D4 Addition of Velocities in the STR
In D1 we showed that the addition of velocities is simple in the mechanics of Galileo and Newton: If B moves with velocity v in the x-direction of A and C moves for B with velocity w' in the same direction, then C moves with velocity v + w' for A. We recognized in A3 that this simple formula cannot apply in the STR: The light of a locomotive standing still must travel forward just as fast as that of one moving forward, i.e. with c.
For the new velocity addition formula it must be true that the sum of v and c results in c. It must also be true that in no case may a velocity be greater than c. If a spaceship flies past us at 0.7 • c and fires off a rocket in the direction of its flight, which itself has velocity 0.8 • c relative to the spaceship, then we would already have a speed of the rocket of 1.5 • c according to Newton …
The derivation of the correct formula for the addition of velocities is quite harmless, if the Lorentz transformations are available:
Let C move with velocity w' in the x'-direction of B, while B moves as usual with a relative velocity v along the x-direction of A. Then for the x'-coordinate of C we have
x’ = a + w’ • t’ where a is some constant
We now simply substitute both x' and t' by expressions with x and t from the Lorentz transformations:
x’ = ( x – v • t ) / √ und t’ = ( t – x • v / c2 ) / √
Thus the equation from above becomes
( x – v • t ) / √ = a + w’ • ( t – x • v / c2 ) / √
Multiplying both sides by the radical we obtain
x – v • t = a • √ + w’ • ( t – x • v / c2 ) or
x = a • √ + v • t + w' • ( t – x • v / c2) = a • √ + v • t + w' • t - w’ • x • v / c2
From this we obtain
x + x • w’ • v / c2 = a • √ + v • t + w' • t or x • ( 1 + w' • v / c2 ) = a • √ + ( v + w’ ) • t
Dividing by the bracketed term on the left we obtain
x = a • √ / ( 1 + w’ • v / c2 ) + ( v + w’ ) / ( 1 + w’ • v / c2) • t
Since both the radical and the bracketed term are constant we can read from this that C moves for A with the constant velocity of
w = (v + w') / (1 + v • w' / c2)
along the x-axis!
If we use the symbol ⊕ to represent the relativistic addition of speeds that are parallel to the relative velocity v, then we can summarize:
In exercise 5 we check that this formula supplies reasonable values in all cases. Thus one obtains for the above example of the spaceship with a rocket a resulting velocity of 0.7 • c ⊕ 0.8 • c = (1.5/1.56) • c ≈ 0.962 • c
Addendum from october 2012: Jerzy Kocik shows in Am. J. Phys., Vol.80, No.8, august 2012 how to add parallel velocities in STR with a simple compass-and-ruler construction. As a small extension of this idea Alfred Hepp and Martin Gubler show in a small paper how to add the corresponding Epstein-angles. The short text is abvailable at http://www.physastromath.ch/uploads/myPdfs/Relativ/Relativ_06_en.pdf .