Addition of Velocities (Velocity Composition) in Special Relativity

[robphy]

The “Addition of Velocities” formula (more correctly, the “Composition of Velocities” formula) in Special Relativity
$$\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c} }{1 + \frac{v_{AB}}{c} \frac{v_{BC}}{c}}$$
is a non-intuitive result that arises from a “hyperbolic-tangent of a sum”-identity in Minkowski spacetime geometry, with its use of hyperbolic trigonometry.
However, I claim it is difficult to obtain this by looking at the Galilean version of this formula and then motivating the special-relativistic version.
Instead, one should start with the Euclidean analogue (in what could be mistakenly called the “addition of slopes” formula… “composition of slopes” is better),
then do the special-relativistic analogue, then do the Galilean analogue (to obtain the familiar but unfortunately-“our common sense” formula).
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[Mister T]

Composition of slopes is also called the stacked wedge analogy.

 

[A T Wilson]

There is a strong analogy between relativistic boosts in a single direction and Euclidean rotations in a single plane. Such boosts or rotations may be composed, but whether they add or not is a property of the representation. In what follows, we deal first with composition of Euclidean rotations, and represent that in three different ways, using slopes, using angles, and using complex rotors. Then we deal with composition of boosts, using velocities, using rapidities, and using velocity factors.

The message here is the rotations can be composed, and boosts can be composed. Rotations don't really add. Addditive representations may be possible, but also multiplicative representations. Boosts don't really add either. Again, additive representations are possible, but also multiplicative representation.

COMPOSING EUCLIDEAN ROTATIONS IN A SINGLE 2D PLANE

Slopes $\frac{y_j}{x_j}$ are somewhat inconvenient representation for composing rotations in a plane. One composes slopes under rotation by using what might be called the slope addition formula, e.g. $$\frac yx=\frac{\frac{y_1}{x_1}+\frac{y_2}{x_2}}{1-\frac{y_1\cdot y_2}{x_1\cdot x_2}}.$$ This is immediate from the trigonometric addition formulation for tangents, e.g. $$\tan(\theta_1+\theta_2)=\frac{\tan\theta_1+\tan\theta_2}{1-(\tan\theta_1)\,(\tan\theta_2)}.$$

To compose a rotation that has a slope of 0.2 with a rotation that has a slope of 0.3, we calculate $$\frac{0.2+0.3}{1-0.2\times0.3}\approx0.532$$ as the slope of the composite rotation.

Rotations in a single plane are commutative and associative, and satisfy the any order property, regardless of representation. It can be convenient to use a representation that maps rotation onto a more familiar any-order operation, such as addition or multiplication.

Additive representation of rotations using angles
One may use angles $$\theta_j=\arctan\!2(y_j,x_j)$$ defined so that $y_j/x_j=\tan\theta_j$ and also so that the signs of $x_j$ and $y_j$ match the corresponding signs of $\cos\theta_j$ and $\sin\theta_j$. (It is necessary in general to use $\arctan2$ because, if $x<0$, i.e. if the angle is not acute, then usual arctan function will give an angle that is wrong by 180 degrees.)

When rotations compose, angles add, $$\theta=\sum_j\theta_j.$$

We shall, for our example, suppose that the slopes are correspond to acute angles.. Then the corresponding angles to the slopes 0.2 and 0.3 are the arctangents of these numbers, i.e. the angles 11.31 degrees and 16.70 degrees (or 0.197 radians and 0.292 radians). Adding these, we get 28.01 degrees (or 0.489 radians), which we note is also acute. The tangent of this angle is the slope, i.e. approximately 0.532. These angles are a little larger than small, and so the slopes are a little larger than the radian measure of the angles.

Multiplicative representation of rotations using complex numbers
Or one may use complex numbers $$z_j=x_j + i\cdot y_j.$$ It would be usual to to normalise these, $$\hat z_j=\frac{x_j + i\cdot y_j}{\sqrt{x_j^2+y_j^2}},$$ so that $|\hat z_j|=1$. This is called a rotor, and it is unique. But we don't actually need to normalise. What matters is that that signs of the real and imaginary parts are each correct, and also that their ratio is correct.

Whether normalised or not, we compose rotations by multiplying the complex numbers, $$z=\prod_j z_j\text{ or }\hat z=\prod_k\hat z_j.$$

For (normalised) rotors, there is a simple relation between this multiplicative relation and the preceding additive representation $$\hat z_j=\exp(i\,\theta_j)=\cos\theta_j+i\cdot\sin\theta_j.$$

The slopes 0.2 and 0.3 can quickly be realised exactly as the (unnormalised) complex numbers $1+0.2\cdot i$ and $1+0.3\cdot i$. To compose rotations, multiply the complex numbers, getting $(1+0.2\cdot i)\times(1+0.3\cdot i)=0.94+0.5\cdot i$ exactly. The slope of this is exactly $0.5/0.94$, i.e. approximately 0.532 again

In any case, beyond a single plane, one must use matrix multiplication or some equivalent to compose rotations, because composition is no longer commutative. Addition or multiplication of scalars is commutative, so it cannot suffice.

We have spent some time setting up the rotational part of the analogy. Now we turn to boosts.

COMPOSING RELATIVISTIC BOOSTS IN A SINGLE 2D HYPERPLANE

Velocities $x_j/t_j$are not a convenient form for composing boosts in a single spatial direction, e.g. the $x\,t$ plane. (We take $c=1$ here throughout, or equivalently, the $t_j$ here can be taken as shorthand for $c\cdot t_j$.)

One can neverthless compose them using the relativistic velocity addition formula, e.g. $$\frac xt=\frac{\frac{x_1}{t_1}+\frac{x_2}{t_2}}{1+\frac{x_1\cdot x_2}{t_1\cdot t_2}},$$ which corresponds to the addition formula for hyperbolic tangents, e.g. $$\tanh(\alpha_1+\alpha_2)=\frac{\tan\alpha_1+\tan\alpha_2}{1+\tan\alpha_1\,\tan\alpha_2}.$$

To compose a boost that has a velocity of 0.2 with a boost in the same direction that has a velocity of 0.3, we calculate $$\frac{0.2+0.3}{1+0.2\times0.3}\approx0.472$$ as the velocity of the composed boost.

Like rotations in a single $x\,y$ plane, relativistic boosts in a single $x\,t$ plane are commutative and associative, and satisfy the any order property, regardless of representation. It can be convenient to use a representation that maps boosts onto a more familiar any-order operation, such as addition or multiplication.

Additive representation of boosts using rapidities
One may use rapidities, $$\alpha_j=\text{arctanh}\!(x_j/t_j)$$.

When boosts compose, properly defined rapidities add to give $$\alpha=\sum_j\alpha.$$

For velocities of 0.2 and 0.3, the rapidities are the inverse hyperbolic tangents 0.203 nepers and 0.310 nepers. These add to give a rapidity of 0.513 nepers. The hyperbolic tangent of this rapidity is approximately 0.472, which is the velocity of the composed boost. The velocities are a little larger than small, and so the neper measure of the rapidities are somewhat larger, numerically, than the corresponding velocities.

Multiplicative representation of boosts using velocity factors
Or one may use velocity factors,

$$f_j=\frac{1+\frac{x_j}{t_j}}{1-\frac{x_j}{t_j}}=\frac{t_j+x_j}{t_j-x_j}.$$

We compose boosts by multiplying velocity factors,

$$f=\prod_j f_j.$$

For velocities of $0.2=\frac2{10}$ and $0.3=\frac3{10}$, the velocity factors are $\frac{10+2}{10-2}=\frac32$ and $\frac{10+3}{10-3}=\frac{13}7$.

To compose, we multiply, $\frac32\times\frac{13}7=\frac{39}{14}$, which gives the velocity factor of the composed boost. The velocity is then given by reverting (with the difference now in the numerator), e.g $\frac{39-14}{39+14}=\frac{25}{53}.$ This fractional result is exact if 0.2 and 0.3 are exact. Performing the division to 3 decimal places, we get 0.472 again.

We note that the velocity factor $f$ defined here is a two-way Doppler factor. It is the square of the $k$ found in Bondi's $k$-calculus, which is the usual one-way relativistic Doppler factor. Since $k$ is non-negative, there is no loss of information. Multiplication survives application of a constant power the way that additivity survives application of a multiplicative scale factor. The square roots just make the calculations a bit more difficult.

Of course, with more than one spatial direction, one needs matrix multiplication (or equivalent) because boost composition is no longer commutative.

SUMMARY
Rotations in a single plane, or boosts in one space dimension, can be represented somewhat inconveniently by slopes or velocities, and somewhat more conveniently by an additive representation, angles or rapidities, or by a multiplicative representation, complex rotors or velocity factors. Interconverting between slope or velocity and thee the corresponding multiplicative representation is actually easier than interconverting with the corresponding additive representation.

In more dimensions, composing rotations and composing boosts become non-commutative, so no commutative scalar representation can work.

For details on the uses of velocity factors, including pedagogically, see my "Using ordinary multiplication to do relativistic velocity addition" available at https://arxiv.org/pdf/physics/0611192.