Velocity addition

The classical problem

In Galilean relativity, velocities add by simple subtraction:

$u'*x=u_x-v$

If a projectile moves at speed $u_x$ in frame $S$ and frame $S'$ moves at speed $v$, this rule works well at low speeds. But it fails for light. If $u_x=c$, it predicts $u'*x=c-v$, contradicting the light-speed postulate.

Special relativity replaces this with a new velocity transformation.

Deriving the formula

Start with the Lorentz transformation:

$x'=gamma(x-vt)$

ight)$$ For a particle, velocity in $S$ is $$u_x=rac{dx}{dt}$$ Velocity in $S'$ is $$u'*x=rac{dx'}{dt'}$$ Differentiate the Lorentz transformations: $$dx'=gamma(dx-vdt)$$ $$dt'=gammaleft(dt-rac{v dx}{c^2} ight)$$ Then $$u'*x=rac{dx-vdt}{dt-vdx/c^2}$$ Divide numerator and denominator by $dt$: $$u'*x=rac{u_x-v}{1-u_xv/c^2}$$ ## Inverse velocity addition Solving for $u_x$ gives $$u_x=rac{u'*x+v}{1+u'*xv/c^2}$$ This is the relativistic version of adding velocities. At low speeds, the denominator is approximately 1, and the Galilean formula returns. ## Light speed remains light speed If $u_x=c$, then $$u'*x=rac{c-v}{1-v/c}=c$$ Thus every inertial observer measures light in vacuum moving at $c$. This is not a coincidence. The velocity transformation is built from Lorentz transformations, which preserve the spacetime structure required by constant light speed. ## Combining high speeds Suppose a spaceship moves at $0.8c$ relative to Earth and fires a probe forward at $0.8c$ relative to the spaceship. Earth does not measure the probe moving at $1.6c$. Instead, $$u=rac{0.8c+0.8c}{1+0.64}=rac{1.6c}{1.64}approx0.976c$$ The result is less than $c$. ## Transverse velocities Velocity components perpendicular to the relative motion also transform. If $S'$ moves along x, then $$u'*y=rac{u_y}{gamma(1-u_xv/c^2)}$$ and similarly for $u'*z$. The transverse transformation depends on the longitudinal velocity component because time itself transforms. ## No massive object reaches $c$ Relativistic velocity addition ensures that combining sub-light velocities never produces a superluminal result. A massive object can get closer and closer to $c$, but not reach or exceed it through ordinary acceleration. The speed $c$ acts as an invariant boundary, not just the speed of one particular object. ## The big idea Velocities do not add linearly at relativistic speeds. The correct formula $u'*x=(u_x-v)/(1-u_xv/c^2)$ preserves the speed of light and prevents sub-light velocities from combining into faster-than-light motion. Classical velocity addition is recovered only when speeds are much smaller than $c$.