Playground
Two velocities u' and v as arrows, and their relativistic sum u = (u'+v)/(1+u'v/c^2) shown below.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Lab-frame Velocityoutput Velocity of object in unprimed frame | m/s | LT^-1 | -299792458 – 299792458 | |
| Rest-frame Velocity Velocity of object in primed frame | m/s | LT^-1 | -299792458 – 299792458 | |
| Frame Velocity Velocity of primed frame in lab frame | m/s | LT^-1 | -299792458 – 299792458 |
Deep dive
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).
Experimental verification
Fizeau's 1851 water experiment foreshadowed the effect; modern particle collisions respect the rule to better than 10^-6.
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
Real-world applications
- Doppler shift calculations in astrophysics
- Particle beam targeting
- Relativistic jet kinematics in quasars
Worked examples
Two rockets head-on
Given:
- u_prime:
- 269813212
- v:
- 269813212
Find: u
Solution
u = (0.9 + 0.9)c / (1 + 0.81) = 1.8/1.81 c ≈ 0.9945c.
Light in moving medium
Given:
- u_prime:
- 299792458
- v:
- 100
Find: u
Solution
u = (c + 100)/(1 + 100/c) = c — light speed invariant.
Scenarios
What if…
- scenario:
- What if we add 1000 boosts of 0.5c?
- answer:
- The composition asymptotes to c but never reaches it — rapidities add linearly instead.
- scenario:
- What if v = -u'?
- answer:
- u = 0 — the primed frame is the object's rest frame.
- scenario:
- What if motions are perpendicular?
- answer:
- Use vector form: u_y = u'_y/(gamma·(1 + u'_x v/c^2)) — transverse velocities pick up 1/gamma.
Limiting cases
- condition:
- u', v << c
- result:
- u → u' + v
- explanation:
- Recovers Galilean addition.
- condition:
- u' = c
- result:
- u = c
- explanation:
- Speed of light is invariant in every frame.
- condition:
- u' = v = c/2
- result:
- u = 4c/5 = 0.8c
- explanation:
- Classical would give c — relativity caps it below c.
Context
Albert Einstein · 1905
Einstein derived the relativistic addition rule as a direct corollary of the Lorentz transformation.
Hook
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.
Dimensions:
- lhs:
- u → [LT^-1]
- rhs:
- (u' + v)/(1 + u'v/c^2) → [LT^-1]/[1] = [LT^-1]
- check:
- Both sides velocity. ✓
Validity: Valid along a common axis; for general directions use vector form with gamma corrections for transverse components.