis. The correct theoretical basis is Einstein’s special theory of relativity.

The reverse transformation expresses the variables in S in terms of those in $S\prime .$ Simply interchanging the primed and unprimed variables and substituting gives: