The Velocity Addition Formula

Velocity Addition Formula

Updated: May 31, 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 )

#### 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 )

#### 1920-Chapter VI

Chapter VI has a simple rendering of right-hand-side speeds v and w.  A “carriage” has speed-v on an “embankment.”  A “man” has speed-w on the carriage.  Chapter VI has the carriage and the man moving alike in one unit of time, and then variables v and w double as distance numbers.

Distances v and w are in effect serial distances, not parallel distances.  Rename the embankment as “pavement.”   At time t =1, distance-v is distance of carriage-zero from pavement origin-zero.  At time t =1, distance-w is distance on the carriage floor, pavement point of serial distance v + w.  Distance-w is not distance from pavement origin-zero, not a distance parallel to distance-v.

Switch variables v and w back to speeds and you have what Section §5 calls the “composition of velocities.”

V = v + w

#### 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 )

#### Three SRT problems

V* = ( v + w ) / ( 1 + vw/c² )

Use the special case of w = c shown in Section §5.

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

V* = c

rf reduction factor = ( 1 + v/c )

V* = ( v + c ) / rf

In terms of Chapter VI, speed w = c converts the “man” on the carriage floor into a photon on the carriage, a photon of speed-c, a figure of speech because a photon does not really exist.  The photon is fired off by a laser gun bolted to carriage-zero, not bolted to pavement-zero, and such a laser gun moves on the pavement at speed-v.  Classical mechanics gives the photon ( aka the man ) pavement speed:

V = v + c.


Problem #1: Symmetric

Section §5 quote: “v and w enter into the resultant velocity in a symmetrical manner.”  It’s a version of the distributive law:

Symmetric: V* = c  ←  v* + c* ← ( v + c ) / rf

Asymmetric: V* = c  ←  V / rf ← ( v + c ) / rf

In our special case of w = c, symmetric calls for addition of v* + c*.  Speed-c* asterisk, not doable.

c* = c / rf  ( c* less than c ) cannot exist.

Also: Symmetric transforms input variables before they are added. The arrow cannot transform its own right-hand-side.

Problem #2: V* = c ← ( v + c ) / rf

If you transform one or both of distance-time doing the photon’s destination on the pavement, you must transform everything on the pavement. You must transform distance-time doing speed-v and you have:

No good: V* = c ← ( v* + c ) / rf

Problem #3:  In Section §4, gamma is basically a length contraction effect, although it appears in time dilation also, reducing time dilation from ( 1 – v²/c² ) to the sqrt ( 1 – v²/c² ) which is 1/gamma.  H&R Chapter 37, deriving the v-a-f, has the quotient of plus-sign spatial and temporal, where gamma divides out.  Question: length contraction is simply not there, or it is still there but can be ignored?

When the carriage goes down the pike at speed-v, it must suffer length contraction of second order effect v²/c², which is much smaller than first order effect v/c in the reduction factor.  Is length contraction so small that it can be ignored?

1920-Chapters VI and XIII doing the v-a-f make no mention of the carriage affected by length contraction.

#### A numbers game

The speed-of-light is 12 inches per second.  Lay a foot-rule on the table top.

## Minus-sign subtraction:  A photon goes 12 inches on the table top.  During that interval of time, the ruler goes 4 inches on the table top: ruler-0 goes from table-0 to table-4.  The ruler observes the photon.  At time t = 1 sec, the ruler finds the photon at ruler-8, an on-ruler speed of 8 inches per second.  Too slow.  Doctor up time on the ruler so for example the photon duration is only 2/3 second.  The ruler then says photon-on-ruler has speed c = 12 ips.

You can doctor up space-time on the moving ruler without changing any of the table top numbers: time of 1 sec, photon location of table-12, ruler-0 location of table-4.

## Plus-sign addition.  A photon goes 12 inches on the RULER, in 1 sec according to the ruler clock.  During that interval of time, the ruler goes 4 inches on the table top.  The table top observes the photon.  At time t = 1 sec, the table top finds the photon at table-16, a distance addition of 4 + 12, for a table top speed of 16 ips, too fast.  You can transform the stationary table clock at table-16 so that it reads 4/3 sec, not 1 sec, and then photon speed on the table is c = 12 ips.

You cannot transform the stationary table clock at inch-16 unless you also transform the stationary table clock at inch-4, in which case the ruler has not moved at speed of 4 ips.

Addition is not the inverse of subtraction.

Ruler ← table minus table

The fictional ruler is output.

Table ← table plus ruler

The fictional ruler is one of the two inputs.  It won’t work.

#### Conclusion:

The special case v-a-f with w = c is obviously correct in some sense.  No photon can have speed V = v + c.

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

V* = c

Problem: The LT’s switched over from minus-sign to plus-sign conjure up untenable propositions.  The plus-sign pair is shown neither in 1905 nor in 1920.

How do you get the v-a-f if you can’t use the plus-sign LT’s?   Here’s the introductory equation of Section §5:

x = ( w′ + v ) • t / rf

It is the composition of distances.  How do you get it?  You look at the minus-sign LT’s of Section §3 and “solve for x.”  The algebra of “solve for” is not shown.  The previous version of this blog tried to figure it out.  We spilled a lot of ink.  The fact is, nobody knows what’s going on here.