Relativistic Velocity Addition

Equivalent forms

u = \frac{u' + v}{1 + \beta' \beta}

No matter how many boosts you stack, the result is bounded by c.

Symbol	Name	SI unit	Dimension	Range
$u$	Lab-frame Velocityoutput Velocity of object in unprimed frame	m/s	LT^-1	-299792458 – 299792458
$u_prime$	Rest-frame Velocity Velocity of object in primed frame	m/s	LT^-1	-299792458 – 299792458
$v$	Frame Velocity Velocity of primed frame in lab frame	m/s	LT^-1	-299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$u$	m/s	cm/s	ft/s
$u_prime$	m/s	cm/s	ft/s
$v$	m/s	cm/s	ft/s

Where it holds

Valid along a common axis; for general directions use vector form with gamma corrections for transverse components.

Dimensional analysis

u \to [LT^-1]

(u' + v)/(1 + u'v/c^2) \to [LT^-1]/[1] = [LT^-1]

Both sides velocity.

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Discovery

Albert Einstein · 1905

Einstein derived the relativistic addition rule as a direct corollary of the Lorentz transformation.

Try this

Two rockets approach each other each at 0.9c in Earth's frame — how fast does one see the other?

Apply u = (u' + v)/(1 + u'v/c^2) with u' = 0.9c and v = 0.9c to get u ≈ 0.9945c.

Research status: stable

Real-world applications

Doppler shift calculations in astrophysics
Particle beam targeting
Relativistic jet kinematics in quasars

Common misconceptions

Two objects approaching each other at 0.9c do not exceed c relative to one another
Velocity addition is non-commutative for non-collinear boosts (Wigner rotation)
Photons still travel at c in every frame regardless of source motion

Experimental verification

Fizeau's 1851 water experiment foreshadowed the effect; modern particle collisions respect the rule to better than 10^-6.

Derivation

Differentiate the Lorentz transformation

x = \gamma (x' + vt')

,

t = \gamma (t' + vx'/c^2)

.

Then

u = dx/dt = (dx'/dt' + v)/(1 + (v/c^2) dx'/dt') = (u' + v)/(1 + u'v/c^2)

.

Limiting cases

u', v << c⟶

u \to u' + v

Recovers Galilean addition.

u' = c

⟶

u = c

Speed of light is invariant in every frame.

u' = v = c/2

⟶

u = 4c/5 = 0.8c

Classical would give c — relativity caps it below c.

What if…

What if we add 1000 boosts of 0.5c?

The composition asymptotes to c but never reaches it — rapidities add linearly instead.

What if

v = -u

'?

$u = 0$ — the primed frame is the object's rest frame.

What if motions are perpendicular?

Use vector form: $u_y = u'_y/(\gamma \cdot (1 + u'_x v/c^2))$ — transverse velocities pick up 1/gamma.

1

Two rockets head-on

Given ·

u prime:: 269813212
v:: 269813212

Find · u

Steps

$u' = 0.9c$ , $v = 0.9c$
$u'\cdot v/c^2 = 0.81$
$u = (0.9 + 0.9)/(1 + 0.81) \times c \approx 0.9945c$

Result ·

u = (0.9 + 0.9)c / (1 + 0.81) = 1.8/1.81\,\mathrm{c} \approx 0.9945c

.

2

Light in moving medium

Given ·

u prime:: 299792458
v:: 100

Find · u

Steps

$u' = c$ exactly
Numerator $= c + v$ , denominator $= 1 + v/c = (c + v)/c$
$u = c$

Result ·

u = (c + 100)/(1 + 100/c) = c

— light speed invariant.

Lorentz Factor→Relativistic Momentum→Relativistic Doppler Effect→