Translation:Remarks on the Law of the Lever in the Theory of Relativity

by Max von Laue

Remarks on the Law of the Lever in the Theory of Relativity.

M. Laue (Munich)

One of the most beautiful results of relativity theory is surely the theorem of the inertia of energy; its most general formulation was stated by Planck 3 years ago in Cologne.^[1] It claims, that wherever an energy current of density ${\mathfrak {S}}$ arises, a momentum is always connected with it, having the amount (related to unit volume):

${\mathfrak {g}}={\frac {\mathfrak {S}}{c^{2}}}$

(1)

where $c$ is the speed of light in empty space. As to electrodynamics, this relation was already stated and discussed by Abraham in 1903^[2]; its application upon mechanics was given by Planck in his Cologne lecture. Today, I would like to present to you an additional conclusion from this theorem.

For that, we start with a comparison of relativistic mechanics with that of Newton. At the top of the latter stands the theorem of momentum, which connects the force ${\mathfrak {K}}$ with the momentum ${\mathfrak {G}}$ of a body by

${\mathfrak {K}}={\frac {d{\mathfrak {S}}}{dt}}$

(2)

Not independent from it (although coordinated in many respects) is the surface theorem, i.e. that the increase of angular momentum ${\mathfrak {L}}$ is equal to torque ${\mathfrak {R}}$ exerted upon the body:

${\mathfrak {K}}={\frac {d{\mathfrak {L}}}{dt}}$

(3)

Both relations remain in the theory of relativity. Even the definition of angular momentum can be given in a way which is valid for both theories. Namely, if ${\mathfrak {r}}$ is the radius vector from any fixed spacepoint in the direction of the material volume element $dV$, we have

${\mathfrak {L}}=\int [{\mathfrak {rg}}]dV$

(4)

The entire difference between the theories lies in the statements concerning momentum. According to Newton's mechanics, it is connected with mass density $\mu$ and velocity ${\mathfrak {q}}$ by the relation

${\mathfrak {g}}=\mu {\mathfrak {q}}$

(5)

while in relativity theory, the theorem of the inertia of energy (1) takes its place.

According to (5), the momentum density is always parallel to the velocity. However, nothing similar is valid with respect to the energy current ${\mathfrak {S}}$ . For example, think about a transmission belt; here, the energy current conveyed by its stress arises opposite to the velocity; or think about a rotating drive shaft; it conveys (by its torsion) an energy transport perpendicular to the velocity of the material parts. In general, the conduction current of the mechanical energy caused by the elastic stresses, can in general have any direction with respect to the velocity. Certainly, it is by far outweighed by the convection current of the energy-forms that are resting in matter, so that the parallelism of momentum density and velocity can remain as an approximation, which is quite sufficient in most cases. Nevertheless it is of interest, for example regarding the Trouton-Noble experiment, to follow the consequences caused by this deviation. According to (4), the angular momentum of an uniformly and purely translatory moving body, with total momentum of

$\int {\mathfrak {g}}dV={\mathfrak {G}}$,

changes per unit time by

${\frac {d{\mathfrak {L}}}{dt}}=\left[{\mathfrak {qG}}\right]$

(6)

Because the radius vector ${\mathfrak {r}}$ increases by ${\mathfrak {q}}dt$ for every volume element $dV$ in time $dt$. In Newton's mechanics if follows from that by (5):

$R=\int \mu dV\cdot \left[{\mathfrak {qq}}\right]=0$;

no torque is necessary to maintain a translatory and uniform motion. It is different in relativity theory: with respect to a body with elastic tensions, a torque is generally necessary in this case.

An example shall illustrate this.^[3] In the valid reference system $K^{0}$, an angle-lever $ABC$ with two equally long arms – being mutually perpendicular $\left(AB=BC=l^{0}\right)$ – it at rest. It is rotatable in $B$ around an axis perpendicular to its plane. $A$ and $C$ are affected by two forces ${\mathfrak {K}}_{1}^{0}$ and ${\mathfrak {K}}_{2}^{0}$ of equal magnitude; ${\mathfrak {K}}_{1}^{0}$ is parallel to ${\overrightarrow {BC}}$, and ${\mathfrak {K}}_{2}^{0}$ to ${\overrightarrow {BA}}$. The torques of these forces are $\pm \left|{\mathfrak {K}}_{1}^{0}\right|l^{0}$ and they compensate each other.

Let us consider this state from another reference system $K$, relative to which $K^{0}$ has the velocity ${\mathfrak {q}}$ and the direction ${\overrightarrow {BC}}$. If one transforms the forces into this system, then one finds under consideration of the Lorentz contraction of arm $BC$, that the torque is not at all zero, but is having the value $l^{0}\left|{\mathfrak {K}}_{1}^{0}\right|{\tfrac {q^{2}}{c^{2}}}$ and the sense of rotation given in the figure. Despite of this, no rotation arises, since this would be identical to a rotation in $K^{0}$, while we know that (with respect to $K^{0}$) the lever is in statical equilibrium. Exactly this torque is necessary to move in a translatory way.

Also equation (3) can be easily confirmed here, by calculating the component of momentum ${\mathfrak {G}}$ perpendicular to the velocity, and from that we calculate the increase of angular momentum according to (6). Namely, force ${\mathfrak {K}}_{1}$ performs the work $q\left|{\mathfrak {K}}_{1}\right|$ per unit time in $A$; thus an energy current enters at $A$ into the lever, end leaves it at $B$ where the rotation axis exerts the force ${\mathfrak {K}}_{1}+{\mathfrak {K}}_{2}$ upon it. According to (1), it corresponds to a momentum (perpendicular to ${\mathfrak {q}}$) of amount

$\left|{\mathfrak {G}}\right|={\frac {1}{c^{2}}}l^{0}q\left|{\mathfrak {K}}_{1}^{0}\right|$

(7)

Force ${\mathfrak {K}}_{2}$ doesn't perform work, and also the convection current of energy (being parallel to ${\mathfrak {q}}$) provides no part of the momentum component to be calculated here. According to (6), the amount of the increase of angular momentum is

$q\left|{\mathfrak {G}}\right|={\frac {q^{2}}{c^{2}}}l^{0}\left|{\mathfrak {K}}_{1}^{0}\right|$

and, as one easily convince himself, it also has the sense of rotation given in the figure. Thus it is equal to the torque, as required by equation (3).

The momentum component perpendicular to the velocity, is proportional to ${\mathfrak {q}}$ according to equation (7). If we accelerate the lever without changing its inner state in the longitudinal direction, i.e. parallel to the velocity, then its increase is:

${\frac {d\left|{\mathfrak {G}}\right|}{dt}}={\frac {\dot {q}}{c^{2}}}l^{0}\left|{\mathfrak {K}}_{1}^{0}\right|{\frac {dq}{dt}}$

and according to the momentum theorem (2), a transverse force component belongs to it. In Newton's mechanics, however, it is totally impossible that the longitudinal acceleration requires a transverse component. This component by no means vanishes in the limit of very small velocity; it is rather totally independent from velocity, thus Newton's mechanics of elastic stressed bodies is not even valid as an approximation for small velocities. As I have shown at another place^[4], this is only the case for bodies which are unaffected by external forces, and which are in static equilibrium in their rest system. For such ones, the dynamics of the mass point (as it was developed by Einstein and Planck) is valid, and this is completely independent from other properties of their constitution.

Finally, I want to make a practical application of the things said, upon the theory of the Trouton-Noble experiment. It's known that this experiment is about finding the torque, which a uniformly and translatory moving condenser experiences from its electromagnetic field, according to the concordant statement of all electromagnetic theories. It is experimentally given that no rotation occurs. However, one may not conclude from that, that the mentioned torque is not present. The material parts of the condenser indeed contain elastic stresses and thus require a torque to move in a translatory way without rotation. The torque performed by the field, is exactly the one required for that. – In this sense, the Trouton-Noble experiment decides in favor of the dynamics of the theory of relativity, and against Newton's dynamics.

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1. M. Planck, this journal 9, 828, 1908; Verhandl. d. D. phys. Ges. 6, 728, 1908.
2. M. Abraham, Ann. d. Phys. 10, 105.
3. See M. Laue, Verh. d. Deutsch. phys. Ges. 13, 513, 1911.
4. M. Laue, Das Relativitätsprinzip. Braunschweig 1911; Ann. d. Phys. 35, 524, 1911.