Relativistic composition of velocities

We have mentioned several times that the Galilean composition $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$ does not work at relativistic speeds: two summed velocities must stay below $c$, whereas Galileo does not guarantee that at all. The remarkable thing is that the correct formula is derived with the same ingredient used for the Doppler effect: a scientist firing light signals and reasoning event by event.

Consider three reference frames aligned along the same line. SRA is the laboratory of the scientist emitting signals; SRB is at rest relative to an object $B$ moving at speed $v_{B(A)}$ relative to SRA; SRC is at rest relative to a second object $C$ moving at speed $u_{C(B)}$ relative to SRB and $u_{C(A)}$ relative to SRA. The scientist fires two light flashes a time $\Delta t_{(A)}$ apart, and we follow them through the three systems.

Arrival at $B$. From the Doppler-effect analysis we know that $B$ receives the two flashes with interval

$\Delta t'_{(B)} = \Delta t_{(A)}\,\sqrt{\frac{1 + v_{B(A)}/c}{1 - v_{B(A)}/c}}$

$B$ relays them to $C$. Imagine that $B$, as soon as it receives them, “forwards” them to $C$. The interval with which $B$ emits them towards $C$ coincides, in SRB, with the interval at which it received them: $\Delta t_{(B)}^\text{emiss} = \Delta t'_{(B)}$. Applying the Doppler formula again, this time between SRB and SRC:

$\Delta t'_{(C)} = \Delta t'_{(B)}\,\sqrt{\frac{1 + u_{C(B)}/c}{1 - u_{C(B)}/c}}$

Direct link $A \to C$. But the same flashes, viewed directly with the speed $u_{C(A)}$ of $C$ relative to $A$, still obey Doppler:

$\Delta t'_{(C)} = \Delta t_{(A)}\,\sqrt{\frac{1 + u_{C(A)}/c}{1 - u_{C(A)}/c}}$

Equating the two expressions for $\Delta t'_{(C)}$ and cancelling the common factor $\Delta t_{(A)}$ gives the velocity-composition law in multiplicative form: two Doppler factors compose into a third.

$\ev{\sqrt{\frac{1 + u_{C(A)}/c}{1 - u_{C(A)}/c}} = \sqrt{\frac{1 + v_{B(A)}/c}{1 - v_{B(A)}/c}}\cdot \sqrt{\frac{1 + u_{C(B)}/c}{1 - u_{C(B)}/c}}}$

Squaring and solving for $u_{C(A)}$ gives back the more familiar form usually found in textbooks:

$\ev{u_{C(A)} = \frac{u_{C(B)} + v_{B(A)}}{1 + u_{C(B)}\,v_{B(A)}/c^2}}$

Principle — Relativistic composition (1D)

If $C$ has velocity $u'$ in SRB and SRB moves at velocity $v$ relative to SRA, the velocity of $C$ in SRA is $\ev{u = \frac{u' + v}{1 + u'\,v/c^2}}$

The merit of this formula is that it behaves well in both regimes that interest us. For $u', v \ll c$ the product $u'v/c^2$ is tiny, the denominator tends to $1$ and we recover the good old Galilean law $u \approx u' + v$: everyday velocities add up as always. At the other extreme, for $u' = c$ the denominator becomes $1 + v/c$ and

$u = \frac{c + v}{1 + v/c} = c$

light remains $c$ in every reference frame, as required by the second postulate — indeed in the Doppler derivation itself we had imposed $c$ equal in SRA and SRB. It is the denominator $1 + u'v/c^2 > 1$ that automatically restrains the result: $c$ acts as an insurmountable asymptote, and two velocities both close to $c$ add up to something still close to $c$, never beyond.

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Topics: Special relativity Concepts: Relativistic composition of velocities Skills: Changing reference frame

Related exercises: Problem — Composition via Doppler · Worked exercise — Condor versus rocket · Problem — Composition of velocities (electron)