The Velocity Addition Formula

Velocity Addition Formula

Updated: June 24, 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 )

#### Speed quotients

Shown next, minus-sign LT pair and plus-sign LT pair:

x′ = gamma • ( x minus vt )

t′ = gamma • ( t minus vx / c² )

x = gamma • ( x′ plus vt′ )

t = gamma • ( t′ plus vx′ / c² )

Change right-hand-side distance x-or-x′ to right-hand-side distance wt-or-wt′. Leave left-hand-side x′-or-x unchanged.

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 announced in Section §5. 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* = x′/ t′ = ( c – v ) / ( 1 – v/c )

The plus-sign pair creates an additive speed quotient:

V* = x / t = ( 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.  In both models, speed-w is the special case of speed-c equal to 12 ips.  In both models, the ruler moves 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 at 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: At time t = 1, observer at ruler-zero ( table-4 ) in the sub-model can reconsider his 8-inch ruler distance any way he wants, in terms of time and distance.  It has no effect on his location at table-4 nor on the photon location at table-12.

At time t = 1, observer in the add-model at table-zero changes time at table-16 from t = 1 to t = 4/3.  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.

#### Terminal time, addition

In terms of classical mechanics, you have two terminal events at table-16: arrival of the man-photon at time t = 1 sec, speed c = 16 ips, and arrival of ruler-front at time t = 1 sec, speed of 4 ips.  Switch over to SRT.  The photon clock is transformed, reading 4/3 sec and the man-photon has speed 12 ips.  No can do.  You can’t have two stationary clocks, at one place and one moment, reading different times.

Section §4 has two clocks at one place, one moment, reading different times.  But one of the clocks is moving, in another frame of reference.  It works. The destination here is round trip, return to ruler-zero ( not the starting line at time-t ).  The man-photon is a palindrome going out and back.

A man a plan a canal, Panama.

#### 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.  “Not even wrong.”  You can’t transform speed-c down to less than speed-c, from 12 ips to 9 ips.

#### 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 derived from the temporal LT in the speed quotient?   Or is it a more general speed factor such as we used in the distributive law?