Translation:On the Composition of Velocities in the Theory of Relativity

by Arnold Sommerfeld

On the Composition of Velocities in the Theory of Relativity;

by A. Sommerfeld.

Minkowski has taught us to interpret the Lorentz-Einstein transformation as a "space-time rotation", i.e., as a transformation of the character of an ordinary rotation, not in space xyz but in the four-dimensional manifold of the magnitudes xyzl, where l = ict also means a length, namely the "light path" multiplied by the imaginary unit. If the primed reference frame carries out a translation of uniform velocity v in the direction of the x-axis, and β denotes the velocity ratio v/c, then the transformation equations are:

$\left.{\begin{array}{lrl}x'=&x\ \cos \varphi +l\ \sin \varphi ,&y'=y\\l'=&-x\ \sin \varphi +l\ \cos \varphi ,&z'=z\end{array}}\right\}$

1)

and there exists a relation between the imaginary rotation angle $\varphi$ and the velocity ratio β:

$\operatorname {tg} \,\varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}$

2)

I want to show by some examples, how useful this analogy (or analytically spoken, the identity) between space-time rotations and ordinary rotations is for the kinematics of relativity theory.

If we compose two rotations of equal axis, or better spoken, of equal rotation plane, then the rotation angles sum up, not their trigonometric functions.

The same is true for two translations of equal direction x (two space-time rotations in the same plane xl); if $\varphi _{1}$, $\varphi _{2}$ are the (imaginary) rotation angles, $\varphi$ the angle resulting from the composition, $\beta _{1},\beta _{2},\beta ;\ v_{1},v_{2},v$ the corresponding velocity ratios and velocities, then

$\varphi =\varphi _{1}+\varphi _{2}$

and therefore

$\beta ={\frac {1}{i}}\operatorname {tg} \,\left(\varphi _{1}+\varphi _{2}\right)={\frac {1}{i}}{\frac {\operatorname {tg} \,\varphi _{1}+\operatorname {tg} \,\varphi _{2}}{1-\operatorname {tg} \,\varphi _{1}\operatorname {tg} \,\varphi _{2}}}={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}$

or in addition

$v={\frac {v_{1}+v_{2}}{1+v_{1}v_{2}/c^{2}}}$

That is the famous addition theorem for velocities of Einstein; in Minkowski's interpretation it looses all its strangeness.

Two rotations of equal rotation plane are commutative, i.e., their result is independent of the order of the individual operations. The same is true for two translations of equal direction, because $\varphi _{1}+\varphi _{2}=\varphi _{2}+\varphi _{1}$. However, two rotations in different planes are not commutative, as well as two translations of different direction. The reason for this is obviously the fact, that by the first rotation the plane of the second rotation will be displaced in general, namely always then when the two planes don't coincide.

If we, for example, carry out a first space-time rotation in the xl-plane of rotation angle $\varphi _{1}$, and relative to the thus moved system we carry out a second space-time rotation in the yl-plane, and its rotation angle $\varphi _{1}$ is estimated within the moving system, then by scheme 1) it is given:

${\begin{array}{l}x_{1}=x\ \cos \varphi _{1}+l\ \sin \varphi _{1},\\y_{1}=y,\\l_{1}=-x\ \sin \varphi _{1}+l\ \cos \varphi _{1}\end{array}}$

and

${\begin{array}{l}x_{2}=x_{1}=x\ \cos \varphi _{1}+l\ \sin \varphi _{1},\\y_{2}=y_{1}\cos \varphi _{2}+l_{1}\sin \varphi _{2}=-x\ \sin \varphi _{1}\sin \varphi _{2}+y\ \cos \varphi _{2}+l\ \cos \varphi _{1}\cos \varphi _{2},\\l_{2}=-y_{1}\sin \varphi _{2}+l_{1}\cos \varphi _{2}=-x\ \sin \varphi _{1}\cos \varphi _{2}-y\ \sin \varphi _{2}+l\ \cos \varphi _{1}\cos \varphi _{2}.\end{array}}$

A point $x_{2}=$ const, $y_{2}=$ const participating with this composed motion $\varphi _{1},\varphi _{2}$, consequently describes a straight line in system xy, and its direction is defined by:

${\begin{array}{l}0=dx\ \cos \varphi _{1}+dl\ \sin \varphi _{1}\\0=-dx\ \sin \varphi _{1}\sin \varphi _{2}+dy\ \cos \varphi _{2}+dl\ \cos \varphi _{1}\cos \varphi _{2}\end{array}}$

to:

$\left.{\begin{array}{l}{\frac {dx}{dl}}=-\operatorname {tg} \,\varphi _{1},\\\\{\frac {dy}{dl}}=\operatorname {tg} \,\varphi _{2}\left({\frac {dx}{dl}}\sin \varphi _{1}-\cos \varphi _{1}\right)=-{\frac {\operatorname {tg} \,\varphi _{2}}{\cos \varphi _{1}}}\end{array}}\right\}$

3)

because of 2):

${\frac {1}{c}}{\frac {dx}{dt}}=\beta _{1},\ {\frac {1}{c}}{\frac {dy}{dt}}=\beta _{2}{\sqrt {1-\beta _{1}^{2}}}$

4a)

However, if the order is reversed we obtain:

${\frac {1}{c}}{\frac {dy}{dt}}=\beta _{2},\ {\frac {1}{c}}{\frac {dx}{dt}}=\beta _{1}{\sqrt {1-\beta _{2}^{2}}}$

4b)

Thus if we have a right angled ruler in the drawing plane of Fig. 1, and its two sides at the starting point coincide with OA, OB, and if we direct a pencil along the side OB with velocity $v_{2}=\beta _{2}c$, while we simultaneously move the ruler in the direction of its second side OA with velocity $v_{1}=\beta _{1}c$, then the pencil describes another path, as if we would lead it along OA with $v_{1}$ and move the ruler in the direction OB with $v_{2}$. The deviation of both paths in unit time (OC in the first, OD in the second case) is small of second order, if $\beta _{1},\beta _{2}$ are small of first order. The reason for this is, that in the first case the velocity $v_{1}$ from the moving system (ruler) is differently estimated than it is estimated from the stationary system (drawing plane), because of the motion dependence of time; and correspondingly in the second case this applies to velocity $v_{1}$. Exactly the same fact we have expressed above, that by execution of one space-time rotation in space xyl, the plane of the other space-time rotation will be displaced. In the figure, the velocities AC, BD are drawn in such a way, as they appear from the viewpoint of the stationary drawing plane; the attributed magnitudes denote the estimated velocities from the viewpoint of the moving ruler (above) and the viewpoint of the stationary drawing plane (below).

For the resultant velocity $v=\beta c$, as it is estimated in the stationary system, the same value for both cases is given by 4a) and 4b):

$\beta ^{2}={\frac {1}{c^{2}}}\left[\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}\right]=\beta _{1}^{2}+\beta _{2}^{2}-\beta _{1}^{2}\beta _{2}^{2}$

or

$1-\beta ^{2}=\left(1-\beta _{1}^{2}\right)\left(1-\beta _{2}^{2}\right)$

or, if besides $\varphi _{1},\varphi _{2}$ we take the resultant rotation angle $\varphi$, by 2):

$\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}$

5)

If $\alpha _{1}$ and $\alpha _{2}$ are the obliquities denoted in the figure, then it follows from 3):

$\left.{\begin{array}{ll}&\operatorname {tg} \,\alpha _{1}={\frac {\operatorname {tg} \,\varphi _{2}}{\sin \varphi _{1}}}\\\mathrm {and} \\&\operatorname {tg} \,\alpha _{2}={\frac {\operatorname {tg} \,\varphi _{1}}{\sin \varphi _{2}}}\end{array}}\right\}$

6)

where it is always given:

$\alpha _{1}+\alpha _{2}<{\frac {\pi }{2}}$

7)

But these at first strange results are becoming completely clear from the Minkowskian standpoint. If we namely compose (Fig. 2) the rotation angles $\varphi _{1},\varphi _{2}$ as arcs on a unit sphere, so that in triangle OCA the angle at A and in triangle OBD the angle at B is a right one, then the resultant rotation angle $OC=OD=\varphi$ is immediately given from the cosine rule as the hypotenuse of the two spherical triangles (that are congruent to each other) in accordance with equation 5). That the resultant rotation plane depends on the order of the composing rotations is obvious; because the greatest circle through B (perpendicular to OB) obviously doesn't go through C but intersects AC. The so called Napier rule for right angled spherical triangles gives formula 6) for the angle of the sides $\alpha _{1},\alpha _{2}$. The sum of the angles in a spherical triangle is greater then two right angles, the sum of the angle of the sides in a right angled spherical triangle is thus greater then a right angle, as it is shown in Fig. 2. Specifically, the (spherical) excess becomes equal to the content of the triangle on the unit sphere. Now the sides of our spherical triangle are imaginary in case of a space-time rotation, its content is thus negative. The spherical excess thus becomes a spherical defect, it

causes the inequality 7) and the divergence of the paths OC, OD in Fig. 1.^[1]

In summarizing we can say: For the composition of velocities in the theory of relativity, not the formulas of the plane, but the formulas of the spherical trigonometry (with imaginary sides) are valid. By this remark the complicated transformation calculus becomes dispensable, and can be replaced by a lucid construction on a sphere. Another example to illustrate this:

If the two velocities $v_{1},v_{2}$ under the arbitrary angle α are inclined to each other, then in the spherical triangle OAC of Fig. 2 the exterior angle at A becomes α as well, and the resultant rotation angle $\varphi$ can be calculated according to the general cosine rule of spherical trigonometry:

$\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}\cos \alpha$

8)

If we want this to transform into the velocities $v_{1},v_{2},v$, then by 2) it is given:

${\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1+\beta _{1}\beta _{2}\cos \alpha }{{\sqrt {1-\beta _{1}^{2}}}{\sqrt {1-\beta _{2}^{2}}}}}$ $v^{2}={\frac {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -{\frac {1}{c^{2}}}v_{1}^{2}v_{2}^{2}\sin ^{2}\alpha }{\left(1+{\frac {1}{c^{2}}}v_{1}v_{2}\cos \alpha \right)^{2}}}$

9)

a formula that was already derived by Einstein^[2] from the transformation equations. As we can see, formula 8) is more lucid than 9), as well as Fig. 2 was more lucid than Fig. 1. This follows from the fact that it apparently better corresponds to the meaning of the theory of relativity, to calculate and (by consideration of the reality relations) to construct by rotation angles, instead of only using its tangents, the velocities.

The single reason for this short paper was to show, that the deep space-time interpretation of Minkowski not only simplifies the general construction of the theory of relativity, but it is also reliable as a guidance concerning specific questions.

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1. To illustrate that relation of both figures, the following might serve: Fig. 1 emerges when we centrally project Fig. 2 from the center M of the sphere on the tangential plane at point O. In that course the angle at O remains, as do the right angles at A and B; the sides $\varphi _{1}\varphi _{2}\varphi$ propagating from O were replaced by its tangents or by the velocities $v_{1}v_{2}v$ that are proportional to them. But the angles at C and D will be altered by this projection (they are in Fig. 2 equal to $\alpha _{2}$ or $\alpha _{1}$, but in Fig.1 greater than $\alpha _{2}$ or $\alpha _{1}$) and the sides CA and BD were not simply replaced by its tangents. Fig. 3 shows how this projection for the triangle OAC can be created as real.
2. Jahrbuch der Radioaktivität 4, 423, 1908