The Velocity Addition Formula

Velocity Addition Formula

Updated: January 5, 2025

On the Electrodynamics of Moving Bodies

Einstein-EDoMB-1905

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

Section §5 presents the velocity-addition-formula used today.

Shown: V = ( v + w ) / ( 1 + vw/c² )

#### Lorentz Transforms

MINUS-sign Lorentz Transforms of EDoMB-1905-Sections §3 and §4:

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

PLUS-sign transforms of modern textbooks not shown in Section §5:

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

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

The numerator in the v-a-f is the plus-sign LT spatial, an addition of distances.

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 • ( wt′ + vt′ ) / gamma • ( 1 + vw/c² ) • t′

V = ( v + w ) / ( 1 + vw/c² )

Gamma divides out, and then does not influence quotient-V on the left-hand-side.  But it is still doing its thing on the pavement.

[ Note: Algebra shown here is from modern text books, neither Section §5-1905 nor Relativity, The Special and the General Theory, 1920, Chapter XIII. ]

#### Subtract or add distances, not speeds

Equations will be minus-sign for subtraction of distances and plus-sign for addition of distances.  Always fixed distances.  No time, no speed.

On the right-hand-side of the subtraction equation, distances x and p are both mileposts relative to [ x = 0 ].

On the right-hand-side of the addition equation, distance-p is a milepost relative to [ x = 0 ].  Distance-x′ is an increment relative to milepost-p, not a milepost.

Factor-1.6 is kilometers per mile.  Our kilometer factor-1.6 is something simpler than the gamma factor in the LT’s.  Even so, a simple kilometer factor shows that switching over from subtraction to addition can present a problem.

#### Subtract

x′ = x – p

x′ = 80 miles – 20 miles

x′ = 60 miles

Right-hand-side milepost x = 80 miles is fixed distance 80 miles from the starting line.  Post p = 20 is fixed distance 20 miles from the starting line.  Subtraction ( x – p ) on the highway yields variable-x′, a distance 60 miles.  Left-hand-side distance x′ = 60 miles is the gap between post p = 20 miles and post x = 80. It is also a new axis, with axis origin [ x′ = 0 ] located at p = 20 miles and terminal point x′ = 60 miles at highway terminal x = 80 miles.  

Bring in kilometer factor-1.6.

x′ = 1.6 • ( x – p )

x′ = 1.6 • ( 80 miles – 20 miles )

x′ = 96 kilometers

The equation rewritten with factor-1.6 renders left-hand-side variable-x′ as 96 kilometers rather than 60 miles.  Now origin [ x′ = 0 ] is a kilometer origin, located as before at p = 20 miles.   The 60 mile gap between mile-p and mile-x is reconsidered as a kilometer distance between x′ = zero kilometers and x′ = 96 kilometers.  No harm done. Reconsidering the gap to be determined by kilometer posts rather than mileposts does not impinge on original mileposts p and x.

#### Add

x = p + x′

x = 20 miles + 60 miles

x = 80 miles

Stipulation: right-hand-side value-x′ is a 60 mile increment, not a milepost.  Segment x′ = 60 miles is added to point p = 20 miles, yielding milepost x = 80 miles on the highway.

x = 1.6 • ( p + x′ )

x = 1.6 • ( 20 miles + 60 miles )

x = 128 kilometers

The equation rewritten with factor-1.6 renders left-hand-side variable-x as 128 kilometers.  You simply reconsider highway distance x = 80 miles to be distance x = 128 kilometers.

Problem-A:

Add: x = 20 miles + 60 miles

The equation can easily consider both right-hand numbers to be mileposts, as is the case in subtraction.  Algebraic addition 20 + 60 = 80 miles exists, but number-80 is not milepost-80.  A preliminary stipulation is required: that mile-60 be an increment with respect to milepost-20, not a milepost with respect to [ x = 0 ].  Then algebraic addition creates milepost-80 on the left-hand-side, as required.  Galilean transformation of r-h-s increment-60 into l-h-s milepost-80 is enabled.  But relativistic transformation of the r-h-s increment-60 miles into l-h-s terminal post-128 kilometers is not completely facilitated.

The subtractive model needs no preliminary stipulation.

Problem-B:

Add:  x = 1.6 • ( 20 miles + 60 miles )

Before the equation is solved for kilometers on the left-hand-side, point-p on the right-hand-side is miles.  When left-x is expressed as kilometers, so also the entire highway is kilometers, and then point-p must be changed to kilometers.  No can do. 

No good:  x = 1.6 • ( 36 kilometers + 60 miles )

The subtractive model does not need to change point-p into kilometers.

#### Section §4

Section §4 uses subtraction to do time dilation and length contraction.  Contraction is done at ground time t = 0 and dilation is done at a later time.

Length contraction uses the spatial LT only ( no temporal LT ), and a certain special situation is at hand.  Our point-p is point-vt.  Section §4 does length contraction at time t = zero, and then vt = 0, or, in our model, p = 0.  Then our gap-( x – p ) is equal simply to x, and with some justification we say, at time t = 0, that x = 60 miles not x = 80 miles.

x′ = 1.6 • 60 miles

x′ = 96 kilometers

New axis-X′ has a new value for point-x = 60 miles, namely x′ = 96 kilometers, same location.  But point x′ = 96 on axis-X′ is not the point of interest.  The point of interest is kilometer point x′ = 60, which is found on axis-X at milepost x = 37.5 miles.  Actually, LT-gamma doesn’t do kilometers.  Gamma transforms unprimed miles ( right-hand-side ) into primed miles ( left-hand-side ) and a primed-mile is shorter than an unprimed-mile.  That is distance x′ = 60 primed-miles “appears shorter” than distance x = 60 unprimed-miles.