Lorentz Transformation

Let us see if we can derive the a transformation of coordinates that is consistent with relativity. In frame $S$, at time $t$, the origin of the $x'$-axis is located a perpendicular distance $v\,t$ from the origin of the $x$-axis. Moreover, in $S$, our event is located a perpendicular distance $x'/\gamma$ from the origin of the $x'$-axis, because of length contraction. [See Equation (3.81).] Thus,

$$x = v\,t + \frac{x'}{\gamma},$$ (3.95)

which yields

$$x' = \gamma\,(x-v\,t).$$ (3.96)

Because there is no motion between the two frames in the $y$-direction or the $z$-direction, and because there is no contraction of lengths perpendicular to the $x$-axis, we deduce that

$$y' =y,$$ (3.97)

$$z' =z.$$ (3.98)

Finally, in $S'$, a clock located at the origin reads $\gamma\,t$ when a clock at the origin of $S$ reads $t$, as a consequence of time dilation (note that, in $S'$, a clock in $S$ appears to run slowly by a factor $\gamma$). [See Equation (3.73).] Furthermore, a second clock in $S'$, displaced from the origin a distance $x$ (measured in $S$) in the $x$-direction, appears to read $\gamma\,t-\gamma\,v\,x/c^2$ as a consequence of clock error. [See Equation (3.87).] Hence,

$$t'= \gamma\,t-\frac{\gamma\,v\,x}{c^2} = \gamma\left(t-\frac{v\,x}{c^2}\right).$$ (3.99)

We deduce that the transformation of coordinates that is consistent with the theory of relativity is

$$x' = \gamma\,(x-v\,t),$$ (3.100)

$$y' =y,$$ (3.101)

$$z' = z,$$ (3.102)

$$t' = \gamma\left(t-\frac{v\,x}{c^2}\right).$$ (3.103)

This transformation is know as the Lorentz transformation. Note that, in the limit in which the relative velocity of frames $S$ and $S'$ is non-relativistic (i.e., much smaller in magnitude than the speed of light in vacuum), so that $v/c\ll 1$, and $\gamma\rightarrow 1$, the Lorentz transformation morphs into the Galilean transformation, (3.88)–(3.91). Thus, the common sense transformation, (3.88)–(3.91), holds as long as the relative velocity between the two frames of reference is much smaller than the velocity of light in vacuum. Incidentally, it is easily shown from Equations (3.100)–(3.103) that

$$x = \gamma\,(x'+v\,t'),$$ (3.104)

$$y =y',$$ (3.105)

$$z =z',$$ (3.106)

$$t = \gamma\left(t'+\frac{v\,x'}{c^2}\right).$$ (3.107)

In the Galilean transformation, (3.88)–(3.91), the transformation of time is completely independent from that of space. This is no longer the case in the Lorentz transformation, (3.100)–(3.103). In fact, the transformations of space and time are mixed together in special relativity in such a manner that, rather than thinking of space and time as separate concepts, it makes more sense to talk about a generalized concept that Einstein called spacetime.