Transformation of Velocities

Consider two frames, $S$ and $S'$ , in the standard configuration. Let ${\bf u}$ be the velocity of a particle in $S$ . What is the particle's velocity in $S'$ ? The components of ${\bf u}$ are

$$u_1 = \frac{dx}{dt},$$ (1649)

$$u_2 = \frac{dy}{dt},$$ (1650)

$$u_3 = \frac{dz}{dt}.$$ (1651)

Similarly, the components of ${\bf u}'$ are

$$u_1' = \frac{dx'}{dt'},$$ (1652)

$$u_2' = \frac{dy'}{dt'},$$ (1653)

$$u_3' = \frac{dz'}{dt'}.$$ (1654)

Now, we can write Equations (1643)-(1646) in the form $dx' = \gamma\,(dx-v\,dt)$ , $dy'=dy$ , $dz'=dz$ , and $dt'= \gamma\,(dt-v\,dx/c^{\,2})$ , where

$$\gamma = (1-v^{\,2}/c^{\,2})^{-1/2}$$ (1655)

is the well-known Lorentz factor. If we substitute these differentials into Equations (1651)-(1653), and make use of Equations (1654)-(1656), we obtain the transformation rules

$$u_1' = \frac{u_1-v}{1-u_1\, v/c^{\,2}},$$ (1656)

$$u_2' = \frac{u_2}{\gamma\,(1-u_1 \,v/c^{\,2})},$$ (1657)

$$u_3' = \frac{u_3}{\gamma\,(1-u_1 \,v/c^{\,2})}.$$ (1658)

As in the transformation of coordinates, we can obtain the inverse transform by interchanging primed and unprimed symbols, and replacing $+v$ with $-v$ . Thus,

$$u_1 = \frac{u'_1+v}{1+u'_1 \,v/c^{\,2}},$$ (1659)

$$u_2 = \frac{u'_2}{\gamma\,(1+u'_1 \,v/c^{\,2})},$$ (1660)

$$u_3 = \frac{u'_3}{\gamma\,(1+u'_1 \,v/c^{\,2})}.$$ (1661)

Equations (1661)-(1663) can be regarded as giving the resultant, ${\bf u} =(u_1, u_2, u_3)$ , of two velocities, ${\bf v} = (v, 0, 0)$ and ${\bf u}' = (u'_1, u'_2, u'_3)$ , and are therefore usually referred to as the relativistic velocity addition formulae. The following relation between the magnitudes $u=(u_1^{\,2} + u_2^{\,2}+ u_3^{\,2})^{1/2}$ and $u'=({u'_1}^{2} + {u'_2}^{2}+ {u'_3}^{2})^{1/2}$ of the velocities is easily demonstrated:

$$c^{\,2} - u^{\,2} = \frac{c^{\,2} \,(c^{\,2}-u'^{\,2}) \,(c^{\,2}-v^{\,2})}{(c^{\,2} + u'_1\, v)^2}.$$ (1662)

If $u'<c$ and $v<c$ then the right-hand side is positive, implying that $u<c$ . In other words, the resultant of two subluminal velocities is another subluminal velocity. It is evident that a particle can never attain the velocity of light relative to a given inertial frame, no matter how many subluminal velocity increments it is given. It follows that no inertial frame can ever appear to propagate with a superluminal velocity with respect to any other inertial frame (because we can track a given inertial frame using a particle which remains at rest at the origin of that frame).

According to Equation (1664), if $u'=c$ then $u=c$ , no matter what value $v$ takes: that is, the speed of light is invariant between different inertial frames. Note that the Lorentz transform only allows one such invariant speed [i.e., the speed $c$ that appears in Equations (1643)-(1646)]. Einstein's relativity principle tells us that any disturbance that propagates through a vacuum must appear to propagate at the same speed in all inertial frames. It is now evident that all such disturbances must propagate at the speed $c$ . It follows immediately that all electromagnetic waves must propagate through the vacuum with this speed, irrespective of their wavelength. In other words, it is impossible for there to be any dispersion of electromagnetic waves propagating through a vacuum. Furthermore, gravity waves must also propagate at the speed $c$ .

The Lorentz transformation implies that the propagation speeds of all physical effects are limited by $c$ in deterministic physics. Consider a general process by which an event $P$ causes an event $Q$ at a velocity $U>c$ in some frame $S$ . In other words, information about the event $P$ appears to propagate to the event $Q$ with a superluminal velocity. Let us choose coordinates such that these two events occur on the $x$ -axis with (finite) time and distance separations ${\mit\Delta} t>0$ and ${\mit\Delta} x>0$ , respectively. The time separation in some other inertial frame $S'$ is given by [see Equation (1646)]

$${\mit\Delta} t' = \gamma\,({\mit\Delta} t - v\,{\mit\Delta} x/c^{\,2}) = \gamma\,{\mit\Delta} t\, (1-v\, U/c^2).$$ (1663)

Thus, for sufficiently large $v<c$ we obtain ${\mit\Delta} t'<0$ : that is, there exist inertial frames in which cause and effect appear to be reversed. Of course, this is impossible in deterministic physics. It follows, therefore, that information can never appear to propagate with a superluminal velocity in any inertial frame, otherwise causality would be violated.