Causality

Let events $1$ and $2$ have spacetime coordinates ($x_1$, 0, 0, $t_1$) and ($x_2$, 0, 0, $t_2$) in some inertial reference frame, $S$. Suppose that event $1$ causes event $2$. It follows that $t_1<t_2$. In other words, event $1$ necessarily precedes event $2$ in time. Let

$$u = \frac{x_2-x_1}{t_2-t_1}$$ (3.126)

be the velocity with which information flows from event $1$ to event $2$ in order to allow the former event to cause the latter. Let us observe the two events in a second inertial frame, $S'$, that moves at velocity ${\bf v} = v\,{\bf e}_x$ with respect to $S$, and is in a standard configuration with respect to $S$. According to Equation (3.103),

$$t_2'-t_1' = \gamma\left(t_2-\frac{v\,x_2}{c^2}\right)-\gamma\left(t_1-\frac{v\,x_1}{c^2}\right),$$ (3.127)

or

$$t_2'-t_1' = \gamma\,(t_2-t_1)\left(1-\frac{u\,v}{c^2}\right).$$ (3.128)

Now, irrespective of the value of $v$, whose magnitude can never exceed $c$, event $2$ can never occur prior to event $1$ in $S'$, otherwise we could classify inertial frames into two groups; those in which event $1$ appears to cause event $2$, and those in which event $2$ appears to cause event $1$. However, this state of affairs is forbidden by Einstein's first postulate. Thus, we require $t_2'-t_1'>0$ for all $\vert v\vert<c$. It is clear from Equation (3.128) that this is only possible if $\vert u\vert<c$. Hence, we deduce that information can never propagate faster than the speed of light in vacuum, in any inertial reference frame, otherwise it is possible to find other inertial reference frames in which causality appears to be violated.