= Velocity-addition formula
{wiki}
= Relativistic velocity-addition formula
{synonym}
= Relativistic velocity addition
{synonym}
The <velocity-addition formula> translates a particle's <velocity> between two specified <inertial frames>. Let $S'$ move at signed <velocity> $V$ along the $x$-axis of $S$, with parallel axes and $|V|<c$, where $c$ is the <speed of light>. Define $u_x=dx/dt$ and $u'_x=dx'/dt'$ for the same particle's <world line>, using position and time measured within the corresponding <inertial frame>. With <Lorentz factor> $\gamma_V=(1-V^2/c^2)^{-1/2}$, the forward <Lorentz boost> gives
$$
u'_x=\frac{u_x-V}{1-Vu_x/c^2},\qquad u'_y=\frac{u_y}{\gamma_V(1-Vu_x/c^2)},\qquad u'_z=\frac{u_z}{\gamma_V(1-Vu_x/c^2)}.
$$
For collinear motion, write $u=u_x$ and $u'=u'_x$. The inverse <velocity-addition formula> is
$$
u=\frac{u'+V}{1+Vu'/c^2}.
$$
This is the composition of the particle's <velocity> in $S'$ and that frame's <velocity> relative to $S$. The <velocities> must belong to this same chain of <reference frames>, with directions specified. Two arbitrary quoted speeds cannot be inserted blindly. Each <velocity> measurement uses distances and clocks in a single <inertial frame>, with <Einstein synchronisation> for separated clocks.
For example, let $V=0.6c$ and let the particle travel at $u'=0.6c$ in $S'$. Its <velocity> in $S$ is
$$
u=\frac{0.6c+0.6c}{1+0.6\times0.6}=\frac{15}{17}c\approx0.88235294c.
$$
The <Galilean transformation> would instead give $u=u'+V=1.2c$, which fails here. For <light> moving along the positive axis, $u'=c$ gives $u=c$; $u'=-c$ similarly gives $u=-c$. Negative directions also work for massive particles: at $V=0.6c$, $u'=-0.6c$ gives $u=0$, while $u'=-0.8c$ gives $u=-5c/13$. Thus “addition” means composition of signed <velocities>, rather than addition of their magnitudes.
\Image[/relativistic-velocity-addition.png]
{title=Relativistic and Galilean velocity composition for a frame moving at 0.6 times the speed of light}
{id=image-relativistic-velocity-addition}
{description=The horizontal coordinate is the particle's <velocity> in $S'$, and the vertical coordinate is its <velocity> in $S$. The <velocity-addition formula> maps $-1<u'/c<1$ into $-1<u/c<1$; the endpoint values $\pm1$ describe <light>. The marked example is $u'/c=0.6$, giving $u/c=15/17$.}
A short derivation differentiates the <Lorentz transformation>: $dx'=\gamma_V(dx-Vdt)$ and $dt'=\gamma_V(dt-Vdx/c^2)$. Dividing these differentials gives the longitudinal <velocity-addition formula>. The transverse formulas use $dy'=dy$ and $dz'=dz$, but the same transformed time denominator; omitting it would ignore the <relativity of simultaneity>. Replacing $V$ by $-V$ gives the inverse. When both relative and particle <velocities> are small compared with the <speed of light>, the denominator is close to one and the <nonrelativistic limit> recovers the <Galilean transformation>.
The formula lets us compare particle motion in a moving apparatus and a laboratory without losing the invariant <speed of light>. Einstein's 1905 paper derives the coordinate change in §3 and the composition of <velocities> in §5, including the collinear formula, its light-speed case, and composition of parallel transformations. https://www.fourmilab.ch/etexts/einstein/specrel/www/[Albert Einstein, On the Electrodynamics of Moving Bodies (1905), §§3 and 5; English translation by W. Perrett and G. B. Jeffery, electronic edition by John Walker]{ref}. The <rapidity> parametrization makes the collinear composition particularly simple.
Back to article page
