The Velocity Addition Formula

Velocity Addition Formula

Updated: March 28, 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 )

#### Lorentz Transforms

Lorentz Transforms – subtraction, 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 – addition, shown in H&R as equations 37-22.  Shown neither in Section §5 ( v-a-f ) nor in Chapter XIII ( v-a-f ):

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

#### Transition from subtraction to addition

Halliday & Resnick, 10th Edition, Chapter 37-2. 

H&R has the two LT sets shown here: subtraction with unprimed on the right-hand-side and addition wirh primed on the right-hand-side. Given the subtractive set, how do you get the additive set?  Quote: “Simply solve the subtractives for x and t.”  The implied algebra is not shown.  Spatial, without gamma:
 
x′ = x – vt

4 = 10 – 6

4 + 6 = 10

10 = 4 + 6

x = x′ + vt

Subtract: value-4 = axis-10 minus axis-6

What is output value-4 on the left-hand-side?  It is a value, not an axis location.  But that’s all it needs to be: the separation of point-10 and point-6. 

Add: Value-10 = value-4 plus point-6

For the v-a-f, doing “composition” of distances, output value-10 needs to be location-10 on the axis but it isn’t necessarily.  If input value-4 is with respect to axis origin-zero rather than point-6, output value-10 is not location-10.   A monumental crisis of institutions.

#### Textbook derivation of the v-a-f

Halliday & Resnick, 10th Edition, Chapter 37-4.

Let x′ be of speed-w:  x′ = wt′

Spatial:  x = gamma • ( wt′ + vt′ )

Temporal:  t = gamma • ( t′ + vwt′ / c² )

Speed is distance divided by time:

V = x / t = gamma • ( w + v ) • t′ / gamma • ( 1 + vw/c² ) • t′

gamma divides out:

V = x / t = ( w + v ) • t′ / ( 1 + vw/c² ) • t′

Time-t′ divides out:

V = ( v + w ) / ( 1 + vw/c² )

EDoMB-1905 Section §5 and Relativity-1920 Chapter XIII invoke the Lorentz Transforms but neither provide an LT derivation of the v-a-f.   In particular, Section §5 begins straightaway with addition of speeds, not addition of distances.

#### Abbreviations and notation

SRT = spezielle Relativitätstheorie, Special Relativity

CLS = classical mechanics, a term from Chapter XIII.

rf = reduction factor, denominator ( 1 + vw/c² ) > 1

Let’s change a notation without changing anything on the ground.  Bring in an asterisk for the v-a-f output:

CLS sum:  V  ← ( v + w )

SRT sum:  V*  ← ( v + w ) / rf

#### Chapter VI

Note: As per modern notation and Section §5 we use symbol-V for output velocity. Chapters VI and XIII use symbol-W.

A carriage “travels” an embankment at speed-v.   A man “traverses” the floor of the carriage at speed-w.  That same man has velocity-V on the embankment.

New vocabulary:  The carriage is still the “carriage.”  The embankment is the “pavement.”  The man of speed-w on the carriage is “walker-w” of speed-w.  

Chapter VI stipulation:  Time is 1.  Let’s have miles, hours and miles per hour, so time is 1 hour.  The carriage-rear goes a distance on the pavement at speed-v, elapsed time being 1 hour.  Speed-v is then also a number which represents distance on the pavement.  For example: 5 mph for 1 hour is 5 miles. 

Another stipulation:  Walker-w does the length of the carriage ( from back to front ) during that same 1 hour.

Blog number: The carriage has length 10 miles.  Walker-w traverses the length of the carriage ( 10 miles ) in 1 hour. Walker-w then has speed 10 mph on the carriage floor.

Blog number:  The carriage has speed v = 1 mph.  The carriage-rear necessarily travels a distance of 1 mile, from mile-0 to mile-1, and the carriage-front travels from mile-10 to mile-11.

Classical:  Walker-w must end up at the carriage-front-10 which location at time-1 is pavement-11.  Walker-w, who has speed w = 10 on the carriage, has velocity V = 11 on the pavement:

CLS sum:  V  ← ( v + w )

11 mph ← ( 1 mph + 10 mph )

It’s hard to see why output velocity-V on the pavement should be anything other than V = 11 mph, unless it so happens that the speed-of-light, ie speed-c, is 10 mph. Then it’s obvious that velocity-V ( ie speed-c ) on the pavement cannot be 11 mph.  We must have speed V* = 10 mph.

SRT sum: V* ← ( v + w ) / rf

rf = 1.1 ← 1 + 1/10 ← 1 + v/c

10 mph ← ( 1 mph + 10 mph ) / 1.1

Walker-w is walker-c, a photon.   Walker-w goes from carriage-rear to carriage front at speed w = c = 10 mph which is just right. 

Walker-w-c cannot go from pavement-0 to pavement-11 in 1 hour, velocity V = 11 mph.  Not allowed.

You can change the photon’s velocity by changing its time of journey, duration of journey.  You can change that duration by “adjusting” ( c.f. Section §4 ) terminal clock time.  Change terminal time from 1 hour to 1.1 hours.  Quotient 11 miles over 1.1 hours is 10 mph.

Question: Which terminal clock has face-time increased by the factor 1.1 ?  Which terminal clock does velocity-V* ?

Is it the carriage front clock, always at carriage-10, or the stationary clock at pavement mile-11 ?

Changing the carriage clock ( which has pavement speed-v ) is just like time dilation in Section §4, except that face-time on the clock of speed-v is adjusted upward not downward.  Might be something that can be done, but adjustment of the carriage clock does not adjust velocity-V on the pavement.

So you adjust the pavement clock.  Double jeopardy. As before, the destination clock ( stationary on the pavement ) must be adjusted upward not downward, a reversal of Section §4.  Then, also contrary to Section §4, the moving clock must adjust the stationary clock.

Notice in our discussion of Chapter VI we have made no use of unprimed~primed symbols.  That notation is not in Chapter VI, only later in Chapter XIII where the v-a-f is supposedly derived.  It’s complete chaos.  The visceral impact of elemental chaos.

H&R Chapter 37-4 has the primed axis on the right-hand-side.  The r-h-s axis is necessarily the input axis and reasonably the stationary axis.  Possibly in Chapter 37 the primed r-h-s axis is moving, and that axis transforms the stationary ( unprimed ) axis which calculates velocity-V*.

####  Length contraction

The carriage goes down the pike at speed-v.  Length contraction, based on speed-v and gamma, is in effect.  At time of 1 hour, when the rear of the carriage is at mile-1, the front of the carriage is shy of mile-11.

The H&R derivation of the v-a-f transitions from a quotient with gamma in numerator and denominator to a quotient with no gammas.  When the two gammas are not shown, is that to say they are present but to null effect in the quotient, or more simply, the two gammas have simply disappeared?  Length contraction and time dilation have disappeared. 

Bertrand Russell:  “Mathematics is the subject where we never know what we are talking about, nor whether what we are saying is true.”