The Velocity Addition Formula

Velocity Addition Formula

Updated: June 13, 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 )

Bibliography: comment #1

#### 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 left ← to ← right equation logic ( an assignment operator so-called ) rather than high school left = to = right, an identity.

5 ← 2 + 3

2 + 3 = 5

On occasion we change notation without changing anything on the ground.  Bring in an arrow ← for assignment and an asterisk* for relativistic output velocity-V*.

Classical addition:  V  ← ( v + w )

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

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

rf = ( 1 + v/c )

#### Lorentz Transforms

Lorentz Transforms: minus-sign, shown in EDoMB-1905 Section §3 and used in Section §4 ( length contraction and time dilation. ) Shown in Halliday & Resnick as equations 37-2I:

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

Lorentz Transforms: plus-sign, shown in H&R as equations 37-22. 

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

This plus-sign pair is shown neither in Section §5 ( v-a-f ) nor in 1920-Chapter XIII ( v-a-f ):

#### Derive the v-a-f Section §5

Section §5 has a statement which invokes the minus-sign pair of Section §3.   Quote: “With the help of Section §3 equations we solve for x and t.”   The statement does not lead to a display of the plus-sign pair.  A single additive-spatial is shown. It is not an LT. Gamma is missing:

x = ( w′ + v ) • t / rf

It is this equations which leads to the v-a-f:

Shown:  V* = ( v + w ) / ( 1 + vw/c² )

Special case: w = c

V* = ( v + c ) / ( 1 + vc/c² )

V* = ( v + c ) / ( 1 + v/c )

V* = c

#### Derive the v-a-f H&R Chapter 37

H&R Chapter 37-3 has a similar statement:  Quote: “Simply solve equations 37-21 for x.” The plus-sign LT pair is then shown but the algebra of “solve for” is not shown.  Chapter 37-4 then has derivation of the v-a-f from the plus-sign pair, that derivation shown as Equations 37-29.

The general idea is simple.  Speed is distance divided by time.  In the plus-sign LT pair, change distance-x′ to distance-wt′:

x = gamma • ( wt′ + vt′ )

t = gamma • ( t′ + vwt′ / c² )

Divide:

V* = x / t

Gamma divides out.  Time-t′ divides out.

V* = ( v + w ) / ( 1 + vw/c² )

#### Subtraction and addition

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

In both the minus-sign pair and the plus-sign pair change distance-x to distance-wt:

x′ = gamma • ( wt – vt )

t′ = gamma • ( t – vwt / c² )

x = gamma • ( wt′ + vt′ )

t = gamma • ( t′ + vwt′ / c² )

In both the minus-sign pair and the plus-sign pair use the special case of w = c which is shown in Section §5.

V* = ( v + c ) / ( 1 + v/c )

V* = c

With w = c:

x′ = gamma • ( c – v ) • t

t′ = gamma • ( 1 – v / c ) • t

x = gamma • ( c + v ) • t′

t = gamma • ( 1 + v / c ) • t′

The minus-sign pair creates a subtractive speed quotient:

V* = ( c – v ) / ( 1 – v/c )

The plus-sign pair creates an additive speed quotient:

V* = ( v + c ) / ( 1 + v/c )

In both cases, V* = c.

#### A numbers game

The set-up here is an adaptation of Chapter VI: an embankment, a carriage of speed-v, and on-carriage a “man” of speed-w.
 
Take a foot ruler to the table top.  The ruler is marked off in 12 inches.  Mark off the table at inch-0-4-12-16-18.  A time-zero, the ruler is located between table-0 and table-12. 

We do both subtraction and addition. In both models, the speed of light is 12 inches per second.  A photon has speed 12 ips.  In both models, the ruler will move from table 0-12 to table 4-16 in 1 second, a speed of 4 ips.

Sub-model:  The photon does table distance-12 in 1 second, speed c = 12 ips.  The observer is a ruler-zero and moves from table-zero to table-4 in 1 second, speed of 4 ips.

Add model;  The photon is on the ruler and does ruler distance-12 in 1 second, speed of 12 ips.  The observer is at table-zero, and at time t = 1 second, observes ruler-zero at table-4 and ruler-12 at table-16.

Sub-model: At time-1, the observer at ruler-zero sees the photon at ruler-8, speed-on-ruler of 8 ips, too slow.

Add-model:  At time-1, the observer at table-zero sees the photon at table-16, speed-on-table of 16 ips, too fast.

Sub-model:  Observer at ruler-zero ( table-4 ) solves his too-slow problem by changing denominator time at ruler-8 by the factor ( 1 – v/c ) which is ( 1 – 4/12 ) = 2/3, and denominator time of 1 sec is changed to 2/3 sec.  Ruler distance-8 in 2/3 sec is 12 ips, restoring ( increasing ) photon-on-ruler to the speed-of-light.

Add-model:  Observer at table-zero solves his too-fast problem by changing denominator time at table-16 by the factor ( 1 + v/c ) which is ( 1 + 4/12 ) = 4/3, and denominator time of 1 sec is changes to 4/3 sec.  Table distance-16 in 4/3 sec is 12 ips, restoring ( reducing ) photon-on-table to the speed-of-light.

Compare: Observer-v in the sub-model can reconsider his 8-inch ruler any way he wants, in terms of time and distance.  It has not effect on his location at table-4 nor on the photon location at table-12.

Observer in the add-model at table-zero changes time at table-16.  That change requires change to ALL table times, including time at table-4.  If time at table-4 is changed to 4/3 sec, speed-v at table-4 is v = 3 ips.   It’s what you might call a negative feedback effect. The equation, in cranking out transformed speed-V* on the left-hand-side, must also transform input speed-v on the right-hand-side.  No good.

Section §5 and Chapter XIII, deriving the v-a-f, neither use nor make reference to the plus-sign pair.

#### The distributive law

We do the distributive law for both velocity subtraction and addition with a couple of stipulations.  First, bring back the arrow to emphasize transformation.  Second, invert denominator factors 2/3 and 4/3 into numerator factors 3/2 and 3/4.  It makes the algebra easier. 

Subtract: V* ← ( c – v ) / ( 1 – v/c )

Version-A:  12 ← 8 • 3/2 ← ( 12 – 4 ) • 3/2

Version-B:  12 ← 18 – 6 ← 12 • 3/2 – 4 • 3/2  ← ( 12 – 4 ) • 3/2

Add:  V* ← ( v + c ) / ( 1 + v/c )

Version-A: 12 ← 16 • 3/4 ← ( 4 + 12 ) • 3/4

Version-B: 12 ←  3 + 9 ← 4 • 3/4 + 12 • 3/4 ← ( 4 + 12 ) • 3/4

Velocity subtraction is obviously not Version-B of the distributive law.  Is velocity addition Version-B of the distributive law?  Section §5 says “symmetric” which suggests Version-B.

####  Fizeau’s interferometer

Chapter XIII ends up with the Fizeau’s water in a pipe.  The pipe is “embankment,” the water is the “carriage” and a photon is the “man.”  The photon has speed-w, not speed-c.  Fair enough.  Everyone knows that the speed-of-light is slower in water than in air, slower than in free-space.

Even so, whether the Fizeau device can deal with a single one-way photon in a single tube of water is a dubious proposition.  The device does not do the speed of one photon.  It compares two photons ( both going speed-positive ) in two tubes of water, one water of positive speed-v and one water of negative speed-v. 

#### Conclusion:

The special case v-a-f ( w = c ) is obviously correct in some sense.  A photon can have neither of speed V = v + c nor V = c – v.  Only V* = c.

What is the denominator factor ( 1 ± v/c ) ?  Is it exclusively a time factor in the time denominator of the plus-sign quotient?   Or is it a more general speed factor such as we used in the distributive law?