Translation:On the Discussion Concerning Rigid Bodies in the Theory of Relativity

by Max von Laue

On the Discussion Concerning Rigid Bodies in the Theory of Relativity.

By M. Laue.

Since the dynamics of the mass point was adapted to the relativity principle by Einstein and Planck, several authors have tried the additional step, to accordingly reshape the dynamics of rigid bodies. Since the previous dynamical conditions of rigidity are not consistent with the Lorentz transformation, it was at first important to create a new definition for it. It is the merit of M. Born^[1], that he was the first one to come forward with such a definition, and by that he brought up the problem. After Ehrenfest,^[2] Herglotz,^[3] Noether^[4] have demonstrated, that this first definition allowed only 3 degrees of freedom for motion, he additionally published a second definition allowing 6 degrees. Furthermore, Noether makes the proposal following his criticism of Born's definition, to define the rigid body as a structure having nine kinematic degrees of freedom. And this idea is especially near at hand when Minkowski's representation of the world is considered. Because a three-dimensional geometric structure being steadily congruent, has $4+3+2+1=10$ degrees of freedom; though an arbitrary parameter distinguishing the location must be subtracted from the corresponding space-time variables, so that 9 are remaining now. All three proposals have the (quite justified) idea in common, that a rigid body, contrary to the infinitely many degrees of freedom of the deformable ones, has only a finite number of them (there, the body is of course imagined as a continuum). We now want to show, that the relativity principle excludes this possibility due to dynamical causes, so that any attempt in that direction is hopeless from the outset. However, to prevent misinterpretations, it shall be clearly noticed, that – under certain circumstances – a body of course can move without change of shape related to the employed co-moving system; this is indeed proven by the uniformly accelerated (hyperbolic) motion. Also in classical hydrodynamics, a particle of a fluid can occasionally be in motion as if it were rigid. Yet, this isn't the most generally possible motion for any body; we rather want to show at an example, that processes can be imagined whith respect to every body, in which the number of kinematic degrees of freedom is infinitely great.

There, we start with a theorem already pronounced by Einstein^[5] in the form, that a propagation with superluminal speed is excluded for all physical effects. The consideration is based by us on Minkowski's four-dimensional world ($u=ct$ as fourth coordinate), in which the world-point represents an event in terms of space and time, and we claim: all events which can be effects of an event represented by world-point $O$, are represented by world-points of the after-cone or such points beyond $O$; all events that can come into consideration as causes of event $O$, are represented by world-points of the for-cone or by such points lying on this side of $O$. Points of the intermediary area of $O$, however, cannot be causally connected with $O$. The proof reads: A point $P$ of the intermediary area arises earlier or later as $O$ depending on the choice of reference system. A natural law which should connect $O$ and $P$ thus should state in one part of the valid systems, that $O$ is the cause and $P$ is the effect; and in another part, that $P$ is the cause and $O$ the effect. This contradicts the relativity principle, according to which the natural laws should have the same form in all valid systems.

Now, a body shall at first be at rest in System $K^{0}$, so that the world-lines of its points are mutually parallel lines. The world-line leading to $O$ shall begin to gradually deviate from its previous direction only at world-point $O$. This disturbance will cause corresponding disturbances in the course of all other world-lines, but not before they enter the after-cone of $O$. Until then, they namely lie in the intermediary area or even at this side of $O$, thus they cannot be influenced by the disturbance in $O$.

Disturbances of such kind may not only occur in a point $O$, but in $n$ points $O_{i}$, all of which are lying in a plane space $u^{0}=U$. Then there is always a space $u^{0}=U+\Delta U\ (\Delta U>O)$, in which its $n$ intersections do not yet coincide with the after-cones of the points $O_{i}$, but are completely excluding each other. The motion of the body at instant $t^{0}={\tfrac {U+\Delta U}{c}}$ then has surely $n$ degrees of freedom. Because a part of it which is not yet belonging to any of these intersections, is still at rest in $K^{0}$, while in the parts represented by the intersections, a motion dominates that is independent from the other disturbances, and which naturally has at least one degree of freedom. The number $n$ of disturbance points, however, can be arbitrarily increased; thus the number of kinematic degrees of freedom of a body has no upper limit.

Now, in Newton's mechanics there is of course another definition of rigidity, which fits even better to the physical state of facts; there, one considers the rigid body as the limiting case of a deformable body with a very great elasticity coefficient. This passage to the limit can in any case also be executed in the theory of relativity, although it wouldn't lead to the rigid body, but to a body deformable as less as possible, which possibly has especially simple properties as well. In any case, this problem – and by that we come to the conclusion drawn by Planck^[6] – can only be solved by the theory of elasticity (which is to be adapted to the theory of relativity).

The necessity to admit the possibility of shape changes at any body, however its constitution may be, is also explained in a useful way by an example given by Einstein in conversation, which was communicated by Sommerfeld in a discussion at the meeting of natural scientists in Königsberg.^[7] There, we always use one and the same reference system. A ruler is at rest here, while another one (forming an angle $\alpha$ with the first) moves with velocity $q$ perpendicular to its direction. The intersection point of both rulers then travels over the first with velocity $V={\tfrac {q}{\sin \alpha }}$, and also when $q$ has moderate values (ca. 1 cm/sec) one can achieve that $V$ becomes greater than the speed of light $c$ by the choice of sufficiently small values for $\alpha$. Nothing regarding this purely geometric conclusion can of course be changed by any physical theory. Is this fact in contradiction with Einstein's theorem, that physical effects cannot propagate with superluminal speed?

Namely, it is easy to mount devices at different points of the resting ruler, which trigger arbitrary physical events at the moment when the intersection point moves over them. However, no contradiction lies in this against the theorem in question. Because the events are not mutually connected as cause and effect, but they are all effects of the causes which give rise to the uniform motion of the second ruler. For example, if one removes an arbitrary number of the mentioned devices, the remaining ones will function exactly in the same way as before.

It would be different, when the following case could be realized: We again mount one of the mentioned devices at a point $B$ of the resting ruler. At the beginning, also the second ruler is at rest; the intersection point of both of them may in this stage coincide with point $A$ of the first one. However, then an arbitrary event happens in $A$, which is the cause that the second ruler is almost momentarily set into motion in its entire length with velocity $q$ previously defined. Then, the intersection point traverses the whole ruler as the signal of event $A$, and the event triggered in $B$ is an effect of the cause that happened in $A$; the transmission from $A$ to $B$ takes place with superluminal speed $V$.

Why is it impossible to realize this process? Now, the mechanism of transmission necessarily fails here. The second ruler, which should be momentarily brought into motion by an impulse emanating from $A$, doesn't remain straight but bulges in $A$. The velocity, with which the disturbance emanating in $A$ is communicated to the other points of the ruler, is itself at most equal to that of light; the distant parts at first remain at rest, and only after a certain time they are also affected by the disturbance. The previous conclusion, $V>c$, only holds for motions at which the ruler is straight; it doesn't say anything at all about the process previously discussed. Its theory could only be mathematically worked out on the basis of the theory of elasticity.

Here, a far reaching analogy to the case of the propagation of light already used by Sommerfeld (l. c.) exists. Purely periodic and thus temporally infinite wave trains cannot serve as a signal for the occurrence of an event. Accordingly, it is no contradiction against relativity theory, that under some circumstances (in dispersing bodies and for anomalous spectrum areas) the phase velocity of periodic sinus waves is greater than $c$. However, a wave-train broken at one or at both sides – more generally spoken any interruption in the periodicity $c$ of the oscillations – can surely represent a signal. However, the front of the disturbance exactly propagates with velocity $c$ in all such cases, and the disturbance is so small in the beginning, so that its noticeable bulk mostly arrives considerably later.

In our example, to the unlimited periodic wave train corresponds a uniform and thus also temporally unlimited motion of the ruler; and to the interruption of regular periodicity corresponds the acceleration from the state of rest (or more general: a velocity change). The first processes are both equally inappropriate as signals, the two others can surely serve as such, but they at best transmit it only with the speed of light.

Munich, Institute for Theoretical Physics, January 1911.

(Received January 21, 1911.)

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1. M. Born, Ann. d. Phys. 30, 1, 1909 and Göttinger Nachr. 1910, p. 161; also W. v. Ignatowski, Ann. d. Phys. 33, 607, 1910.
2. P. Ehrenfest, this journal. 10, 918, 1909; 11, 1127, 1910.
3. G. Herglotz, Ann. d. Phys. 31, 393, 1910.
4. F. Noether, Ann. d. Phys. 31, 919, 1910.
5. A. Einstein, Ann. d. Phys. 17, 891, 1905.
6. M. Planck, this journal 11, 294, 1910.
7. This journal, 11, 975, 1910. I wouldn't like to let it unmentioned, that I had several times the opportunity to discuss these matters with Prof. Sommerfeld.