The Velocity Addition Formula

Velocity Addition Formula

Updated: August 3, 2026

On the Electrodynamics of Moving Bodies

Einstein-EDoMB-1905-Section §5

https://www.fourmilab.ch/etexts/einstein/specrel/www/

Einstein: Relativity, The Special and the General Theory, 1920, Chapter VI page 19, Chapter XIII page 45

https://www.google.com/books/edition/Relativity/n8QKAAAAIAAJ?hl=en&gbpv=1

Both sources show the velocity-addition-formula as it is today.

Shown: V = ( v + w ) / ( 1 + vw/c² )

Chapter VI uses upper case W on the left-hand-side.

#### Lorentz Transforms

Lorentz Transforms, minus-sign. Derived in EDoMB-1905 Section §3 and used in Section §4, length contraction and time dilation.

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

Lorentz Transforms, plus-sign.  Used and shown by various text books ( Halliday & Resnick ) in connection with v-a-f derivation.  Section §5 explicitly evokes the minus-sign transforms of Section §3, but no LT’s are shown or used.  Chapter XIII contains, quote: “Solve for x,” which in a way calls forth the plus-sign pair but no transforms are shown.  Solving the minus-sign pair for x ( unprimed ) reverses primed-unprimed.  Nobody can possibly figure it out.

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

Right-hand-side of the temporal ( without gamma ) closely resembles the denominator in the v-a-f.

#### Abbreviations and notation

SRT = spezielle Relativitätstheorie, Special Relativity

rf = reduction factor ( 1 + vw/c² ) in the v-a-f.

As in most of SRT, we use right-to-left equation logic ( an assignment operator so-called ) rather than left-to-right equality, an identity.

Assignment: 5 = 2 + 3

Identity: 2 + 3 = 5

Arrow: Bring in an arrow ← for right-to-left assignment:

5 ← 2 + 3

Asterisk:  Use V-blank for classical mechanics and V*-asterisk for SRT.

Classical addition:  V ← ( v + w )

SRT addition: V* ← ( v + w ) / ( 1 + vw/c² )

#### Special case

In the set-up that follows we use only the special case of input speed-w equal to speed-c, the speed-of-light.  That simplifies the reduction factor to:

rf = ( 1 + v/c )

The special case is shown in Section §5, when input speed-w is equal to speed-c.  Use asterisk.

Shown:  V* = ( v + c ) / ( 1 + v/c )

V* ← ( v + c ) / ( 1 + v/c )

V* = c

The special case v-a-f ( w = c ) is obviously correct in some sense.  Nothing can have speed V = c + v.  A photon in particular must have speed V* = c, not speed V = c + v.

Section §5:  V* ← V / factor

Bad speed V > c on the right-hand-side is transformed into good speed V* = c on the left-hand-side.

#### Chapter VI nomenclature

Chapter VI: has an embankment, a carriage of speed-v, and on-carriage a “man” of carriage floor speed-w.

The special case may envision the “man” as a “photon” of speed-c on the carriage.

Change “embankment” to “pavement,” a highway, a road.

Verbiage: Carriage, photon, pavement.

#### The starting line

Space-time on the moving carriage might not be like space-time on the pavement.  However starting line place-time on the carriage ( zero-zero ) is the same thing as starting line place-time ( zero-zero ) on the pavement.

The starting has, explicitly, only two events: the carriage-rear leaves pavement-zero at speed-v, and man-photon-c leaves the carriage-rear at speed-c.

Point-of-view allows departure of carriage photon-c to be a departure also on the pavement.  According to classical mechanics, that on-pavement departure is speed V = c + v.

Let’s not do the LT’s.  Let’s not do primed-unprimed.  Cognitive dissonance.  Instead, bring a third event.   Put a stationary laser gun at pavement-zero and at time-zero have it fire a photon down the pavement.  Obviously the photon has speed-c on the pavement.  Call it photon-P of speed P = c, observed on the pavement.

Photon-c on the carriage, reconsidered as classical motion on the pavement, has speed V = c + v, obviously out in front of pavement photon P = c.

Like classical photons V and P on the pavement, SRT photon V* and pavement photon P = c are both observed on the pavement.  How does photon V* go down the pike relative to P?  As a front runner or double file?

Take a kilogram of U235, and write an equation:

E = mc²

We know what the facts are even though the equation does not tell us what the facts are.  Mass-m is one gram not one kilogram.

The v-a-f is similar. When photon-c on the carriage is reconsidered as photon-V* on the pavement, we need to know what the facts are:  Is photon-V* double-file with photon-P or out in front of photon-P?